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A user's guide to optimal transport

1. Now we discuss the regularity properties of Kantorovich potentials which follows from Theorem 2 18 Corollary 2 23 Regularity properties of the interpolated potentials Let be a c convex po tential for uo p and let p Hi 4 Define pi Hi Y pi Ht p and choose a geodesic u from po to u1 Then for every t 0 1 it holds i We gt p and both the functions are real valued ii Ye pi on supp p iii p and p are differentiable in the support of u and on this set their gradients coincide Proof For i we have ye Hi p Hi o Ho 4 Hi o H oH lt Hoh p SS lt Id Now observe that by definition Yy x lt 00 and y x gt oo for every x E M thus it holds 00 gt v x gt plx gt o0 Va e M To prove ii let ys be the unique plan associated to the geodesic u via 2 7 recall Proposi tion 2 16 for uniqueness and pick y supp 2 Recall that it holds pilie 90 Y0 piln H Ys m1 Thus from y1 70 71 Y y0 we get that ys y Yr 7e Since pe ee eH the compactness of M gives supp Ht t yesupp p SO that ii follows Now we turn to iii With the same choice of t gt 7 as above recall that it holds piln c Y0 14 yhy pile lt c 40 2 Yo Vr M and that the function x gt c 49 4 Yo is superdifferentiable at y Thus the function x gt qp is superdifferentiable at 2
2. a pr a ax T u ve pr 0 3 41 iff Vo 1 tha tT x T x x dt p ax dx 4 f Vole T 2 pr a de Rem CE App Rems where the remainder term Rem is bounded by PDA _ lip V menje PA I iT 2 2l at p 2 de EEP woo p Since heuristically speaking W2 p0 p has the same magnitude of 7 we have Rem o 7 and the proof is complete 68 3 3 1 Elements of subdifferential calculus in 2 R7 W2 Recall that we introduced a weak Riemannian structure on the space 2 M W2 in Subsec tion 2 3 2 Among others this weak Riemannian structure of Y2 M W2 allows the development of a subdifferential calculus for geodesically convex functionals in the same spirit and with many formal similarities of the usual subdifferential calculus for convex functionals on an Hilbert space To keep the notation and the discussion simpler we are going to define the subdifferential of a geodesically convex functional only for the case Y IR and for regular measures Definition 1 25 but everything can be done also on manifolds or Hilbert spaces and for general y Yo M Recall that for a convex functional F on an Hilbert space H the subdifferential OT F x at a point x is the set of vectors v H such that X F x v y 2 le yl lt Fy Vy H Definition 3 26 Subdifferential in IR7 W2 Let E P2 R 4 gt RU be a geodesically convex and lower se
3. O C SUP z amp together with the fact that s gt T t s L u is absolutely continuous gives that is absolutely continuous along u Then a direct application of the definition gives that its total derivative is given by d aus Oi V i Ve a e t 6 7 which shows that the total derivative is nothing but the convective derivative well known in fluid dynamics E For y Pa R we denote by P r Tan 2Z2 R1 the orthogonal projection and we put ey Id P Definition 6 8 Covariant derivative Let u be an absolutely continuous and tangent vector field along the regular curve Its covariant derivative is defined as D d Bu Pus Zur 6 8 shows that the covariant derivative is an L vector field In order to prove that the covariant derivative we just defined is the Levi Civita connection we need to prove two facts compatibiliy with the metric and torsion free identity Recall that on a standard Riemannian manifold these two conditions are respectively given by The trivial inequality lt a Ht Ht K YOA TX Y X TYI X Y VxY VyX where X Y are smooth vector fields and y is a smooth curve on M The compatibility with the metric follows immediately from the Leibnitz rule 6 6 indeed if ut u are tangent absolutely continuous vector fields we have d d d ula Gulu ut at Ht Ht d d er 5 a2
4. dr lt 2T E T inf E 3 19 t n gt n gt Jt Now passing to the limit in 3 16 written for t 0 we get the first inequality in 3 8 Also from 3 19 we get that the L norm of f t limpo VE x7 on 0 co is finite Thus A f lt oo has full Lebesgue measure and for each t A we can find a subsequence Tn 0 such that sup VE lt oo Then the third assumption in 3 13 guarantees that E a E x and the lower semicontinuity of E that E x lt lim o E 2s for every s gt t Thus passing to the limit in 3 16 as Tn 0 and using 3 18 and 3 19 we get 1 1 E x sf r dr J VE x dr lt E x vte A Vs gt t We conclude with an example which shows why in general we cannot hope to have equality in the EDI Shortly said the problem is that we don t know whether t gt E 2 is an absolutely continuous map Example 3 15 Let X 0 1 with the Euclidean distance C C X a Cantor type set with null Lebesgue measure and f 0 1 gt 1 20 a continuous integrable function such that f a 00 58 for any x C which is smooth on the complement of C Also let g 0 1 0 1 be a Devil staircase built over C i e a continuous non decreasing function satisfying g 0 0 g 1 1 which is constant in each of the connected components of the complement of C Define the energies E 0 1 gt R by SE i Fod E a f f y dy It is immedi
5. holds in the sense of distributions recall Theorem 2 29 and Definition 2 31 We say that p is regular provided 1 f EES 62 and 1 i Lip vu dt lt oo 6 3 0 Observe that the validity of 6 3 is independent on the parametrization of the curve thus if it is fulfilled it is always possible to reparametrize the curve e g with constant speed in order to let it satisfy also 6 2 Now assume that u is regular Then by the classical Cauchy Lipschitz theory we know that there exists a unique family of maps T t s supp w gt supp 4s satisfying d Gs L t 52 v T t s 2 Vt 0 1 x E supp pr a e s 0 1 6 4 T t t 2 2 Vt 0 1 supp pu4 Also it is possible to check that these maps satisfy the additional properties T r s o T t r T t s Vt r s 0 1 Ti S ede Hs Vt gE 0 1 We will call this family of maps the flow maps of the curve u Observe that for any couple of times t s 0 1 the right composition with T s provides a bijective isometry from L2 to Lis Also notice that from condition 6 2 and the inequalities ITE s Ts IZ lt J Payee f ler 2 ll dr 1 v T t r aar diz x 89 we get that for fixed t 0 1 the map s T t s L is absolutely continuous It can be proved that the set of regular curves is dense in the set of absolutely continuous curves on P2 R with respect to uniform c
6. 2 E y lt 1 t E z0 tE a1 5t 1 t d z0 x1 P t y lt 1 t d xo y td a1 y t 1 t d x0 21 3 29 for every t 0 1 Observe that there is no compactness assumption of the sublevels of If X is an Hilbert space and more generally a NPC space Definition 2 19 then the second inequality in 3 29 is satisfied by geodesics Hence A convex functionals are automatically compatible with the metric Following the same lines of the previous section it is possible to show that this assumption im plies both Assumption 3 8 and if the sublevels of E are boundedly compact Assumption 3 13 so that Theorem 3 14 holds Also it can be shown that formula 3 21 is true and thus that Proposition 3 19 holds also in this setting so that Theorem 3 20 can be proved as well However if Assumption 3 24 holds it is better not to follow the general theory as developed before but to restart from scratch indeed in this situation much stronger statements hold also at the level of discrete solutions which can be proved by a direct use of Assumption 3 24 We collect the main results achievable in this setting in the following theorem Theorem 3 25 Gradient Flows for compatible FE and d EVI Assume that X E satisfy As sumption 3 24 Then the following hold e For every x D E and 0 lt T lt 1 X7 there exists a unique discrete solution a7 as in Definition 3 7 e Let x D E and a7 any f
7. Adm u v is optimal if and only if it holds 3 zi a gt x Batol for any N N xi yi supp and o permutation of the set 1 N Expanding the squares we get N N 5 Ti Yi gt Ti Yo i A i 6 which by definition means that the support of y is cyclically monotone Let us recall the following theorem Theorem 1 6 Rockafellar A set T C R x R is cyclically monotone if and only if there exists a convex and lower semicontinuous function yp R RU 00 such that T is included in the graph of the subdifferential of We skip the proof of this theorem because later on we will prove a much more general version What we want to point out here is that under the above assumptions on u and v we have that the following three things are equivalent e y Adm v is optimal e supp y is cyclically monotone e there exists a convex and lower semicontinuous function y such that y is concentrated on the graph of the subdifferential of vy The good news is that the equivalence between these three statements holds in a much more general context more general underlying spaces cost functions measures Key concepts that are needed in the analysis are the generalizations of the concepts of cyclical monotonicity convexity and subdifferential which fit with a general cost function c The definitions below make sense for a general Borel and real valued cost Definition 1 7 c cyclical mo
8. Intrinsicness The definition is based only on the property of the space itself that is is not something like if the space is the limit of smooth spaces Compatibility If the metric measure space is a Riemannian manifold equipped with the volume measure then the bound provided by the abstract definition coincides with the lower bound on the 108 Ricci curvature of the manifold equipped with the Riemannian distance and the volume measure Stability Curvature bounds are stable w r t the natural passage to the limit of the objects which define it Interest Geometrical and analytical consequences on the space can be derived from curvature dimension condition In the next section we recall some basic concepts concerning convergence of metric measure spaces which are key to discuss the stability issue while in the following one we give the definition of curvature dimension condition and analyze its properties All the metric measure spaces X d m that we will consider satisfy the following assumption Assumption 7 1 X d is Polish the measure m is a Borel probability measure and m Po X 7 1 Convergence of metric measure spaces We say that two metric measure spaces X dx mx and Y dy my are isomorphic provided there exists a bijective isometry f supp mx gt supp my such that fymx my This is the same as to say that we don t care about the behavior of the space X dx where there is no mass Thi
9. and thus Vivier veVe Pu V Pulver Vpn V2 Ve 6 35 Subtracting 6 35 and 6 34 from 6 33 and observing that V V7 3 Vy Veo V V7 p3 Veo Vier V793 V7 1 Veo V7 3 V7 92 Ver we get the thesis Observe that equation 6 32 is equivalent to R Ve1 Vier Ves Vea Nu Ver Ves Na V2 Vea Ni V p2 Ves Nu V ei V a 6 36 2 Ni V 1 Viva Nul V es Vpa 3 for any ys E CZ M From this formula it follows immediately that the operator R is actually a tensor Proposition 6 26 Let u A R The curvature operator given by formula 6 36 is a tensor on Vey 4 ie its value depends only on the j1 a e value of the 4 vector fields Proof Clearly the left hand side of equation 6 36 is a tensor w r t the fourth entry The conclusion follows from the symmetries of the right hand side 105 We remark that from 6 36 it follows that R has all the expected symmetries Concerning the domain of definition of the curvature tensor the following statement holds whose proof follows from the properties of the normal tensor NV Proposition 6 27 Let p A2 R Then the curvature tensor thought as map from Vy to R given by 6 36 extends uniquely to a sequentially continuous map on the set of 4 ples of vector fields in L in which at least 3 vector fields are Lipschitz where we say that v v2 v3 v is converging to v v2 v3 vt if there
10. m n dula and therefore by 7 21 we have m lt Ht S tNm B All these arguments can be repeated symmetrically with 1 t in place of t because the push forward of u via the map which takes y and gives the geodesic t 7_ is ps itself thus we obtain m m r 2N m tNm B 1 t Nm B J 7 m B ba lt min f Vt 0 1 To conclude it is sufficient to prove that u is concentrated on 2B for all t 0 1 But this is obvious as u is concentrated on B B and a geodesic whose endpoints lie on B cannot leave 2B As we said we will use this lemma together with the doubling property which is a consequence of the Bishop Gromov inequality to prove a local Poincar inequality For simplicity we stick to the case of Lipschitz functions and their local Lipschitz constant although everything could be equivalently stated in terms of generic Borel functions and their upper gradients For f X R Lipschitz the local Lipschitz constant V f X R is defined as lf a f y d x y IV fI lim 121 For any ball B such that m B gt 0 the number f p is the average value of f on B 1 Ne sop ff fam Proposition 7 20 Local Poincar inequality Assume that X d m is a non branching CD 0 N space Then for every ball B such that m B gt 0 and any Lipschitz function f X gt R it holds 92N 1 amy MO Waldm e lt r fim r being the radius a Proof Notice tha
11. y yo in place of d amp x xo show that if K C A X and K2 C P Y are 2 uniformly integrable so is the set fa E P X xY TAY Eki THY E Ka We say that a function f X R has quadratic growth provided f x lt a d x 20 1 2 2 for some a Rand zp X It is immediate to check that if f has quadratic growth and u E P X then f L1 X p The concept of 2 uniform integrability in conjunction with tightness in relation with conver gence of integral of functions with quadratic growth plays a role similar to the one played by tight ness in relation with convergence of integral of bounded functions as shown in the next proposition Proposition 2 4 Let un C A X be a sequence narrowly converging to some u Then the fol lowing 3 properties are equivalent i un is 2 uniformly integrable ii f fdn gt f fdu for any continuous f with quadratic growth iii f d 2 dpin gt f x o du for some zo X Proof i gt ii It is not restrictive to assume f gt 0 Since any such f can be written as supremum of a family of continuous and bounded functions it clearly holds J tans tmint f fam n o0 Thus we only have to prove the limsup inequality Fix gt 0 zo X and find R gt 1 such that Ji Bn Eo d o dun lt for every n Now let x be a function with bounded support values in 0 1 and identically 1 on Br and notice that for every n N it holds teem J
12. 0 where the infimum is taken among all weakly continuous distributional solutions of the continuity equation u vg such that po u and py pt Proof We start with inequality lt Let u v be a solution of the continuity equation Then if J v z2 there is nothing to prove Otherwise we may apply part B of Theorem 2 29 to get that u is an absolutely continuous curve on 22 M The conclusion follows from 1 1 Wau f ilde f oeli 0 0 where in the last step we used part B of Theorem 2 29 again To prove the converse inequality it is enough to consider a constant speed geodesic u connect ing u to ut and apply part A of Theorem 2 29 to get the existence of vector fields v such that the continuity equation is satisfied and v 2 lt fe Wo u ut for ae t 0 1 Then we have 1 Wau gt f vel 22 44 EE 0 as desired This proposition strongly suggests that the scalar product in L should be considered as the metric tensor on Z M at u Now observe that given an absolutely continuous curve u C gt M in general there is no unique choice of vector field v such that the continuity equation 2 20 is satisfied Indeed if 2 20 holds and w is a Borel family of vector fields such that V wut 0 for a e t then the continuity equation is satisfied also with the vector fields v w It is then natural to ask whether there is some natural selection principle
13. 27 9 27137 T lt E E 273 lt E E x Adding up these two inequalities and observing that E x7 D lt E Ga we obtain d 27 7 2 9 T lt 2 E g E a 65 On the other hand equation 3 33 with x y and y at reads as 2 yt 07 d2 y 07 7 T lt 2 E 7 E7 Adding up these last two inequalities we get d yt 7 2 9 lt 2 E G E yz G3 Discrete estimates Pick t nt lt mt s write inequality 3 33 for zj i n m 1 and add everything up to get d z7 y d 23 y 2 ae DY y E j EN eo SD Pagny lt EW saz de Ee 639 i n 1 i n 1 T 2 Similarly pick t nr write inequality 3 34 for z x and J yj for i 0 n 1 and add everything up to get d x yf PG X 7 lt 2 E Elyz Now let y 7 to get d 7 7 lt 2r E T E 27 lt 2rE 3 36 having used the fact that E gt 0 Conclusion of passage to the limit Putting 7 2 instead of 7 in 3 36 we get at 7 2 ed s gn 1 therefore a 7 Bat a7 lt 2 2 Ez Yn lt mEN which tells that n gt a gt isa Cauchy sequence for any t gt 0 Also choosing n 0 and letting m co we get the error estimate 3 32 We pass to the EVI Letting 7 0 in 3 35 it is immediate to verify that we get Peu o gt f ena lt 8 2 f Blea 2 s t 2 s t
14. Pf T thus it remains to prove that its orthogonal is sent isometrically onto Tanps 7 2 R by dPf Fix T BM let p Pf T Typ and observe that Tanz Pf p vector fields w ica dp 0 Vu s t V uwoT p o vector fields w vw o T7 uo T7 dp 0 Vu s t V uo T7 p o vector fields w wo T7 Vy for some vy CPR Now pick w Tanz PEt p let p CX R be such that w o T7 Vy and observe that d d T tw 4p Id twoT 4 Typ gt Pf T tw P Fico ice Id tV 4p d dt lt 0 l which means by definition of Tan 2 IR and the action of tangent vectors that the differential dPf T w of Pf calculated at T along the direction w is given by Vy The fact that this map is an isometry follows once again by the change of variable formula f lw ap wo T dp Vol dp 86 6 2 On the notion of tangent space Aim of this section is to quickly discuss the definition of tangent space of 7 IR at a certain measure u from a purely geometric perspective We will see how this perspective is related to the discussion made in Subsection 2 3 2 where we defined tangent space as L R R u Tan P2 R4 Ve pE CP R Recall that this definition came from the characterization of absolutely continuous curves on P R Theorem 2 29 and the subsequent discussion Yet there is a completely different and purely geometrical approach which leads to a defini
15. an m Gin eu i 0 converges to some plan w r t the distance Wz on X Now fix n N and notice that for t 2 i 1 2 and y 7 Geod X it holds d e w lt dijar Ya 1 27 J Z d J1 d 7o in and therefore squaring and then taking the sup over t 0 1 we get 2 1 5 1 ee ah r P t 1 L2 XO P Oir Hisvyan aa 400 11 P F0 71 2 10 te0 i 0 Choosing to be a constant geodesic and using 2 9 we get that y Yo Geod X for every m N Now for any givenv v A Geod X by a gluing argument Lemma 2 12 below with v v in place of v Y Geod X Z X we can find a plan B Y Geod X such that nif v 13 i an an eo ins KOBE ie sen ont co eu SaR sen o 7 BE opt leize gv ei i 0 i 0 where optimality between TE e yon v and Hoe jon v is meant w r t the Wasserstein dis tance on Y X Using 8 to bound from above W2 v and using 2 10 we get that for every couple of measures v Yg Geod X it holds gn on W3 v v mie Cian EV He eij2n e fe yo 1 dv y far F0 71 dv tas for Theorem 2 7 everything is simpler if closed balls in X are compact Indeed observe that a geodesic connecting two points in Br xo lies entirely on the compact set B2r xo and that the set of geodesics lying on a given compact set is itself compact in Geod X so that the tightness of u follows directly fro
16. any y CX R it holds TO J epo _ faeo o r 3 38 T because this identity tells us that p is a first order approximation of the distributional solution of the Heat equation starting from fp and evaluated at time T 67 To prove 3 38 fix y C R and perturb p in the following way Id EV Pr The density of pF can be explicitly expressed by pr x det Id eV y x p x eVy 2 Observe that it holds E p Je log p fe log p o Id eV fo log ora E p fo log det Id eV7y E p e mA o where we used the fact that det Id A 1 etr A o e To evaluate the first variation of the distance squared let T be the optimal transport map from p to po which exists because of Theorem 1 26 and observe that from Typ po d EV Pr p and inequality 2 1 we have W3 po p lt IT Id EVelli2 p 3 39 therefore from the fact that equality holds at 0 we get W3 po P Ja We p0 Pr S lt IT Id Volle IT Idli 2e f T Id Vp pr o From the minimality of p for the problem 3 37 we know that W2 p po W2 pr po ee AE UE Bs ge ye Eo ae pr 5 Ve T 3 40 E p so that using 3 39 and 3 40 dividing by rearranging the terms and letting 0 and f 0 we get following Euler Lagrange equation for p7 frase f E Now observe that from Typ po we get Jerr Jer I f CATE
17. converges in the Gromov Hausdorff topology to some space X d It is well known that in this situation there exists a compact space Y dy anda family of isometric embeddings fn supp m gt Y f X Y such that the Hausdorff distance between f supp m and f X goes to 0 as n gt oo The space f supp m dy fn 7Mn is isomorphic to Xn dn Mn by construction for every n N and f X dy is isometric to X d so we identify these spaces with the respective subspaces of Y dy Since Y dy is compact the sequence m admits a subsequence not relabeled which weakly converges to some m A Y It is immediate to verify that actually m E P X Also again by compactness weak convergence is equivalent to convergence w r t W2 which means that there exists plans y Z Y admissible for the couple m Mm such that e dey 0 2 0 Therefore n dy 7 is a sequence of admissible couplings for X d m and Xn dn Mn whose cost tends to zero This concludes the proof O Now we prove the HWI which relates the entropy often denoted by H the Wasserstein distance Wz and the Fisher information J and the log Sobolev inequalities To this aim we introduce the Fisher information functional I A X 0 oo on a general metric measure space X d m as the squared slope of the entropy v I u lim W2 u v if amp o H lt 00 00 otherwise The functional J is called Fis
18. notice that n lt N and thus it is not true that sublevels of amp y are tight and therefore boundedly compact Then the inequality R u supp m Br 0 lt P 20 dp shows that the set of ws in X with bounded second moment is tight Hence the conclusion follows as before using this narrow compactness together with the lower semicontinuity of y w r t narrow convergence It remains to discuss the interest from now on we discuss some of the geometric and analytic properties of spaces having a weak Ricci curvature bound Proposition 7 13 Restriction and rescaling Let X d m be a CD K oo space resp CD 0 N space Then i Restriction f Y C X is a closed totally convex subset i e every geodesic with endpoints in Y lies entirely inside Y such that m Y gt 0 then the space Y d m Y mj is a CD K oo space resp CD 0 N space ii Rescaling for every a gt 0 the space X ad m isa CD a K space resp CD 0 N space Proof i Pick po y1 E P Y C A X and a constant speed geodesic u C A X connecting them such that Bo tHe lt 1 1 820 Ho tolh 40 HW o m resp satisfying the convexity inequality for the functional y N gt N We claim that supp u C Y for any t 0 1 Recall Theorem 2 10 and pick a measure u E Y Geod X such that He er 4M where ez is the evaluation map defined by equation 2 6 Since supp jio supp
19. 1 d a n X dG 1 AT o lt L SONOR o Eten 57 E a m d ltn Eri Therefore IFAT 1 25 2 d Em 2 lt es Se a ge 0 Panke i ren Pe Ea TEN and thus the sequence xn is a Cauchy sequence as soon as 0 lt T lt 1 A This shows uniqueness existence follows by the 1 s c of E One step estimates We claim that the following discrete version of the EVI 3 30 holds for any LEX d 17 y d G L eur SP 2 y lt Ey Ee Wwex 3 33 m where x is the minimizer of 3 12 Indeed pick a curve y satisfying 3 29 for o 7 x y and y x and use the minimality of x to get lt Eim 2 lt 1 NBE By E PaT y d x x7 2T E a 2T PS 1 t d x 27 td x y t 1 t d a7 y 2T i Rearranging the terms dropping the positive addend td x 7 and dividing by t gt 0 we get Ca OE ey ED 5X EE lt Bly Ble so that letting t 0 we get 3 33 Now we pass to the discrete version of the error estimate which will also give the full conver gence of the discrete solutions to the limit curve Given 7 D E and the associate discrete solutions x7 yf we are going to bound the distance d at y7 in terms of the distance d z 7 Write two times the discrete EVI 3 33 for r 7 2 and y J first with x 7 then with C ce to get we use the assumption gt 0 a Cae y _ a z y T d
20. A dimension free condition on u We saw that a sufficient condition on u to ensure that is convex along interpolating curves is the fact that the map z gt 2 u z is convex and non increasing so the dimension d of the ambient space plays a role in the condition The fact that the map is non increasing follows by the convexity of u together with u 0 0 while by simple computations we see that its convexity is equivalent to z u z u z zu z gt cs zu 2 3 48 Notice that the higher d is the stricter the condition becomes For applications in infinite dimensional spaces it is desirable to have a condition on u ensuring the convexity of in which the dimension does not enter As inequality 3 48 shows the weakest such condition for which is convex in any dimension is ztu z ul z zu z gt 0 and some computations show that this is in turn equivalent to the convexity of the map z e ule A key example of map satisfying this condition is z gt zlog z a Therefore we have the following existence and uniqueness result 73 Theorem 3 35 Let gt 0 and F be either V W E or a linear combination of them with positive coefficients and d convex along interpolating curves Then for every Ti Po IR there exists a unique Gradient Flow u for F starting from i in the EVI formulation The curve u satisfies is locally absolutely continuous on 0 00 p gt T as t 0
21. Opt u v must be concentrated on the c superdifferential of a c concave function y By Proposition 1 30 we know that y is semiconcave and thus differentiable p a e by our as sumption on u Therefore x gt T x exp V x is well defined ju a e and its graph must be of full y measure for any y Opt u v This means that y is unique and induced by T i gt ii Argue by contradiction and assume that there exists a semiconcave function f whose set of points of non differentiability has positive u measure Use Lemma 1 34 below to find gt 0 such that y ef is c concave and satisfies v 0 y zx if and only exp v x Then conclude the proof as in Theorem 1 26 Lemma 1 34 Let M be a smooth compact Riemannian manifold without boundary and y M gt R semiconcave Then for gt O sufficiently small the function ey is c concave and it holds v O ew ax if and only exp v 0 ev x Proof We start with the following claim there exists gt 0 such that for every 79 E M and every v t p x the function d x eXPro ev 2 rH ey z has a global maximum at x zo Use the smoothness and compactness of M to find r gt 0 such that d 2 x y d x y lt r R is C and satisfies V7d y 2 gt cId for every y M with c gt 0 independent on y Now observe that since y is semiconcave and real valued it is Lipschitz Thus for
22. a Now we pass to the calculus of total and covariant derivatives Let u be a fixed regular curve and let v be its velocity vector field Start observing that if u is absolutely continuous along 98 u then P uz is absolutely continuous as well as it follows from the inequality Py us T t 8 Py ue 1 S Pr s 0 TC 85 Pue Pus u 0 Tt s P Eu us o T t s P uso T t s T Pin us p T t S Pu ut Ht Ht Mee lt Pi Prous 0 TE 8 9 Puc Pi us 0 TE s us o T t s ull 6 20 s s lt asc Lip u dar f t t ae Ht d dr dr Hr Ur 6 21 valid for any t lt s where S sup uzl Thus P u has a well defined covariant derivative for a e t The question is can we find a formula to express this derivative To compute it apply the Leibniz rule for the total and covariant derivatives 6 6 and 6 9 to get that for a e t 0 1 it holds d D D dt Pu ut VO a E DPn ut ve Pr ut ave d d d dt ut VO a E Gu ve u SON Since Vy Tan 72 R for any t it holds P us V ut Ve for any t 0 1 and thus the left hand sides of the previous equations are equal for a e t Recalling formula 6 7 we have Vyp Vy v and Vy P V7y ve thus from the equality of the right hand sides we obtai
23. and us on 0 00 as un z N z z177 and Uso z z log z Then given a metric measure space X d m we define the functionals y amp A X gt RU 00 by n n Elulm where amp is given by formula 7 6 with u uy similarly for x The definitions of weak Ricci curvature bounds are the following Definition 7 6 Curvature gt K and no bound on dimension C_D K 0o We say that a metric measure space X d m has Ricci curvature bounded from below by K R provided the functional Ex P X RU 00 is K geodesically convex on P X W2 In this case we say that X d m satisfies the curvature dimension condition CD K oo or that X d m is a CD K oo space Definition 7 7 Curvature gt 0 and dimension lt N CD 0 N We say that a metric measure space X d m has nonnegative Ricci curvature and dimension bounded from above by N provided the functionals are geodesically convex on P X W2 for every N gt N In this case we say that X d m satisfies the curvature dimension condition C D 0 N or that X d m is a CD 0 N space Note that N gt 1 is not necessarily an integer Remark 7 8 Notice that geodesic convexity is required on 2 supp mx and not on 22 X This makes no difference for what concerns CD K 00 spaces as is 00 on measures having a singular part w r t m but is important for the case of CD 0 N spaces as the functional y has only real valu
24. d y1 z da y 2 to l to d 0 71 du 7 2 16 gt 1 to W2 uo V toW3 p v to 1 to W2 uo m and by the arbitrariness of t we conclude 39 Example 2 21 2 X W2 may be not NPC if X d is Let X R with the Euclidean dis tance We will prove that 2 R W2 is not NPC Define 6 0 0 4 0 4 1 1 1 Ho 5 91 1 4 5 3 Ha 5 9 1 1 teos Y 5 then explicit computations show that W2 uo p1 40 and W2 uo v 30 W2 u1 v The unique constant speed geodesic u from jug to p1 is given by 1 bt 5 Sa 6 1428 6 5 6t 3 21 gt and simple computations show that 30 W T 2 3 X Riemannian manifold In this section X will always be a compact smooth Riemannian manifold M without boundary endowed with the Riemannian distance d We study two aspects the first one is the analysis of some important consequences of Theorem 2 18 about the structure of geodesics in Z M the second one is the introduction of the so called weak Riemannian structure of Y2 M W2 Notice that since M is compact Y2 M Y M Yet we stick to the notation A2 M because all the statements we make in this section are true also for non compact manifolds although for simplicity we prove them only in the compact case 2 3 1 Regularity of interpolated potentials and consequences We start observing how Theorem 2 10 specializes to the case of Riemannian manifolds
25. lt en E 20 E min Observe that we didn t state any result concerning the uniqueness nor about contractivity of the curve x satisfying the Energy Dissipation Equality 3 9 The reason is that if no further assumptions are made on either X or E in general uniqueness fails as the following simple example shows Example 3 23 Lack of uniqueness Let X R endowed with the L norm E X gt R be defined by E x x xt and 7 0 0 Then it is immediate to verify that V E 1 and that any Lipschitz curve t x x x satisfying rl t Vt gt 0 x lt 1 a e t gt 0 satisfies also E x t tz 1 This implies that any such x satisfies the Energy Dissipation Equality 3 9 a 3 2 4 The compatibility of Energy and distance EVI and error estimates As the last example of the previous section shows in general we cannot hope to have uniqueness of the limit curve x obtained via the Minimizing Movements scheme for a generic geodesically convex functional If we want to derive properties like uniqueness and contractivity of the flow we need to have some stronger relation between the Energy functional and the distance d on X in this section we will assume the following 63 Assumption 3 24 Compatibility in Energy and distance X d is a Polish space E X gt RU 00 is a lower semicontinuous functional and for any xo 1 y X there exists a curve t q t such that
26. s lt lt sy lt land u Tya M we have PEN u Re P3 Rew SN 1 SN 2 BER CPSs u Ba PN u PSR CP es ea P32 u lt C lullsn sil sw sw 1 PEN u PiN C P3 u N 1 lt Clu O Jsi sills siil lt C lulls sy 6 15 4 2 With this result we can prove existence of the limit of P w as P varies in 3B 95 Theorem 6 12 For any u Ta M there exists the limit of P u as P varies in Proof We have to prove that given gt 0 there exists a partition P such that P u Q u lt jule YQ gt P 6 16 In order to do so it is sufficient to find 0 to lt t lt lt tn 1 such that gt tj41 t lt C and repeatedly apply equation 6 15 to all partitions induced by Q in the intervals t ti 1 Now for s lt t we can introduce the maps Tf T M T4 M which associate to the vector u T M the limit of the process just described taking into account partitions of s t instead of those of 0 1 Theorem 6 13 For any t lt tg lt t3 0 1 it holds TST 6 17 Moreover for any u Tyo M the curve t u Tj u Ty M is the parallel transport of u along y Proof For the group property consider those partitions of t3 which contain t and pass to the limit first on t t2 and then on t2 t3 To prove the second part of the statement we prove first that u is absolutely continuous
27. s lt 0 t t iv Evolution Variational Inequality and contraction x is the unique solution of the system of differential inequalities ld A sua Fle 5l yl lt FQ Vy H a e t 50 among all locally absolutely continuous curves 4 in 0 00 converging to T as t gt 0 Furthermore if y is a solution of 3 2 starting from Y it holds e y lt e z 7l v Asymptotic behavior Zf gt 0 then there exists a unique minimum min Of F and it holds F a F tmin lt F F min e In particular the pointwise energy inequality Bia EAA z minl Va e H gives tt Zmin lt a ae 3 2 The theory of Gradient Flows in a metric setting Here we give an overview of the theory of Gradient Flows in a purely metric framework 3 2 1 The framework The first thing we need to understand is the meaning of Gradient Flow in a metric setting Indeed the system 3 2 makes no sense in metric spaces thus we need to reformulate it so that it has a metric analogous There are several ways to do this below we summarize the most important ones For the purpose of the discussion below we assume that H R and that E H gt R is A convex and of class C1 Let us start observing that 3 2 may be written as t gt 2 is locally absolutely continuous in 0 00 converges to 7 as t 0 and it holds d 1 1 qe lt 5IVEP x zl a e t gt 0 3 3 Indeed along any a
28. uh Pr 52 6 9 Ht Ht D 1 i 1 D 7 Up Uy Uy Uj 4 it dt we To prove the torsion free identity we need first to understand how to calculate the Lie bracket of two vector fields To this aim let u i 1 2 be two regular curves such that yj 2 p and let ui Tan 72 R be two C vector fields satisfying uj v ug v where v are the velocity vector fields of ut We assume that the velocity fields v of jv are continuous in time in the sense that the map t gt vi is continuous in the set of vector valued measure with the weak topology and tH lvi yi is continuous as well to be sure that 6 7 holds for all t with vs vi and the initial condition makes sense With these hypotheses it makes sense to consider the covariant derivative Fu along p at t 0 for this derivative we write V u Similarly for wt 92 Let us consider vector fields as derivations and the functional u gt F u f pdp for given p E C R By the continuity equation the derivative of F along u is equal to Vo Uz 2 therefore the compatibility with the metric 6 9 gives d utu g Voua eco VP 08108 Ve Wage V7p ub ub Ve Vast Subtracting the analogous term u u F j and using the symmetry of V7 we get ut uP EH Ve Vugt Vage Given that the set VY pec is dense in Tan 22 R the above equation characterizes u u as uu Vazut Vaz 6 10 w
29. v w du 0 Yw L u s t V wu o Thus we now have a definition of tangent space for every y 2 M and this tangent space is natu rally endowed with a scalar product the one of L u This fact Theorem 2 29 and Proposition 2 30 are the bases of the so called weak Riemannian structure of 72 M W2 We now state without proof some other properties of 2 M W2 which resemble those of a Riemannian manifold For simplicity we will deal with the case M R only and we will assume that the measures we are dealing with are regular Definition 1 25 but analogous statements hold for general manifolds and general measures In the next three propositions p is an absolutely continuous curve in 2 IR such that p is regular for every t Also v is the unique up to a negligible set of times family of vector fields such that the continuity equation holds and v Tan Ao R for ae t Proposition 2 32 v can be recovered by infinitesimal displacement Let u and v as above Also let TF be the optimal transport map from u to us which exists and is unique by Theorem 1 26 due to our assumptions on u Then for a e t 0 1 it holds I2 Id v lim sot s t the limit being understood in L ut Proposition 2 33 Displacement tangency Let u and v as above Then for a e t 0 1 it holds W Id h lim o lith Id hvi Jgh lim 5 0 2 24 Proposition 2 34 Deriva
30. 14 we know that V E x7 lt Dsl and because in the limiting process Dsl will produce the slope term in the EDI see the proof of Theorem 3 14 With this notation we have the following result Corollary 3 12 EDE for the discrete solutions Let X E be satisfying Assumption 3 8 T D E 0 lt T lt F and a7 defined via the variational interpolation as in Definition 3 11 above Then it holds 1 ff AUE i E x7 F Dsp7 dr F Dsl7 dr E a7 3 16 t t for everyt ntT s m7 n lt meEN Proof It is just a restatement of equation 3 13 in terms of the notation given in 3 15 Thus at the level of discrete solutions it is possible to get a discrete form of the Energy Dissipa tion Equality under the quite general Assumptions 3 8 Now we want to pass to the limit as 7 0 In order to do this we need to add some compactness and regularity assumptions on the functional Assumption 3 13 Coercivity and regularity assumptions Assume that E X RU 00 satisfies e E is bounded from below and its sublevels are boundedly compact i e E lt c N B x is compact for any c R r gt Qand xz X e the slope V E D E 0 00 is lower semicontinuous e FE has the following continuity property Ln gt T sup VE an E n lt 00 gt E a gt E x Under these assumptions we can prove the following result Theorem 3 14 Gradient Flows in EDI formulation Let X d be a met
31. 2 19 PC and NPC spaces A geodesic space X d is said to be positively curved PC in the sense of Alexandrov if for every constant speed geodesic y 0 1 gt X and every z X the following concavity inequality holds P y 2 gt 1 t d 40 z ta m1 z t Hd 0 11 2 14 Similarly X is said to be non positively curved NPC in the sense of Alexandrov if the converse inequality always holds Observe that in an Hilbert space equality holds in 2 14 The result here is that Y2 X W2 is PC if X d is while in general it is not NPC if X is Theorem 2 20 2 X W2 is PC if X d is Assume that X d is positively curved Then P X W2 is positively curved as well Proof Let u be a constant speed geodesic in Y2 X and v E P2 X Let y PYo Geod X be a measure such that Mt er 4H vt 0 1 as in Theorem 2 10 Fix to 0 1 and choose y Opt uto v Using a gluing argument we omit the details it is possible to show the existence a measure Y Geod X x X such that greed w p ss 2 15 eto T yo y where 76 4U y x y Geod X n 7 2 X and er 7 2 Vto X Thena satisfies also e0 7 pa Adm uo v 2 16 e1 7 pa Adm p1 V and therefore it holds 2 pitt j f a cr 1 z da 7 2 2 14 A 4 gt to d yo z tod 71 2 to 1 to d Yo 1 da y C 1 to i d 40 z da y to
32. 5 Wo Miso H i 1 2 F Wo ij jon Hkjon i j 2 9 Therefore it holds ej 27 eran yH Opt uj jon Me 2n s Vj k 0 se WARE Also since the inequalities in 2 9 are equalities it is not hard to see that for jz a e y the points Yisor 0 2 must lie along a geodesic and satisfy d y 2 Ya 1 27 A yo 71 2 i 0 2 1 Hence p a e yis a constant speed geodesic and thus u Y Geod X Now suppose for a moment that u narrowly converges up to pass to a subsequence to some u E PY Geod X Then the continuity of the evaluation maps e yields that for any t 0 1 the 32 sequence n gt e 4fe narrowly converges to e 4 4 and this together with the uniform bound 2 9 easily implies that pz satisfies 2 7 Thus to conclude it is sufficient to show that some subsequence of jz has a narrow limit We will prove this by showing that u E 2 Geod X for every n N and that some subsequence is a Cauchy sequence in Y2 Geod X W2 Wa being the Wasserstein distance built over Geod X endowed with the sup distance so that by Theorem 2 7 we conclude We know by Remark 1 4 Remark 2 3 and Theorem 2 7 that for every n N the set of plans a P X 1 such that ta pijan for i 0 2 is compact in Y2 X Therefore a diagonal argument tells that possibly passing to a subsequence not relabeled we may assume that for every n N the sequence
33. A n lt y so that is nonnegative B null first and second marginal so that 7 Adm p v C f edn lt 0 so that y is not optimal Let Q IINU x V and P A Q be defined as the product of the measures mee where m y U x V Denote by n7 7 the natural projections of Q to U and V respectively and define N an Silat ato gP a ao gP It is immediate to verify that 77 fulfills A B C above so that the thesis is proven ii gt iii We need to prove that if T C X x Y is a c cyclically monotone set then there exists a c concave function such that f D T and max y 0 L u Fix x y T and observe that since we want to be c concave with the c superdifferential that contains I for any choice of xi yi T i 1 N we need to have p x lt e a y1 p y e z y1 e 1 91 21 lt e Cle yi x1 1 c z1 y2 pt y2 o c z y c x1 4 T c 21 42 c z2 y2 x2 lt ete a1 y1 elzen v2 wa ya claw 9 2 9 2 It is therefore natural to define as the infimum of the above expression as x yi i 1 y Vary among all N ples in I and N varies in N Also since we are free to add a constant to p we can neglect the addendum Y T and define pla inf efx y1 e 1 y1 el21 y2 clez v2 elen 2 9 the infimum being taken on N gt 1 integer and 2 y E T
34. Adm p v foc y dy x 7 1 6 inf yEM4 XxY where x y is equal to 0 if y Adm u v and 00 if y Adm u v and M4 X x Y is the set of non negative Borel measures on X x Y We claim that the function y may be written as xc sup f oadan a vende f ee vdre P where the supremum is taken among all y Y Cy X x Ca Y Indeed if y Adm u v then x y 0 while if y Adm u v we can find p Y Cy X x C Y such that the value between the brackets is different from 0 thus by multiplying y Y by appropriate real numbers we have that the supremum is 00 Thus from 1 6 we have inf J c x y dy a y YE Adm p v pills sup cle u er e 9 lejana vavt f e dva Call the expression between brackets F y p Y Since y gt F y p Y is convex actually linear and y Y F y 9 w is concave actually linear the min max principle holds and we have inf sup F su inf p Y E Y pH a et F x p Y Thus we have inf J ola y dy a y YE Adm p v sup int foment f eaaa f vender f o tvore sup elaran a f voir int f olen ele vvdarte b Now observe the quantity P J e z y ela vivre 12 If y x w y lt c x y for any x y then the integrand is non negative and the infimum is 0 achieved when y is the null measure Conversely if y x w y gt c x y for some x y X x Y then choose y nd z
35. Corollary 2 22 Geodesics in 42 M W2 Let m C Ao M Then the following two things are equivalent i m is a geodesic in P2 M W2 ii there exists a plan y P TM TM being the tangent bundle of M such that ji v dy z v W3 po u1 Exp t 4 ht 2 17 Exp t TM M being defined by x v exp tv Also for any p v E Pa M such that p is a regular measure Definition 1 32 the geodesic con necting u to v is unique Notice that we cannot substitute the first equation in 2 17 with 7 exp Y E Opt tUo H1 be cause this latter condition is strictly weaker it may be that the curve t gt exp tv is not a globally minimizing geodesic from x to exp v for some a v supp 7 40 Proof The implication i ii follows directly from Theorem 2 10 by taking into account the fact that t gt y is a constant speed geodesic on M implies that for some x v TM it holds Yt exp tv and in this case d yo 1 v For the converse implication just observe that from the second equation in 2 17 we have W2 111s fa lt J d exp tv exp sv dy v lt t s J v dy v t 8 W2 uo 11 having used the first equation in 2 17 in the last step To prove the last claim just recall that by Remark 1 35 we know that for u a e x there exists a unique geodesic connecting x to T x T being the optimal transport map Hence the conclusion follows from ii of Theorem 2 10
36. E is Gaussian and H is its Cameron Martin space namely H heE mhiy I In this case 2 z yl c x y 2 00 otherwise if yeH The issue of regularity of optimal maps would nowadays require a lecture note in its own A rough statement that one should have in mind is that it is rare to have regular even just continuous optimal transport maps The key Theorem 1 27 is due to L Caffarelli 22 21 23 Example 1 36 is due to G Loeper 55 For the general case of cost squared distance on a com pact Riemannian manifold it turns out that continuity of optimal maps between two measures with smooth and strictly positive density is strictly related to the positivity of the so called Ma Trudinger Wang tensor 59 an object defined taking fourth order derivatives of the distance function The 23 understanding of the structure of this tensor has been a very active research area in the last years with contributions coming from X N Ma N Trudinger X J Wang C Villani P Delanoe R McCann A Figalli L Rifford H Y Kim and others A topic which we didn t discuss at all is the original formulation of the transport problem of Monge the case c x y x y on R The situation in this case is much more complicated than the one with c x y x y 2 as it is typically not true that optimal plans are unique or that optimal plans are induced by maps For example consider on R any two probability me
37. O u are absolutely continuous and their total derivatives are given by dt d m d 50 us O te 02 Fue 05 O 0 O On Pulte Proof The first formula follows directly from Theorem 6 22 the second from the fact that OF is the adjoint of O d d On U Oy te On Fe On On W0 Pu 03 On 0 6 29 An important feature of equations 6 27 and 6 29 is that to express the derivatives of Nu Ur We Ov ue and OF u no new operators appear This implies that we can re cursively calculate derivatives of any order of the vector fields P uz Cian ue Ov ue and O uz provided of course that we make appropriate regularity assumptions on the vector field u and on the velocity vector field v An example of result which can be proved following this direction is that the operator t gt P is analytic along the restriction of a geodesic Proposition 6 24 Analyticity of t gt P Let u be the restriction to 0 1 of a geodesic de fined in some larger interval 1 Then the operator t P is analytic in the following sense For any to 0 1 there exists a sequence of bounded linear operators Ay Liss gt Las such that the following equality holds in a neighborhood of to Pap u gt CE ias T to t 0 T t to Vu L 6 30 n 103 Proof From the fact that u is the restricti
38. Riemannian manifolds Geom Funct Anal 20 2010 pp 124 159 N Fusco F MAGGI AND A PRATELLI The sharp quantitative isoperimetric inequality Ann of Math 2 168 2008 pp 941 980 W GANGBO The Monge mass transfer problem and its applications in Monge Amp re equa tion applications to geometry and optimization Deerfield Beach FL 1997 vol 226 of Con temp Math Amer Math Soc Providence RI 1999 pp 79 104 W GANGBO AND R J MCCANN The geometry of optimal transportation Acta Math 177 1996 pp 113 161 N GIGLI On the geometry of the space of probability measures in R endowed with the quadratic optimal transport distance 2008 Thesis Ph D Scuola Normale Superiore Second order calculus on Y2 M W2 Accepted by Memoirs of the AMS 2009 On the heat flow on metric measure spaces existence uniqueness and stability Calc Var Partial Differential Equations 2010 On the inverse implication of Brenier McCann theorems and the structure of P2 M W2 accepted paper Meth Appl Anal 2011 R JORDAN D KINDERLEHRER AND F OTTO The variational formulation of the Fokker Planck equation SIAM J Math Anal 29 1998 pp 1 17 electronic N JUILLET On displacement interpolation of measures involved in brenier s theorem ac cepted paper Proc of the AMS 2011 L V KANTOROVICH On an effective method of solving certain classes of extremal problems D
39. Va R a e t 0 1 T t z a Va R t 0 1 A simple computation shows that the curve t gt Af T t 4 106 solves d dt which is the same equation solved by jf It is possible to show that this fact together with the smoothness of the vf s and the equality ug 46 gives that 4 yu for every t see Proposition 8 1 7 and Theorem 8 3 1 of 6 for a proof of this fact Conclude observing that jie V ofii 0 2 28 2 dig x S W3 ue ue lt lt f IT 2 T s x Pdu f f velt a lt t f f vs T r z ease f lor T r Maca dr lt ta KAA adr eee L and that by the characterization of convergence 2 4 W2 us ut 0 as gt 0 for every t 0 1 48 2 4 Bibliographical notes To call the distance W3 the Wasserstein distance is quite not fair a much more appropriate would be Kantorovich distance Also the spelling Wasserstein is questionable as the original one was Vasershtein Yet this terminology is nowadays so common that it would be impossible to change 1t The equivalence 2 4 has been proven by the authors and G Savar in 6 In the same reference Remark 2 8 has been first made The fact that Y2 X W2 is complete and separable as soon as X d is belongs to the folklore of the theory a proof can be found in 6 Proposition 2 4 was proved by C Villani in 79 Theorem 7 12 The terminology displ
40. a det V7y z and then the regularity theory for Monge Ampere developed by Caffarelli and Urbas applies As an application of Theorem 1 26 we discuss the question of polar factorization of vector fields on R Let Q C Rf be a bounded domain denote by jig the normalized Lebesgue measure on 9 and consider the space S Q Borel maps Q gt Q syuo uo The following result provides a nonlinear projection on the nonconvex space Q Proposition 1 28 Polar factorization Let S L uo R be such that v Syp is regular Definition 1 25 Then there exist unique s S Q and V with p convex such that S Vp os Also s is the unique minimizer of J18 3an Proof By assumption we know that both uo and v are regular measures with finite second moment We claim that among all S Q inf S 3 du i y dy x y 1 7 cat f 5 dy ee z y dy x y 1 7 To see why associate to each S Q the plan y 8 S which clearly belongs to Adm uo v This gives inequality gt Now let be the unique optimal plan and apply Theorem 1 26 twice to get that 7 Id Ve gua Ve Id gv for appropriate convex functions y which therefore satisfy Vy o V Id p a e Define s Veo S Then spi uo and thus s S Q Also S Vy o s which proves the existence of the polar factorization The identity i le yey 2 4 Is Pdyn V oS SPdun J ve Id 2av min
41. and if u is regular for every t gt 0 it holds u OY F u a e t 0 00 3 49 where v is the velocity vector field associated to u characterized by d qiet Vv vie 0 v Tanp Po R a e t Proof Use the existence Theorem 3 25 and the equivalence of the EVI formulation of Gradient Flow and the Subdifferential one provided by Proposition 3 28 It remains to understand which kind of equation is satisfied by the Gradient Flow u By equation 3 49 this corresponds to identify the subdifferentials of V W at a generic u P2 R This is the content of the next three propositions For simplicity we state and prove them only under some unneeded smoothness assumptions The underlying idea of all the calculations we are going to do is the following equivalence veWF B im Td beV een Flu e 0 E f 0 Vo Yee CHR 3 50 valid for any A geodesically convex functional where we wrote to intend that this equivalence holds only when everything is smooth To understand why 3 50 holds start assuming that v OW F p fix p CS R and recall that for sufficiently small the map Id eVy is optimal Remark 1 22 Thus by definition of subdifferential we have Fw te f Vo du S IVelayy lt Fd eV9 40 Subtracting F u on both sides dividing by gt 0 and lt 0 and letting 0 we get the implication To prove the converse one
42. and discussion thereafter Parallel transport This is the main existence result on this subject we prove that along regular curves the parallel transport always exists Theorem 6 15 We will also discuss a counterexample to the existence of parallel transport along a non regular geodesic Example 6 16 This will show that the definition of regular curve is not just operationally needed to provide a definition of smoothness 88 of vector fields but is actually intrinsically related to the geometry of A R Calculus of derivatives Using the technical tools developed for the study of the parallel transport we will be able to explicitly compute the total and covariant derivatives of basic examples of vector fields Curvature We conclude the discussion by showing how the concepts developed can lead to a rigor ous definition of the curvature tensor on 2 R We will write v and v w for the norm of the vector field v and the scalar product of the vector fields v w in the space L u which we will denote by L respectively We now start with the definition of regular curve All the curves we will consider are defined on 0 1 unless otherwise stated Definition 6 2 Regular curve Let u be an absolutely continuous curve and let v be its ve locity vector field that is v is the unique vector field up to equality for a e t such that vi Tanp P2 R for a e t and the continuity equation d q V vm 0
43. connecting them i e a constant speed geodesic such that Yo x and J1 y Before entering into the details let us describe an important example Recall that X 3 71 gt y P X is an isometry Therefore if t y is a constant speed geodesic on X connecting lt to y the curve t is a constant speed geodesic on 22 X which connects 6 to y The important thing to notice here is that the natural way to interpolate between 6 and 4 is given by this so called displacement interpolation Conversely observe that the classical linear interpolation try uy 1 t d ty produces a curve which has infinite length as soon as x y because W ut Hs t s d a y and thus is unnatural in this setting We will denote by Geod X the metric space of all constant speed geodesics on X endowed with the sup norm With some work it is possible to show that Geod X is complete and separable as soon as X is we omit the details The evaluation maps e Geod X X are defined for every t 0 1 by ely Y 2 6 Theorem 2 10 Let X d be Polish and geodesic Then P2 X W2 is geodesic as well Further more the following two are equivalent i t gt m E P2 X is a constant speed geodesic ii There exists a measure p E Pa Geod X such that eo e1 4M E Opt o p1 and Ht er 4H 2 7 31 Proof Choose p ut A X and find an optimal plan y Opt u v By Lemma 2 11 be low and class
44. consequence of the previous theorem is that being optimal is a property that depends only on the support of the plan y and not on how the mass is distributed in the support itself if is an optimal plan between its own marginals and is such that supp 7 C supp 7 then is optimal as well between its own marginals of course We will see in Proposition 2 5 that one of the important consequences of this fact is the stability of optimality Analogous arguments works for maps Indeed assume that T X Y is a map such that T x O0 y x for some c concave function y for all x Then for every y A X such that condition 1 4 is satisfied for v Ty the map T is optimal between u and Typu Therefore it makes sense to say that T is an optimal map without explicit mention to the reference measures Remark 1 15 From Theorem 1 13 we know that given pp A X v AY satisfying the assumption of the theorem for every optimal plan there exists a c concave function y such that supp y C 0 y Actually a stronger statement holds namely if supp y C y for some optimal y then supp 7 C y for every optimal plan y Indeed arguing as in the proof of 1 13 one can see that max y 0 L u implies max y 0 L1 v and thus it holds feds ord f oe p y dy z y lt J denare J denare supp y c g J p z 9 y dy x y if pdu J p tdv Thus the inequality must be an equality which is true if
45. e Finally we discuss the minimal regularity requirements for the object found to be well defined Pick y 7 CS R and observe that a curve of the kind t Id tVy 4p is a regular geodesic on an interval T T for T sufficiently small Remark 1 22 and Proposition 6 3 It is then immediate to verify that a vector field of the kind V7 along it is C Its covariant derivative calculated at t 0 is given by P V w Vi Thus we write Vvo VY Pa V Y Vy Vy E CX R 6 31 Proposition 6 25 Let y P2 R and 1 Y2 3 E CL R The curvature tensor R in p calculated for the 3 vector fields Vp i 1 2 3 is given by NET Obe CAES 6 32 O p Nal p2 Vos 205p Nal Ver Vo Proof We start computing the value of Vyp Vvy Vys Let u Id tV 2 4p and observe as just recalled that u is a regular geodesic in some symmetric interval T T The vector field V7 3 Vopr is clearly C along it thus by Proposition 6 24 also the vector field u P V 43 V 1 Vvo Vp3 ut is C The covariant derivative at t 0 of u along u is by definition the value of Vou Vvo V3 at u Applying formula 6 25 we get Vyp Vyp Ves Pu V V7 3 Ver Veo V7 92 Pi V p3 Ve 6 33 Symmetrically it holds Vy Vvo Vo3 Py V V7 93 Vea Vyr V y1 Pi V7H3 V2 6 34 Finally from the torsion free identity 6 10 we have Vyr Vea Py V g1 Veo V p2 Vya
46. f x dv x lt co there exists only one transport plan from p to v and this plan is induced by a map T ii uis regular If either i or ii hold the optimal map T can be recovered by taking the gradient of a convex function Proof ii gt i and the last statement Take a x b x x in the statement of Theorem 1 13 Then our assumptions on u v guarantees that the bound 1 4 holds Thus the conclusions of Theorems 1 13 and 1 17 are true as well Using Remark 1 18 we know that for any c concave Kantorovich po tential y and any optimal plan y Opt j1 v it holds supp y C 0 y Now from Proposition 1 21 we know that Y 2 is convex and that 0 y O Here we use our assumption on p since Y is convex we know that the set E of points of non differentiability of Y is u negligible Therefore the map V R R is well defined ji a e and every optimal plan must be concen trated on its graph Hence the optimal plan is unique and induced by the gradient of the convex function ii i We argue by contradiction and assume that there is some convex function R gt R such that the set E of points of non differentiability of Y has positive u measure Possibly modifying outside a compact set we can assume that it has linear growth at infinity Now define the two maps T x the element of smallest norm in 07 G x S x the element of biggest norm in 07 p x and the plan 1 y 5 Ud
47. function not identically equal to oo Then is Lipschitz semiconcave and real valued Also assume that y 0 x Then exp y C O y za Conversely if p is differentiable at x then exp Vy x 0 y x 18 Proof The fact that is real valued follows from the fact that the cost function d x y 2 is uni formly bounded in x y M Smoothness and compactness ensure that the functions d y 2 are uniformly Lipschitz and uniformly semiconcave in y M this gives that is Lipschitz and semiconcave Now pick y 0 y x and v exp y Recall that v belongs to the superdifferential of d y 2 at x ie zy x y 2 2 Thus from y p x we have lz oa lt lt Haw _ Fey v exp 2 o d a z lt v exp z o d z z that is v OT y x To prove the converse implication it is enough to show that the c superdifferential of at x is non empty To prove this use the c concavity of y to find a sequence yn C M such that gt d as c p z lim gt 9 Yn d ri z lt P Yn VzE M neN By compactness we can extract a subsequence converging to some y M Then from the continuity of d z 2 and y it is immediate to verify that y 0 y z Remark 1 31 The converse implication in the previous proposition is false if one doesn t assume y to be differentiable at x i e it is not true in general t
48. graph and denote by T the corresponding map By the inner regularity of measures it is easily seen that we can also assume I U I to be a compact Under this assumption the domain of T i e the projection of T on X is o compact hence Borel and the restriction of T to the compact set 7x Tn is continuous It follows that T is a Borel map Since y T x y a e in X x Y we conclude that a y dyle y f p z T e dy e y J p z T 2 du z so that y Id x T gp Thus the point is the following We know by Theorem 1 13 that optimal plans are concentrated on c cyclically monotone sets still from Theorem 1 13 we know that c cyclically monotone sets are obtained by taking the c superdifferential of a c concave function Hence from the lemma above what we need to understand is how often the c superdifferential of a c concave function is single valued There is no general answer to this question but many particular cases can be studied Here we focus on two special and very important situations 14 e X Y R and c z y x y 2 e X Y M where M is a Riemannian manifold and c x y d x y 2 d being the Riemannian distance Let us start with the case X Y R and c z y x y 2 In this case there is a simple characterization of c concavity and c superdifferential Proposition 1 21 Let y R RU oo Then y is c concave if and only if x gt G x x 2 p x
49. ie uPere u YE Adm 1 7 17 shows inequality lt in 1 7 and the uniqueness of the optimal plan ensures that s is the unique minimizer To conclude we need to show uniqueness of the polar factorization Assume that S V o5 is another factorization and notice that VO po VZ o 5 guo v Thus the map V is a transport map from upg to v and is the gradient of a convex function By Proposition 1 21 and Theorem 1 13 we deduce that V is the optimal map Hence VY Vy and the proof is achieved Remark 1 29 Polar factorization vs Helmholtz decomposition The classical Helmoltz decom position of vector fields can be seen as a linearized version of the polar factorization result which therefore can be though as a generalization of the former To see why assume that 2 and all the objects considered are smooth the arguments hereafter are just formal Let u Q R be a vector field and apply the polar factorization to the map Se Id eu with e small Then we have Se Vy 0 se and both Vy and s will be perturbation of the identity so that Vo Id ev ole Se Id ew o e The question now is which information is carried on v w from the properties of the polar factoriza tion At the level of v from the fact that V x V 0 we deduce V x v 0 which means that v is the gradient of some function p On the other hand the fact that se is measure preserving implies that w satisfies V wxg 0 in
50. if wz is The identity iiim uth o T t t h Ut m Urth T to t h uso T to t o T t to h0 h h0 h Fu Tost o T t to shows that the total derivative is well defined for a e t and that is an L vector field in the sense that it holds t 0 dt s lus 9 Ti S Ulla lt J t dt lt Ht Notice also the inequality dr Mr d g 2 T t T a f t An important property of the total derivative is the Leibnitz rule for any couple of absolutely contin uous vector fields u u along the same regular curve p the map t gt uz Ur p is absolutely continuous and it holds diye d i 2 1d o T ut ut Suha uz Het a e t 6 6 Ht Ht Indeed from the identity up ue uy o T to t ue o T to t ur Ht dt a Hto it follows the absolute continuity and the same expression gives d d dt ut Ue a dt uy o T to t 3 u 9 T to t d 50 o T to t u o T to t D d 2 d 2 A Ut Ut Up yu quel ub get Example 6 7 The smooth case Let x t gt amp x be a CX vector field on R u a regular curve and v its velocity vector field Then the inequality Ss 0 T t 8 Sellue lt Mes amp Hto ut o T to t ul o T to t Hto Hto Hs T Il Q T t S a Eel lt Cls ES t C T S an Id u 91 with C sup
51. in the Heisenberg group J Funct Anal 208 2004 pp 261 301 K BACHER AND K T STURM Localization and tensorization properties of the curvature dimension condition for metric measure spaces J Funct Anal 259 2010 pp 28 56 J D BENAMOU AND Y BRENIER A numerical method for the optimal time continuous mass transport problem and related problems in Monge Amp re equation applications to geometry and optimization Deerfield Beach FL 1997 vol 226 of Contemp Math Amer Math Soc Providence RI 1999 pp 1 11 P BERNARD AND B BUFFONI Optimal mass transportation and Mather theory J Eur Math Soc JEMS 9 2007 pp 85 127 M BERNOT V CASELLES AND J M MOREL The structure of branched transportation networks Calc Var Partial Differential Equations 32 2008 pp 279 317 S BIANCHINI AND A BRANCOLINI Estimates on path functionals over Wasserstein spaces SIAM J Math Anal 42 2010 pp 1179 1217 A BRANCOLINI G BUTTAZZO AND F SANTAMBROGIO Path functionals over Wasserstein spaces J Eur Math Soc JEMS 8 2006 pp 415 434 124 20 21 22 23 32 33 34 Sy 35 L BRASCO G BUTTAZZO AND F SANTAMBROGIO A benamou brenier approach to branched transport Accepted paper at SIAM J of Math Anal 2010 Y BRENIER D composition polaire et r arrangement monotone des champs de vecteurs C R Acad Sci Paris S r I Math 305 1987 pp 80
52. ixdunt f 1a xdun s f trdin f fdin lt f Fan Pas X Br a being given by 2 2 Since fx is continuous and bounded we have f fydun gt f fxd and therefore lim T A T PIE f fap Qae Since gt 0 was arbitrary this part of the statement is proved ii gt iii Obvious iii gt i Argue by contradiction and assume that there exist gt 0 and o X such that for every R gt 0 it holds suppen Sx Ba Go d Zo dpn gt Then it is easy to see that it holds lim d x9 dpn gt 2 3 noo X Br 20 27 For every R gt 0 let yr be a continuous cutoff function with values in 0 1 supported on Br xo and identically 1 on Br 2 xo Since d 7 R is continuous and bounded we have 6 20 xedu lim d 20 xRdUn n co eee d 220 dpin I d x9 1 xn din f Pe zodu lim f 00 1 xen noo lt f 20 du tim f P 00 dpin noo X Br 2xo f Pe zo du lim d 29 din noo X Br xo lt f Pe zodu having used 2 3 in the last step Since J 26 20 qu sup f P zo xndu lt 2 t0 du e we got a contradiction Proposition 2 5 Stability of optimality The distance W2 is lower semicontinuous w r t narrow convergence of measures Furthermore if y C P X is a sequence of optimal plans which narrowly converges to y Pa X then y is optimal as well Proof Let un Un C A X be two sequences of measures narrowly converging t
53. metric setting on the other hand we discuss the important application of the abstract theory to the case of geodesically convex functionals on the space 2 R W2 Let us recall that for a smooth function F M gt R on a Riemannian manifold a gradient flow x starting from 7 M is a differentiable curve solving Beri V 3 1 Xo T Observe that there are two necessary ingredients in this definition the functional F and the metric on M The role of the functional is clear The metric is involved to define V F it is used to identify the cotangent vector dF with the tangent vector V F 3 1 Hilbertian theory of gradient flows In this section we quickly recall the main results of the theory of Gradient flow for A convex func tionals on Hilbert spaces This will deserve as guideline for the analysis that we will make later on of the same problem in a purely metric setting 49 Let H be Hilbert and A R A A convex functional F H RU 00 is a functional satisfying F 1 t z ty lt 1 t F x tF y a t z yl Va y H this corresponds to V F gt AJd for functionals on R We denote with D F the domain of F i e D F x F x lt The subdifferential 0 F x of F at a point x D F is the set of v H such that X F x v y 2 l2 yl lt Fly Vy H An immediate consequence of the definition is the fact that the subdifferential of F satisfies the mo
54. not defined j1 a e for arbitrary H and Lipschitz v Rademacher s theorem is of no help here because we are not assuming the measures u to be absolutely continuous w r t the Lebesgue measure To give a meaning to formula 6 23 we need to introduce a new tensor Definition 6 17 The Lipschitz non Lipschitz space Let P2 R The set LNL C L is the set of couples of vector fields u v such that min Lip w Lip v lt 00 i e the set of couples of vectors such that at least one of them is Lipschitz We say that a sequence un Un E LNL converges to u v E LNL provided un ull 0 lun v 0 and sup min Lip u Lip un lt oo The following theorem holds Theorem 6 18 The Normal tensor Let u Y2 R The map N u v C2 R4 R4 gt Tan Po R2 u v Pi Vut v extends uniquely to a sequentially continuous bilinear and antisymmetric map still denoted by N from LNL in Tan P2 R for which the bound Nu u 0 lt min Lip w o Lip v lull 6 24 holds Proof For u v CX R R we have V u v Vut v Vot u so that taking the projections on Tan P2 R we get Ni u v N v u Vu v CX RI R In this case the bound 6 24 is trivial To prove existence and uniqueness of the sequentially continuous extension it is enough to show that for any given sequence n un Un C R R converging to some u v LNL the
55. ooa aaa 69 3 3 2 Three classical functionals o o oaoa a 70 3 4 Bibliographical notes sa s a 5 sok eo g a ae ee 8 a Gos 76 l_ambrosio sns it Tnicola gigli unice fr 4 Geometric and functional inequalities 77 4 1 Brunn Minkowski inequality 2 0 ee 77 4 2 Isoperimetric ineguality s s vb age ate ewe ba pee ae a ee whe 78 4 3 Sobolev Inequality 2 sapis saret Pag glee Se oe ok be es 78 4 4 Bibliographical notes 2 2 00000002 2 eee ee 79 5 Variants of the Wasserstein distance 80 5 1 Branched optimal transportation 2 2 2 2 0 20 2 2 0 0 00000 80 5 2 Different action functional 0 0 022 0 000000 81 5 3 An extension to measures with unequal mass 000 82 5 4 Bibliographical notes 2 2 i a e a a A E 84 6 More on the structure of 72 M W2 84 6 1 Duality between the Wasserstein and the Arnold Manifolds 84 6 2 Onthe notion of tangent space 2 2 20 0 0 0 00000 87 6 3 Second ordercalculus ooa e 88 6 4 Bibliographical notes s oeei e aa ee 106 7 Ricci curvature bounds 107 7 1 Convergence of metric measure spaces 2 2 ee ee 109 7 2 Weak Ricci curvature bounds definition and properties 112 7 3 Bibliographical notes 2 5 5 e ho eB ee ee a S 122 Introduction The opportunity to write down these notes on Optimal Transport has been the CIME course in Cetraro given by the first author in
56. set of optimal plans from n to v for the Kantorovich formulation of the transport problem i e the set of minimizers of Problem 1 2 More generally we will say that a plan is optimal if it is optimal between its own marginals Observe that with the notation Opt ju v we are losing the reference to the cost function c which of course affects the set itself but the context will always clarify the cost we are referring to Once existence of optimal plans is proved a number of natural questions arise e are optimal plans unique e is there a simple way to check whether a given plan is optimal or not e do optimal plans have any natural regularity property In particular are they induced by maps e how far is the minimum of Problem 1 2 from the infimum of Problem 1 1 This latter question is important to understand whether we can really consider Problem 1 2 the re laxation of Problem 1 1 or not It is possible to prove that if c is continuous and u is non atomic then inf Monge min Kantorovich 1 2 so that transporting with plans can t be strictly cheaper than transporting with maps We won t detail the proof of this fact 1 2 Necessary and sufficient optimality conditions To understand the structure of optimal plans probably the best thing to do is to start with an example Let X Y R and c z y x y 2 Also assume that u v A R are supported on finite sets Then it is immediate to verify that a plan y
57. superdifferential of a c concave function This result is part of the following important theorem Theorem 1 13 Fundamental theorem of optimal transport Assume that c X x Y gt Ris continuous and bounded from below and let u P X v P Y be such that c a y lt a x b y 1 4 for some a L u b L v Also let y Adm u v Then the following three are equivalent i the plan is optimal ii the set supp 7 is c cyclically monotone iii there exists a c concave function such that max p 0 L u and supp y C O y Proof Observe that the inequality 1 4 together with J ola yd a y lt I a z b y d7 2 y J a z du 2 b y dv y lt 00 YF Adn m v implies that for any admissible plan 7 Adm u v the function max c 0 is integrable This together with the bound from below on c gives that c L1 for any admissible plan i ii We argue by contradiction assume that the support of y is not c cyclically monotone Thus we can find N E N x yi i lt icn C supp 7y and some permutation o of 1 N such that N N dele y gt Yo elt vows i l i l By continuity we can find neighborhoods U gt xi V 3 y with C Us Vo i clui Vi lt 0 V ui vi E Ui xX Vi 1 lt S lt i lt N Me t 1 Our goal is to build a variation y y n of y in such a way that minimality of is violated To this aim we need a signed measure n with
58. that the vector field v on supp uz is when read in charts Lipschitz Thus passing to local coordinates and recalling that d y is uniformly semiconcave the sit uation is the following We have a semiconcave function f R R and a semiconvex function g RI gt R such that f gt g on Rf f g ona certain closed set K and we have to prove that the vector field u K R defined by u x V f x Vg x is Lipschitz Up to rescaling we may assume that f and g are such that f is concave and g is convex Then for every x K and y R we have u x y x e yl lt gly g x lt FY Fe lt ula y 2 ly 2 and thus for every x K y R it holds If y F x u x y 2 lt x yl Picking x1 x2 K and y R we have f a2 f 1 ular 2 1 lt 1 x2 f z2 y f w2 u x2 y lt yl f 2 y f z1 u z1 2 y z1 lt z2 y 21 Adding up we get u z1 u z2 y lt 1 eal yl lee y z1 lt 3 lz1 z2 yl Eventually choosing y u x1 u z2 6 we obtain ju x1 u z2 lt 36 xz1 xo It is worth stressing the fact that the regularity property ensured by the previous corollary holds without any assumption on the measures 4o H1 Remark 2 25 A much simpler proof in the Euclidean case The fact that intermediate trans port maps are
59. the Wasserstein manifold R We recall that the tangent space Tan 2 R at the measure p is the set Tan P2 R2 Ve pE cx R endowed with the scalar product of L p we neglect to take the closure in L p because we want to keep the discussion at a formal level The perturbation of a measure p in the direction of a tangent vector Vy is given by t gt Id tVy xp The Arnold Manifold Arn p associated to a certain measure p IR is the set of maps S R R which preserve p Arn p s R gt R Sup p We endow Arn p with the L distance calculated w r t p To understand who is the tangent space at Arn p at a certain map S pick a vector field v on R and consider the perturbation t S tv of S in the direction of v Then v is a tangent vector if and only if ino 5 tv 4p 0 Observing that d d d 7 7 elezo Otte 9 z qlecolattves S p Jilo Fitts 1 up V vosto we deduce TansArn p vector fields v on R such that V vo S p o which is naturally endowed with the scalar product in L p We are calling the manifold Arn p an Arnold Manifold because if p is the Lebesgue measure restricted to some open smooth and bounded set Q this definition reduces to the well known definition of Arnold manifold in fluid mechanics the geodesic equation in such space is formally the Euler equation for the motion of an incompressible and inviscid fluid in 2 Finally th
60. the sense of distributions indeed for any smooth f R gt R it holds A q 0 lo Falselutin Zheng Fo sedua f VF wano Then from the identity V p se Id e Vp w o we can conclude that u Vp w a We now turn to the case X Y M with M smooth Riemannian manifold and c x y d x y 2 d being the Riemannian distance on M For simplicity we will assume that M is compact and with no boundary but everything holds in more general situations The underlying ideas of the foregoing discussion are very similar to the ones of the case X Y R4 the main difference being that there is no more the correspondence given by Proposition 1 21 between c concave functions and convex functions as in the Euclidean case Recall however that the concepts of semiconvexity i e second derivatives bounded from below and semiconcavity make sense also on manifolds since these properties can be read locally and changes of coordinates are smooth In the next proposition we will use the fact that on a compact and smooth Riemannian manifold the functions x d x y are uniformly Lipschitz and uniformly semiconcave in y M i e the second derivative along a unit speed geodesic is bounded above by a universal constant depending only on M see e g the third appendix of Chapter 10 of 80 for the simple proof Proposition 1 30 Let M be a smooth compact Riemannian manifold without boundary Let M RU co be a c concave
61. they satisfy the continuity equation Then the measures fiz ig ps and the vector fields di v satisfy the continuity equation on R Therefore is an absolutely continuous curve and it holds lt lllz vell 2u for a e t Notice that 2 is bilipschitz and therefore u is absolutely continuous as well Hence to conclude it is sufficient to show that m 4 a e t To prove this one can notice that the fact that 7 is bilipschitz and validity of d lim sup 2 y 1 r gt 0 evem i ily d x y lt r give that W lim sup O Ns 1 r0 pve Po M Wolixn inv Wal msv lt r We omit the details Part A Fix y C IR and observe that for every y Opt Ht ps it holds i pds J pdu omarten f ewar elu gla dri ev Ae Volz Ay x y x d dyi a i 2 25 f Vele v s anita v ETE i WotePa ean I yl dy x y Rem y t s IV ell teu Wa ue Ls Rem t s IA where the remainder term Rem y t s can be bounded by by Lip V Lip V Rem p t s lt PEA fle yPayz e y PP wiu ps Thus 2 25 implies that the map t f pdp is absolutely continuous for any y CX R Now let D C CX R be a countable set such that Vy y D is dense in Tan A2 R for every t 0 1 the existence of such D follows from the compactness of 4 rejo 1 C P2 R we omit the details The above arguments imply that th
62. to associate uniquely a family of vector fields v to a given absolutely continuous curve There are two possible approaches Algebraic approach The fact that for distributional solutions of the continuity equation the vector field v acts only on gradients of smooth functions suggests that the v s should be taken in the set of gradients as well or more rigorously v should belong to L m 2 22 Ve pe Ce M for a e t 0 1 Variational approach The fact that the continuity equation is linear in v and the L norm is strictly convex implies that there exists a unique up to negligible sets in time family of vector fields ve L u t 0 1 with minimal norm for a e t among the vector fields compatible with the curve u via the continuity equation In other words for any other vector field compatible with the curve p in the sense that 2 20 is satisfied it holds z2 qu vellz2 u for a e t It is immediate to verify that v is of minimal norm if and only if it belongs to the set fo L u fow dui 0 Vw L m s t V gt wui o 2 23 The important point here is that the sets defined by 2 22 and 2 23 are the same as it is easy to check Therefore it is natural to give the following 45 Definition 2 31 The tangent space Let y 2 M Then the tangent space Tan 72 M at P2 M in n is defined as L u Tan P2 M Vo pE ce M v L y J
63. to some function f on 0 1 as e 0 for which it holds 8 FOLS f oar 61 We know that f f on some set A C 0 1 such that 1 0 1 A 0 and we want to prove that they actually coincide everywhere Recall that f is 1 s c and f M is continuous hence f lt f in 0 1 If by contradiction it holds f to lt c lt C lt f to for some to c C we can find 6 gt Osuch that f t gt Cint to tp 6 Thus f t gt C fort tp 6 to 6 A and the contradiction comes from 1 g t dt gt g t dt gt j cia eee es 0 to d to F0 N A to 4 to 6 n lt tol Thus we proved that if g L 0 1 it holds IFA ca ess server vt lt s 0 1 M gt 0 Letting M oo we prove 3 24 and hence the thesis This proposition is the key ingredient to pass from existence of Gradient Flows in the EDI for mulation to the one in the EDE formulation Theorem 3 20 Gradient Flows in the EDE formulation Let X E be satisfying Assumption 3 17 and T X be such that E T lt Then all the results of Theorem 3 14 hold Also any Gradient Flow in the EDI sense is also a Gradient Flow in the EDE sense Definition 3 4 Proof The first part of the statement follows directly from Proposition 3 18 By Theorem 3 14 we know that the limit curve is absolutely continuous and satisfies 1 ff 1 f E as 7 2dr 5J VE a dr lt E z Vs gt 0 3 27
64. u XN such that mi Uy Yn EN Theorem 2 7 Basic properties of the space 72 X W2 Let X d be complete and separable Then n gt pb narrowly Waltin 0 T fE zojdun gt ero for some x E X 2 4 Furthermore the space P2 X W2 is complete and separable Finally K C P2 X is relatively compact w rt the topology induced by W3 if and only if it is tight and 2 uniformly integrable Proof We start showing implication in 2 4 Thus assume that W2 j1n p 0 Then y P e 209 dpin yf eoma To prove narrow convergence for every n N choose y Opt u Hn and use repeatedly the gluing lemma to find for every n N a measure a A X x X such that W2 Hn zo E W t xo lt Wo Hn H gt 0 0 n fas Ty An Yn 0 1 n 1 Ny An An 1 Then by Kolmogorov s theorem we know that there exists a measure A X x XN such that ry a Qn Yn EN By construction we have a r YI r x xx d x 2 z2 x2 7 Walt Hn gt 0 Thus up to passing to a subsequence not relabeled we can assume that 7 x 7 x for a almost any x X x XN Now pick f C X and use the dominated convergence theorem to get z 7 n ne 0 jim f Fain tim ffor da f for da fau if closed balls in X are compact the proof greatly simplifies Indeed in this case the inequality R u X Br xo lt f d o du and the uniform bound on the second momen
65. understood by the following calculation Suppose we want to move a Delta mass 6 into the Lebesgue measure on a unit cube whose center is x Then the first thing one wants to try is divide the cube into 2 cubes of side length 1 2 then split the delta into 2f masses and let them move onto the centers of these 24 cubes Repeat the process by dividing each of the 2 cubes into 2 cubes of side length 1 4 and so on The total cost of this dynamical transfer is proportional to id a _ i d 1 ad De sa SH Dee t 1 number of segments i 1 at the step length of each weighted mass on each segment at the step i segment at the step 7 which is finite if and only if d 1 ad lt 0 that is if and only if a gt 1 L A regularity result holds for a 1 1 d 1 which states that far away from the supports of the starting and final measures any minimal transfer is actually a finite tree Theorem 5 3 Regularity Let u v YA R with compact support a 1 1 n 1 and let T T w be a continuous tree with minimal a cost between u and v Then T is locally a finite tree in R supp U supp v 5 2 Different action functional Let us recall that the Benamou Brenier formula Proposition 2 30 identifies the squared Wasserstein distance between u P Zt u p Z1 E P R by 1 W m f f Ple e a 0 where the infimum is taken among all the distributional solutions of the continuity equation d she V
66. up to negligible sets for which inequality 2 18 is satisfied see e g Theorem 1 1 2 of 6 for the simple proof The link between absolutely continuous curves in 4 J and the continuity equation is given by the following theorem Theorem 2 29 Characterization of absolutely continuous curves in 72 1 W2 Let M bea smooth complete Riemannian manifold without boundary Then the following holds A For every absolutely continuous curve p C Ao M there exists a Borel family of vector fields vi on M such that v 12 u lt li for a e t and the continuity equation d q V vette 0 2 20 holds in the sense of distributions B If uv satisfies the continuity equation 2 20 in the sense of distributions and ie ve 22 4 dt lt 00 then up to redefining t py on a negligible set of times u is an ab solutely continuous curve on P2 M and fu4 lt v l r2 u for ae t 0 1 Note that we are not assuming any kind of regularity on the us We postpone the sketch of the proof of this theorem to the end of the section for the moment we analyze its consequences in terms of the geometry of 2 M The first important consequence is that the Wasserstein distance which was defined via the static optimal transport problem can be recovered via the following dynamic Riemannian like formula Proposition 2 30 Benamou Brenier formula Let 1 ut A2 M Then it holds 1 W int ulead 2 21
67. we put the apexes in Mosco because we prove the I lim inequality only for measures with bounded densities This will be enough to prove the stability of Ricci curvature bounds see Theorem 7 12 Proof For the first statement we just notice that by Lemma 7 4 we have E Un Mn gt E An enlm and the conclusion follows from the narrow lower semicontinuity of amp m For the second one we define un y 4 Then applying Lemma 7 4 twice we get E ulm gt O Un mMn On nlm from which the T lim inequality follows Thus to conclude we need to show that Wa Yn Ln Lt gt 0 To check this we use the Wassertein space built over the pseudo metric space Xn U X d let yu pmy and for any n N define the plan 7 E A X x X by dy y x p x dy y x and notice that 7 Adm un u Thus Wa un u lt yI d x y dY n y lt J eeeaneerrertaa lt VMV Cldn Yn where M is the essential supremum of p By definition it is immediate to check that the density nn of un is also bounded above by M Introduce the plan by d y x mn y dy y x and notice that 7 E Adm Hn Yn Mn so that as before we have Wo tins Yn glin lt i J d2 y dTn y 1 lt i f d2 2 yn Yq y 2 lt VM Cdn Ta In conclusion we have which gives the thesis 111 7 2 Weak Ricci curvature bounds definition and properties Define the functions uy N gt 1
68. we remark that L Brasco G Buttazzo and F Santambrogio proved a kind of Benamou Brenier formula for branched transport in 17 The content of Section 5 2 comes from J Dolbeault B Nazaret and G Savar 33 and 26 of J Carrillo S Lisini G Savar and D Slepcev Section 5 3 is taken from a work of the second author and A Figalli 37 6 More on the structure of 2 gt M W2 The aim of this Chapter is to give a comprehensive description of the structure of the Riemannian manifold 2 IR W2 thus the content of this part of the work is the natural continuation of what we discussed in Subsection 2 3 2 For the sake of simplicity we are going to stick to the Wasserstein space on R but the reader should keep in mind that the discussions here can be generalized with only little effort to the Wasserstein space built over a Riemannian manifold 6 1 Duality between the Wasserstein and the Arnold Manifolds The content of this section is purely formal and directly comes from the seminal paper of Otto 67 We won t even try to provide a rigorous background for the discussion we will do here as we believe that dealing with the technical problems would lead the reader far from the geometric intuition Also we will not use the results presented here later on we just think that these concepts are worth of mention Thus for the purpose of this section just think that each measure is absolutely continuous with sm
69. y Similarly p is subdifferentiable at 7 Choose v OT yy Yt vg E OT yr 72 and observe that Dele v1 exPs 2 o D x 1 Yela ve pilt v2 exp 0 D 2 4 which gives v v2 and the thesis 41 Corollary 2 24 The intermediate transport maps are locally Lipschitz Let m C Y2 M a constant speed geodesic in Y2 M W2 Then for every t 0 1 and s 0 1 there exists only one optimal transport plan from u to us this transport plan is induced by a map and this map is locally Lipschitz Note clearly in a compact setting being locally Lipschitz means being Lipschitz We wrote locally because this is the regularity of transport maps in the non compact situation Proof Fix t 0 1 and without loss of generality let s 1 The fact that the optimal plan from is unique and induced by a map is known by Proposition 2 16 Now let v be the vector field defined on supp f4z by u x Vy Vij we are using part iii of the above corollary with the same notation The fact that 7 is a c t concave potential for the couple u po tells that the optimal transport map T satisfies T x Ot by x for jz a e x Using Theorem 1 33 the fact that Y is differentiable in supp u and taking into account the scaling properties of the cost we get that T may be written as T x exp v x Since the exponential map is C the fact that T is Lipschitz will follow if we show
70. 0 0 2 loc In particular the functions t gt t and t VE belong to L Proposition 3 19 we know that for any s gt 0 it holds 0 00 Now we use sS 1 S 1 S E E e lt f VEs f Parti f VERE 6 0 0 0 Therefore t E x is locally absolutely continuous and it holds Le 1 f E x a t dr a VE a dr E z VWs gt 0 0 0 Subtracting from this last equation the same equality written for s t we get the thesis Remark 3 21 It is important to underline that the hypothesis of A geodesic convexity is in general of no help for what concerns the compactness of the sequence of discrete solutions The A geodesic convexity hypothesis ensures various regularity results for the limit curve which we state without proof Proposition 3 22 Let X E be satisfying Assumption 3 17 and let x be any limit of a sequence of discrete solutions Then 62 i the limit exists for every t gt 0 ii the equation d a VE 2 lit a7 VE ze is satisfied at every t gt 0 iii the map t gt e tE az is convex the map t gt e VE a is non increasing right continuous and satisfies t sIVEP 2 lt 7 wo E 20 tV E lt 1 2A t e72 Elzo inf E where E X R is defined as d x y Ei a inf Ely EH iv ifX gt 0 then E admits a unique minimum min and it holds gE ae Lmin lt E x E amp min
71. 13 where Dp Ao A1 is defined as SUP zo 40 d zo z1 if K lt 0 and as inf zo 4o d x0 1 if z z1 amp K gt 0 ae m it If X d m isa CD 0 N space it holds m Ao 414 gt 1 t Ao tm 41 7 14 Proof We start with i Suppose that Ag A are open satisfying m Ao m A1 gt 0 Define the measures 4 m A M i for 7 0 1 and find a constant speed geodesic u C A X such that Eoln lt 1 1 B2o Ho tll t0 W2 uo m Arguing as in the proof of the previous proposition it is immediate to see that u is concentrated on Ao A l for any t 0 1 In particular m Ao Ai gt 0 otherwise amp 4 would be 00 and the convexity inequality would fail Now let v m Ao 4il m an application of Jensen inequality shows Ao Aile that x Ht gt v thus we have K Eoo 11 lt 1 t Eo 0 t 41 FE t W3 uo m Notice that for a general u of the form m A my it holds Exo Ht log m A log m A and conclude using the trivial inequality inf d 29 21 lt W uo u1 lt sup d xo 21 zxoE Ao zoE Ao z E A1 z EA 117 The case of Ap Ay compact now follows by a simple approximation argument by considering the e neighborhood Af x d x Aj lt i 0 1 noticing that Ao Ai Neso AG Af e for any t 0 1 and that m A gt 0 because A C supp m i 0 1 Part ii follows along the
72. 2009 Later on the second author joined to the project and the initial set of notes has been enriched and made more detailed in particular in connection with the differentiable structure of the Wasserstein space the synthetic curvature bounds and their analytic implications Some of the results presented here have not yet appeared in a book form with the exception of 44 It is clear that this subject is expanding so quickly that it is impossible to give an account of all developments of the theory in a few hours or a few pages A more modest approach is to give a quick mention of the many aspects of the theory stimulating the reader s curiosity and leaving to more detailed treatises as 6 mostly focused on the theory of gradient flows and the monumental book 80 for a much broader overview on optimal transport In Chapter we introduce the optimal transport problem and its formulations in terms of trans port maps and transport plans Then we introduce basic tools of the theory namely the duality formula the c monotonicity and discuss the problem of existence of optimal maps in the model case cost distance In Chapter 2 we introduce the Wasserstein distance W3 on the set Y2 X of probability measures with finite quadratic moments and X is a generic Polish space This distance naturally arises when considering the optimal transport problem with quadratic cost The connections between geodesics in 2 X and geodesics in X and between
73. 5 808 Polar factorization and monotone rearrangement of vector valued functions Comm Pure Appl Math 44 1991 pp 375 417 D BURAGO Y BURAGO AND S IVANOV A course in metric geometry vol 33 of Graduate Studies in Mathematics American Mathematical Society Providence RI 2001 L A CAFFARELLI Boundary regularity of maps with convex potentials Comm Pure Appl Math 45 1992 pp 1141 1151 The regularity of mappings with a convex potential J Amer Math Soc 5 1992 pp 99 104 Boundary regularity of maps with convex potentials IT Ann of Math 2 144 1996 pp 453 496 L A CAFFARELLI M FELDMAN AND R J MCCANN Constructing optimal maps for Monge s transport problem as a limit of strictly convex costs J Amer Math Soc 15 2002 pp 1 26 electronic L CARAVENNA A proof of sudakov theorem with strictly convex norms Math Z to appear J A CARRILLO S LISINI G SAVARE AND D SLEPCEV Nonlinear mobility continuity equations and generalized displacement convexity J Funct Anal 258 2010 pp 1273 1309 T CHAMPION AND L DE PASCALE The Monge problem in R4 Duke Math J The Monge problem for strictly convex norms in R4 J Eur Math Soc JEMS 12 2010 pp 1355 1369 J CHEEGER Differentiability of Lipschitz functions on metric measure spaces Geom Funct Anal 9 1999 pp 428 517 D CORDERO ERAUSQUIN B NAZARET AND C VILLANI
74. 9 gt 0 sufficiently small it holds 9 v lt r 3 for any v O x and any x M Also since vy is bounded possibly decreasing the value of o we can assume that r2 lt eleal lt 2 Fix zo M v Ot y ao and let yo exp eov We claim that for chosen as above the maximum of coy d yo 2 cannot lie outside B xo Indeed if d x xo gt r we have d x yo gt 2r 3 and thus d x Yo ee 2r ie r g xo yo lt rere coe 2 R 9 12 1g Corto 2 Thus the maximum must lie in B o Recall that in this ball the function d yo is C and satisfies V d yo 2 gt cld thus it holds Vv cov z aa lt Eo i c Id where R is such that V2 lt AId on the whole of M Thus decreasing if necessary the value of Eq we can assume that d Vv lt o Tit lt 0 on B zo which implies that eoy d yo 2 admits a unique point x B o such that 0 p d yo 2 x which therefore is the unique maximum Since V 4d yo 0 Eov OT eop xo we conclude that xo is the unique global maximum as claimed 20 Now define the function Y M RU co by ap Py by inf Z cov z if y exp cov for some x M v OT y z and y y oo otherwise By definition we have my d x cople lt gt Wy Va ye M and the claim proved ensures that if yo exp ovo for ro E
75. A S SOLIMINI AND J M MOREL A variational model of irrigation pat terns Interfaces Free Bound 5 2003 pp 391 415 R J MCCANN A convexity theory for interacting gases and equilibrium crystals ProQuest LLC Ann Arbor MI 1994 Thesis Ph D Princeton University R J MCCANN A convexity principle for interacting gases Adv Math 128 1997 pp 153 179 Polar factorization of maps on riemannian manifolds Geometric and Functional Anal ysis 11 2001 pp 589 608 V D MILMAN AND G SCHECHTMAN Asymptotic theory of finite dimensional normed spaces vol 1200 of Lecture Notes in Mathematics Springer Verlag Berlin 1986 With an appendix by M Gromov G MONGE M moire sur la th orie des d eblais et des remblais Histoire de IOAcad mie Royale des Sciences de Paris 1781 pp 666 704 F OTTO The geometry of dissipative evolution equations the porous medium equation Comm Partial Differential Equations 26 2001 pp 101 174 A PRATELLI On the equality between Monge s infimum and Kantorovich s minimum in opti mal mass transportation Annales de l Institut Henri Poincare B Probability and Statistics 43 2007 pp 1 13 S T RACHEV AND L RUSCHENDORF Mass transportation problems Vol I Probability and its Applications Springer Verlag New York 1998 Theory R T ROCKAFELLAR Convex Analysis Princeton University Press Princeton 1970 L RUSCHENDORF AND S T RACHEV A character
76. A mass transportation approach to sharp Sobolev and Gagliardo Nirenberg inequalities Adv Math 182 2004 pp 307 332 C DELLACHERIE AND P A MEYER Probabilities and potential vol 29 of North Holland Mathematics Studies North Holland Publishing Co Amsterdam 1978 Q DENG AND K T STURM Localization and tensorization properties of the curvature dimension condition for metric measure spaces ii Submitted 2010 J DOLBEAULT B NAZARET AND G SAVARE On the Bakry Emery criterion for linear diffusions and weighted porous media equations Comm Math Sci 6 2008 pp 477 494 L C EVANS AND W GANGBO Differential equations methods for the Monge Kantorovich mass transfer problem Mem Amer Math Soc 137 1999 pp viii 66 A FATHI AND A FIGALLI Optimal transportation on non compact manifolds Israel J Math 175 2010 pp 1 59 D FEYEL AND A S USTUNEL Monge Kantorovitch measure transportation and Monge Amp re equation on Wiener space Probab Theory Related Fields 128 2004 pp 347 385 A FIGALLI AND N GIGLI A new transportation distance between non negative measures with applications to gradients flows with Dirichlet boundary conditions J Math Pures Appl 9 94 2010 pp 107 130 125 A FIGALLI F MAGGI AND A PRATELLI A mass transportation approach to quantitative isoperimetric inequalities Invent Math 182 2010 pp 167 211 A FIGALLI AND L RIFFORD Mass transportation on sub
77. Ay pR Ag V e 5Rie Ve Ve dm Using the hypothesis on M and the fact that Ay lt N V2 we get p EN o gt 0 i e the geodesic convexity of y For the converse implication it is possible to argue as above we omit the details also in this case Now we pass to the stability Theorem 7 12 Stability of weak Ricci curvature bound Assume that Xn dn Mn a X d m and that for every n N the space Xn dn Mn is CD K 00 resp CD 0 N Then X d m is a CD K 00 resp CD 0 N space as well Sketch of the Proof Pick pio 41 E Y X and assume they are both absolutely continuous with bounded densities say u p m i 0 1 Choose dp Y E Opt dn Mn d m Define u Yn yui E PI Xn i 0 1 Then by assumption there is a geodesic u C A Xn such that K EHE S 1 t Eo uo tE lut yt Wa Ho HT 7 12 Now let of Yn gut E YF X t 0 1 From Proposition 7 5 and its proof we know that W ui o gt Oas n gt i 0 1 Also from 7 12 ad Lemma 7 4 we know that 63 07 is uniformly bounded in n t Thus for every fixed t the sequence n gt o is tight and we can extract a subsequence not relabeled such that o narrowly converges to some o E 9 supp m for every rational t By an equicontinuity argument it is not hard to see that then o narrowly converges to some gz for any t 0 1 we omit the details We claim t
78. B let f u be the density of exp 4 w r t Vol at the point exp u Then the function f has the following Taylor expansion 1 f u 1 5 Rie u u o ul 7 1 It is said that the Ricci curvature is bounded below by R provided Ric u u gt Alul for every x E M and u T M Several important geometric and analytic inequalities are related to bounds from below on Ricci curvature we mention just two of them e Brunn Minkowski Suppose that M has non negative Ricci curvature and for any Ao Ay C M compact let Ay fx yis a constant speed geodesic s t y Ag 71 A vt 0 1 Then it holds Vol A gt 1 Vol Ao t Vol Ax VEE 0 1 0D where n is the dimension of M 107 e Bishop Gromov Suppose that M has Ricci curvature bounded from below by n 1 k where n is the dimension of M and k a real number Let M be the simply connected n dimensional space with constant curvature having Ricci curvature equal to n 1 k so that Misa sphere if k gt 0 a Euclidean space if k 0 and an hyperbolic space if k lt 0 Then for every x M and M the map Vol B 2 0 CO r gt a Val B 7 3 is non increasing where Vol and Vol are the volume measures on M M respectively A natural question is whether it is possible to formulate the notion of Ricci bound from below also for metric spaces analogously to the definition of Alexandrov spaces
79. B a where F is an arbitrary open set P E its perimeter and B the unitary ball We will prove this inequality via Brenier s theorem 1 26 neglecting all the smoothness issues Let i 1 Yi 2 H ZE e Zip B and T E B be the optimal transport map w r t the cost given by the distance squared The change of variable formula gives det VT x Vrz E E de LE gap Since we know that T is the gradient of a convex function we have that VT x is a symmetric matrix with non negative eigenvalues for every x E Hence the arithmetic geometric mean inequality ensures that T det VT x lt a Va E Coupling the last two equations we get 1 V T 1 2 Va LE T T ZI E a d ZI B a Integrating over E and applying the divergence theorem we get t 1 1 i Z E a lt aT T x dz a T x v dH 1 where v OE R is the outer unit normal vector Since T x B for every x E we have T x lt 1 for x E and thus T x v x lt 1 We conclude with 1 d 1 P E SS Be ae MENT 2 8 aap TOVO He lt a 4 3 Sobolev Inequality apy re The Sobolev inequality in R reads as fin lt oa vir Yf W1P R4 where 1 lt p lt d p re and C d p is a constant which depends only on the dimension d and the exponent p We will prove it via a method which closely resemble the one just used for the isoperimetric inequality Again we will negl
80. Given these analogies we are going to proceed as follows first we give a proof of the existence of the parallel transport along a smooth curve in an embedded Riemannian manifold then we will see how this proof can be adapted to the Wasserstein case this approach should help highlighting what s the geometric idea behind the construction Thus say that M is a given smooth Riemannian manifold embedded on RP t gt y Ma smooth curve on 0 1 and u T M is a given tangent vector Our goal is to prove the existence of an absolutely continuous vector field t uz Ty M such that uo u and d Py u 0 a e t For any t s 0 1 let tr T RP T R be the natural translation map which takes a vector with base point y tangent or not to the manifold and gives back the translated of this vector with base point ys Notice that an effect of the curvature of the manifold and the chosen embedding on R is that tr u may be not tangent to M even if u is Now define P T R gt T M by P u P tra Vu T R 94 An immediate consequence of the smoothness of M and y are the two inequalities tr u P u lt Clul s tl Vt s 0 1 and u T4 M 6 12a P3 u lt Clul s tl Vt s 0 1 and u TM 6 12b where TM is the orthogonal complement of T M in Ty RP These two inequalities are all we need to prove existence of the parallel transport The proof will be con
81. Lipschitz can be proved in the Euclidean case via the theory of monotone op erators Indeed if G R R is a possibly multivalued monotone map i e satisfies yi Y2 1 22 gt O for every z1 x2 R y G a i 1 2 then the operator 42 1 t Id tG is single valued Lipschitz with Lipschitz constant bounded above by 1 1 t To prove this pick z1 2 R y E G x1 y2 G x2 and observe that 1 t ay ea a ty 1 t a tyal 1 4 lay z2 tly yol 2t 1 t z1 22 91 yo gt 1 t lz a2 which is our claim Now pick uo p1 P2 R an optimal plan y Opt uo 41 and consider the geodesic t gt ht 1 t nt tr yy recall Remark 2 13 From Theorem 1 26 we know that there exists a convex function such that supp y C 07 y Also we know that the unique optimal plan from juo to u is given by the formula nt 1 t r tr 4 which is therefore supported in the graph of 1 t Id t07 y Since the subdifferential of a convex function is a monotone operator the thesis follows from the previous claim Considering the case in which u is a delta and jig is not we can easily see that the bound 1 t on the Lipschitz constant of the optimal transport map from p to uo is sharp An important consequence of Corollary 2 24 is the following proposition Proposition 2 26 Geodesic convexity of the set of absolu
82. M vo Ot y xo the inf in the definition of w yo is realized at x zo and thus d xo Yo 5 Wyo EoP Xo Hence coy w and therefore is c concave Along the same lines one can easily see that for y exp 00 p x it holds P x y c sopte TED _ egyye o i e y O eop xo Thus we have 0 e9y D exp O evy Since the other inclusion has been proved in Proposition 1 30 the proof is finished Remark 1 35 With the same notation of Theorem 1 33 recall that we know that the c concave func tion y whose c superdifferential contains the graph of any optimal plan from yp to v is differentiable p a e for regular u Fix xo such that Vy a9 exists let yo exp Vp o E O p w and observe that from d x yo _ d xo yo 2 2 we deduce that Vy xo belongs to the subdifferential of d yo 2 at o Since we know that d yo 2 always have non empty superdifferential we deduce that it must be differentiable at xo In particular there exists only one geodesic connecting xo to yo Therefore if u is regular not only there exists a unique optimal transport map T but also for u a e x there is only one geodesic connecting x to T x a gt p x x0 The question of regularity of optimal maps on manifolds is much more delicate than the cor responding question on R4 even if one wants to get only the continuity We won t enter into the details of the the
83. N space we followed Sturm and asked only the functionals pm gt N f p p N dm 122 N gt N to be geodesically convex Lott and Villani asked for something more restrictive namely they introduced the displacement convexity classes DC n as the set of functions u 0 co gt R continuous convex and such that Zz gt ZNu z is convex Notice that u z N z z N belongs to DCy Then they say that a space is C D 0 N provided pm fuld with the usual modifications for a measure which is not absolutely continuous is geodesically con vex for any u DCy This notion is still compatible with the Riemannian case and stable un der convergence The main advantage one has in working with this definition is the fact that for a CD 0 N space in this sense for any couple of absolutely continuous measures there exists a geodesic connecting them which is made of absolutely continuous measures The distance D that we used to define the notion of convergence of metric measure spaces has been defined and studied by Sturm in 74 This is not the only possible notion of convergence of metric measure spaces Lott and Villani used a different one see 58 or Chapter 27 of 80 A good property of the distance D is that it pleasantly reminds the Wasserstein distance W3 to some extent the relation of D to W2 is the same relation that there is between Gromov Hausdorff distance and Hausdorff distance between compact subs
84. R and if T is invertible it also holds Pu woT u lt lwll Lip T Id Vw Tan P2 R 96 Proof We start with the first inequality which is equivalent to IVeoT P VeoT l lt Vell Lip T Id Ye CX R 6 19 Let us suppose first that T Id C9 R In this case the map y o T is in C9 R too and therefore V y o T VT Vy o T belongs to Tan 2 R From the minimality properties of the projection we get VeoT P VeoT ly lt VeoT VT Vy oT JIE VTO VATE Pana i 1 2 lt Ve T x 7 Vad Doydu lt Vell Lip T Id where I is the identity matrix and V Id T x op is the operator norm of the linear functional from R to R given by v V Id T v Now turn to the general case and we can certainly assume that T is Lipschitz Then it is not hard to see that there exists a sequence Tn Id C CX R such that T gt T uniformly on compact sets and lim Lip T Id lt Lip T Id It is clear that for such a sequence it holds IT Ta 0 and we have VeoT P VeoT lu lt IVeoT V p Tn lt Ve oT Ve 9 Tall T Ve oT Viv Q Tr lla lt Lip Ve IT Talla Veo Tall Lip Tn Id Letting n 00 we get the thesis For the second inequality just notice that Py woT sup woT v sup w voT a e or eee vilp l vilp t sup w voT P voT lt lwllyLip T Id v Tan Po R4 l
85. R BY g dvol gt and let T resp T be the optimal transport map from fvol to gvol resp from gvol to fvol We claim that either T or T is discontinuous and argue by contradiction Suppose that both are continuous and observe that by the symmetry of the optimal transport problem it must hold T x T x for any x M Again by the symmetry of M f g the point T O must be invariant under the symmetries around the x and y axes Thus it is either T O O or T O O and similarly T O O O We claim that it must hold T O O Indeed otherwise either T O O and T O O or T O O and T O O In the first case the two couples O O and O O belong to the support of the optimal plan and thus by cyclical monotonicity it holds d2 O 0 d2 0 0 lt 0 0 2 0 0 0 which is absurdum In the second case we have T x 4 O for all x M which by continuity and compactness implies d T M O gt 0 This contradicts the fact that f is positive everywhere and T4 gvol fvol Thus it holds T O O Now observe that by construction there must be some mass transfer from B A U B A to B B U B B i e we can find x B A U B A and y Be B U B B such that x y is in the support of the optimal plan Since O O is the support of the optimal plan as well by cyclical monotonicity it must hold d x y d O O lt d x 0 d O y w
86. T gn Id 8 4H The fact that Y has linear growth implies that v Tuy has compact support Thus in particular J z dv x lt oo The contradiction comes from the fact that y Adm u v is c cyclically mono tone because of Proposition 1 21 and thus optimal However it is not induced by a map because T S on a set of positive u measure Lemma 1 20 16 The question of regularity of the optimal map is very delicate In general it is only of bounded variation BV in short since monotone maps always have this regularity property and disconti nuities can occur just think to the case in which the support of the starting measure is connected while the one of the arrival measure is not It turns out that connectedness is not sufficient to prevent discontinuities and that if we want some regularity we have to impose a convexity restriction on supp v The following result holds Theorem 1 27 Regularity theorem Assume Q1 Q2 C IR are two bounded and connected open sets p PL Io v neo with O lt c lt p n lt C for some c C R Assume also that Q2 is convex Then the optimal transport map T belongs to C Q1 for some a lt 1 In addition the following implication holds peC Q1 n E C N2 T C Q1 The convexity assumption on Q is needed to show that the convex function y whose gradient provides the optimal map T is a viscosity solution of the Monge Ampere equation p p Ve
87. T from n to v i e all maps T such that Typ v Regardless of the choice of the cost function c Monge s problem can be ill posed because e no admissible T exists for instance if u is a Dirac delta and v is not e the constraint T y u v is not weakly sequentially closed w r t any reasonable weak topology As an example of the second phenomenon one can consider the sequence f x f nz where f R R is 1 periodic and equal to 1 on 0 1 2 and to 1 on 1 2 1 and the measures p Llio 1 and v 6_ 1 2 It is immediate to check that fn u v for every n N and yet fn weakly converges to the null function f 0 which satisfies fu o v A way to overcome these difficulties is due to Kantorovich who proposed the following way to relax the problem Problem 1 2 Kantorovich s formulation of optimal transportation We minimize yo f ce wdr ey XxY in the set Adm u v of all transport plans y A X x Y from u to v i e the set of Borel Probability measures on X x Y such that y AxY p A YAE A X Xx B 0 B VBE Z Y Equivalently THY p my v where r n are the natural projections from X x Y onto X and Y respectively a Transport plans can be thought of as multivalued transport maps y f Y du with y P x x Y Another way to look at transport plans is to observe that for y Adm u v the value of y A x B is the amount of mass initially in A which is sent int
88. To see this pass to the limit in 6 15 with s to and sy t U Ut to get IP uto un lt C fetigl ta to lt C Iuil ta to 6 18 so that from 6 12a we get tris tto un lt tri uto Pad uto Ped uto un lt Clullti tol Cts tol which shows the absolute continuity Finally due to 6 17 it is sufficient to check that the covariant derivative vanishes at 0 To see this put t 0 and t t in 6 18 to get Pi u u lt C lult so that the thesis follows from 6 13 Now we come back to the Wasserstein case To follow the analogy with the Riemannian case keep in mind that the analogous of the translation map tr is the right composition with T s t and the analogous of the map P is P u Pus u S T s t which maps L onto Tan 72 R We saw that the key to prove the existence of the parallel transport in the embedded Riemannian case are inequalities 6 12 Thus given that we want to im itate the approach in the Wasserstein setting we need to produce an analogous of those inequalities This is the content of the following lemma We will denote by Tant P2 R the orthogonal complement of Tan 72 R in L2 Lemma 6 14 Control of the angles between tangent spaces Let p v E Y2 R and T R gt R be any Borel map satisfying Tau v Then it holds lvoT P voT lt lv Lip T Id Ww Tan 22
89. acement interpolation was introduced by McCann 63 for probability mea sures in R Theorem 2 10 appears in this form here for the first time in 58 the theorem was proved in the compact case in 80 Theorem 7 21 this has been extended to locally compact structures and much more general forms of interpolation The main source of difficulty when dealing with general Polish structure is the potential lack of tightness the proof presented here is strongly inspired by the work of S Lisini 54 Proposition 2 16 and Theorem 2 18 come from 80 Corollary 7 32 and Theorem 7 36 respec tively Theorem 2 20 and the counterexample 2 21 are taken from 6 Theorem 7 3 2 and Example 7 3 3 respectively The proof of Corollary 2 24 is taken from an argument by A Fathi 35 the paper being inspired by Bernand Buffoni 13 Remark 2 27 is due to N Juillet 48 The idea of looking at the transport problem as dynamical problem involving the continuity equa tion is due to J D Benamou and Y Brenier 12 while the fact that A2 IR7 W2 can be viewed as a sort of infinite dimensional Riemannian manifold is an intuition by F Otto 67 Theorem 2 29 has been proven in 6 where also Propositions 2 32 2 33 and 2 34 were proven in the case M RI the generalization to Riemannian manifolds comes from Nash s embedding theorem 3 Gradient flows The aim of this Chapter is twofold on one hand we give an overview of the theory of Gradient Flows in a
90. al Differential Equations 20 2004 pp 283 299 83 L ZAJ CEK On the differentiability of convex functions in finite and infinite dimensional spaces Czechoslovak Math J 29 1979 pp 340 348 128
91. amily of discrete solutions starting from it Then x7 converge locally uniformly to a limit curve x as T 0 so that the limit curve is unique Furthermore x is the unique solution of the system of differential inequalities ld zg oY FE Gry BG lt By ae t 20 Wye X 3 30 among all locally absolutely continuous curves 4 converging to Tast 0 Le x isa Gradient Flow in the EVI formulation see Definition 3 5 e LetT J D E and x y be the two Gradient Flows in the EVI formulation Then there is A exponential contraction of the distance i e d a4 yz lt eT 9 3 31 e Suppose that gt 0 that T D E and build x7 x as above Then the following a priori error estimate holds sup d z 27 lt 8 7 E Z E z 3 32 Sketch of the Proof We will make the following simplifying assumptions E gt 0 A gt 0 and Tz D E Also we will prove just that the sequence of discrete solutions n ay A converges to a limit curve as n oo for any given T gt 0 Existence and uniqueness of the discrete solution Pick x X We have to prove that there exists a unique minimizer of 3 12 Let I gt 0 be the infimum of 3 12 Let n be a minimizing 64 sequence for 3 12 fix n m N and let y 0 1 X bea curve satisfying 3 29 for o n T m and y x Using the inequalities 3 29 at t 1 2 we get d 1 2 2 I lt E SERSA lt E y12 37
92. ance that z 2 Since rt mT yo is an optimal plan by the cyclical monotonicity of its support we know that d x z d x z lt d a 2 d x z lt d x y d y zN d x y d y z 1 to d a z tod 2 z 1 to d a z tod z z which after some manipulation gives d x z d a z D Again from the cyclical mono tonicity of the support we have 2D lt d z z d a z thus either d x z or d x z is gt than D Say d x z gt D so that it holds D lt d x z lt d x y d y z 1 or to D toD D which means that the triple of points x y z lies along a geodesic Since x y z lies on a geodesic as well by the non branching hypothesis we get a contradiction 35 Thus the map supp gt x y z gt y is injective This means that there exists two maps f g X X such that x y z supp q if and only if x f y and z g y This is the same as to say that y is induced by f and 4 is induced by g To summarize we proved that given t 0 1 every optimal plan y Opt jio fz is induced by a map from ut Now we claim that the optimal plan is actually unique Indeed if there are two of them induced by two different maps say f and f then the plan S O Id PLE Hang would be optimal and not induced by a map It remains to prove that 2 X is non branching Choose y Y2 Geod X such that 2 7
93. and a finite open cover 9 such that Q d m Q4 mI is a CD K N space for every i Can we deduce that X d m is a CD K N space as well One would like the answer to be affirmative as any notion of curvature should be local For K 0 or N this is actually the case at least under some technical assumptions The general case is still open and up to now we only know that the conjecture 30 34 in 80 is false being disproved by Deng and Sturm in 32 see also 11 The second and final thing we want to mention is the case of Finsler manifolds which are differentiable manifolds endowed with a norm possibly not coming from an inner product on each tangent space which varies smoothly with the base point A simple example of Finsler manifolds is the space R4 where is any norm It turns out that for any choice of the norm the space R4 2 is a CD 0 N space Various experts have different opinion about this fact namely there is no agreement on the community concerning whether one really wants or not Finsler geometries to be included in the class of spaces with Ricci curvature bounded below In any case 123 it is interesting to know whether there exists a different more restrictive notion of Ricci curvature bound which rules out the Finsler case Progresses in this direction have been made in 8 where the notion of spaces with Riemannian Ricci bounded below is introduced shortly said the
94. and only if for y a e x y it holds x y 0 p hence by the continuity of c we conclude supp 7 C 8 y a 1 3 The dual problem The transport problem in the Kantorovich formulation is the problem of minimizing the linear func tional y f cdy with the affine constraints T y j THY vy and y gt 0 It is well known that problems of this kind admit a natural dual problem where we maximize a linear functional with affine constraints In our case the dual problem is Problem 1 16 Dual problem Let u P X v P Y Maximize the value of eaux vavo among all functions p L pu Y L v such that plz Yly Se z y YreX ye Y 1 5 11 The relation between the transport problem and the dual one consists in the fact that eif f dle sup feoda voat where the supremum is taken among all y w as in the definition of the problem Although the fact that equality holds is an easy consequence of Theorem 1 13 of the previous section taking Y y as we will see we prefer to start with an heuristic argument which shows why duality works The calculations we are going to do are very common in linear programming and are based on the min max principle Observe how the constraint y Adm pu v becomes the functional to maximize in the dual problem and the functional to minimize f cdy becomes the constraint in the dual problem Start observing that inf ole wer YE
95. applications 5 1 Branched optimal transportation Consider the transport problem with u 6 and v 4 Sy y for the cost given by the distance squared on R Then Theorem 2 10 and Remark 2 13 tell that the unique geodesic u connecting u to v is given by 1 Ht 5 Su peren af Sa eteys gt so that the geodesic produces a V shaped path For some applications this is unnatural for instance in real life networks when one wants to transport the good located in x to the destinations y and y it is preferred to produce a branched structure where first the good it is transported on a single truck to some intermediate point and only later split into two parts which are delivered to the 2 destinations This produces a Y shaped path If we want to model the fact that it is convenient to ship things together we are lead to the following construction due to Gilbert Say that the starting distribution of mass is given by u J idx and that the final one is v gt bj y with D7 a D7 bj 1 An admissible dynamical transfer is then given by a finite oriented weighted graph G where the weight is a function w set of edges of G gt R satisfying the Kirchoff s rule 5 w e 5 w e ai Vi edges e outgoing from x edges e incoming in gt w e gt wle bj VJ edges e outgoing from yj edges e incoming in yj gt w e gt w e 0 for any internal node z of G edg
96. archives ouvertes A user s guide to optimal transport Luigi Ambrosio Nicola Gigli gt To cite this version Luigi Ambrosio Nicola Gigli A user s guide to optimal transport CIME summer school 2009 Italy lt hal 00769391 gt HAL Id hal 00769391 https hal archives ouvertes fr hal 00769391 Submitted on 31 Dec 2012 HAL is a multi disciplinary open access archive for the deposit and dissemination of sci entific research documents whether they are pub lished or not The documents may come from teaching and research institutions in France or abroad or from public or private research centers L archive ouverte pluridisciplinaire HAL est destin e au d p t et a la diffusion de documents scientifiques de niveau recherche publi s ou non manant des tablissements d enseignement et de recherche fran ais ou trangers des laboratoires publics ou priv s A user s guide to optimal transport Luigi Ambrosio Nicola Gigli Abstract This text is an expanded version of the lectures given by the first author in the 2009 CIME summer school of Cetraro It provides a quick and reasonably account of the classical theory of optimal mass transportation and of its more recent developments including the metric theory of gradient flows geometric and functional inequalities related to optimal transportation the first and second order differential calculus in the Wasserstein space and the synthetic theory of metr
97. asures pu v such that js is concentrated on the negative numbers and v on the positive ones Then one can see that any admissible plan between them is optimal for the cost c x y x yl Still even in this case there is existence of optimal maps but in order to find them one has to use a sort of selection principle A successful strategy which has later been applied to a number of different situation has been proposed by V N Sudakov in 77 who used a disintegration principle to reduce the d dimensional problem to a problem on R The original argument by V N Sudakov was flawed and has been fixed by the first author in 4 in the case of the Euclidean distance Meanwhile different proofs of existence of optimal maps have been proposed by L C Evans W Gangbo 34 Trudinger and Wang 78 and L Caffarelli M Feldman and R McCann 24 Later existence of optimal maps for the case c x y a yll being any norm has been established at increasing levels of generality in 9 28 27 containing the most general result for any norm and 25 2 The Wasserstein distance W The aim of this chapter is to describe the properties of the Wasserstein distance W3 on the space of Borel Probability measures on a given metric space X d This amounts to study the transport problem with cost function c x y d x y An important characteristic of the Wasserstein distance is that it inherits many interesting geo me
98. at the entropy has a global minimum in M Q such minimum is given by the measure with constant density e i e the measure whose density is everywhere equal to the minimum of z gt zlog z On the bad side the entropy is not geodesically convex in M2 Q Wb2 and this implies that it is not clear whether the strong properties of Gradient Flows w r t W 2 as described in Section 3 3 Theorem 3 35 and Proposition 3 38 are satisfied also in this setting In particular it is not clear whether there is contractivity of the distance or not Open Problem 5 7 Let p p two solutions of the Heat equation with Dirichlet boundary condition p e in OO for every t gt 0 i 1 2 Prove or disprove that Wo p5 p3 lt Wha pi pi Yt gt s The question is open also for convex and smooth open sets Q 5 4 Bibliographical notes The connection of branched transport and transport problem as discussed in Section 5 1 was first pointed out by Q Xia in 81 An equivalent model was proposed by F Maddalena J M Morel and S Solimini in 61 In 81 60 and 15 the existence of an optimal branched transport Theorem 5 2 was also provided Later this result has been extended in several directions see for instance the works A Brancolini G Buttazzo and F Santambrogio 16 and Bianchini Brancolini 15 The interior regularity result Theorem 5 3 has been proved By Q Xia in 82 and M Bernot V Caselles and J M Morel in 14 Also
99. ate to verify that E E satisfy all the Assumptions 3 8 3 13 the choice of f guarantees that the slopes of E E are continuous Now build a Gradient Flow starting from 0 with some work it is possible to check that the Minimizing Movement scheme converges in both cases to absolutely continuous curves x and respectively satisfying x VE z a e t VE ae t Now notice that VE 2 VE x f a for every x 0 1 therefore the fact that f gt 1 is smooth on 0 1 C gives that each of these two equations admit a unique solution Therefore this is the key point of the example x and must coincide In other words the effect of the function g is not seen at the level of Gradient Flow It is then immediate to verify that there is Energy Dissipation Equality for the energy E but there is only the Energy Dissipation Inequality for the energy E 3 2 3 The geodesically convex case EDE and regularizing effects Here we study gradient flows of so called geodesically convex functionals which are the natural metric generalization of convex functionals on linear spaces Definition 3 16 Geodesic convexity Let E X RU 00 be a functional and A R We say that E is geodesically convex provided for every x y E X there exists a constant speed geodesic y 0 1 X connecting x to y such that 2 E lt 1 t E x tE y 5t t d x y 3 20 In this section we will as
100. ational Inequality EVD Thus we got three different characterizations of Gradient Flows in Hilbert spaces the EDI the EDE and the EVI We now want to show that it is possible to formulate these equations also for functionals E defined on a metric space X d The object 2 appearing in EDI and EDE can be naturally interpreted as the metric speed of the absolutely continuous curve x as defined in 2 19 The metric analogous of V E x is the slope of E defined as Definition 3 2 Slope Let E X RU 00 and x X be such that E x lt Then the slope VE x of E at x is z E x E y E x E wete i r en a e y gt s d x y gt s d a y The three definitions of Gradient Flows in a metric setting that we are going to use are Definition 3 3 Energy Dissipation Inequality definition of GF EDI Let E X gt RU 00 and let X be such that E lt oo We say that 0 00 3 t a X is a Gradient Flow in the EDI sense starting at T provided it is a locally absolutely continuous curve xo T and 1 f 1 f8 E as F lt dr a VE 2 dr lt E T Ys gt 0 0 0 8 s 3 8 1 1 E x T a dr F VE 2 dr lt E x a e t gt 0 Vs gt t t t Definition 3 4 Energy Dissipation Equality definition of GF EDE Let E X RU 00 and let amp X be such that E T lt oo We say that 0 00 3 t gt a X is a Gradient Flow in the EDE sense starting at T provided it is a
101. ble formula we get that v lt and for its density s it holds i AT VLE the assumption gt 0 is necessary to have the last inequality in 3 46 If A lt 0 A convexity of V along interpolating curves is not anymore true so that we cannot apply directly the results of Subsection 3 2 4 Yet adapting the arguments it possible to show that all the results which we will present hereafter are true for general A R 72 where we wrote T for 1 t To tT Thus en f uatacty fu seers Oaa Therefore the proof will be complete if we show that A u 5 det A is convex on the set of positively defined symmetric matrices for any x supp j1 Observe that this map is the composition of the convex and non increasing map z gt 2 u p 2 z with the map A det A Thus to conclude it is sufficient to show that A det A 4 is concave To this aim pick two symmetric and positive definite matrices Ap and Aj notice that det 1 t Ap tAy det Ao det Id tB where B Ao A1 Ao VAo and conclude by det Id tB det td tB tr B Id tB det Id tB 4 aw B Id tB T E Id tB lt 0 where in the last step we used the inequality tr C lt dtr C for C B Id tB 7 Important examples of functions u satisfying 3 44 and 3 45 are z Zz 1 Sofie WO a Oe ees 3 47 u x zlog z Remark 3 34
102. bsolutely continuous curve y it holds d gE VEH Vi gt VE y y if and only if y is a positive multiple of VE y 3 4 1 1 gt VE v tl if and only if Ivi VE Thus in particular equation 3 3 may be written in the following integral form 1 S 1 S E x gt r dr 7 VE zr dr lt E x aet lt s 3 5 t t which we call Energy Dissipation Inequality EDI in the following Since the inequality 3 4 shows that 2E y lt 4 V E y 4ly never holds the system 3 2 may be also written in form of Energy Dissipation Equality EDE in the following as 1 f 1 f E x 5J ar dr 5J VE dr E x YO lt t lt s 3 6 t t 51 Notice that the convexity of E does not play any role in this formulation A completely different way to rewrite 3 2 comes from observing that if x solves 3 2 and y H isa generic point it holds d lzi yl ae y x4 y t VE 22 lt Ely E x Ize yl 1 2 dt 2 where in the last inequality we used the fact that E is A convex Since the inequality 2 y 2 0 lt Ely E e Sle yP Wye H characterizes the elements v of the subdifferential of at x we have that an absolutely continuous curve x solves 3 2 if and only if ld 2 dt 1 la y 5 Nee y Elz lt Ely a e t gt 0 3 7 holds for every y H We will call this system of inequalities the Evolution Vari
103. by a map then the space of directions coincides with the space of directions induced by maps 6 3 Second order calculus Now we pass to the description of the second order analysis over IR The concepts that now enter into play are Covariant Derivative Parallel Transport and Curvature To some extent the situation is similar to the one we discussed in Subsection 2 3 2 concerning the first order structure the metric space 2 IR W2 is not a Riemannian manifold but if we are careful in giving definitions and in the regularity requirements of the objects involved we will be able to perform calculations very similar to those valid in a genuine Riemannian context Again we are restricting the analysis to the Euclidean case only for simplicity all of what comes next can be generalized to the analysis over A2 M for a generic Riemannian manifold M On a typical course of basic Riemannian geometry one of the first concepts introduced is that of Levi Civita connection which identifies the only natural natural here means compatible with the Riemannian structure way of differentiating vector fields on the manifold It would therefore be natural to set up our discussion on the second order analysis on 2 IR by giving the definition of Levi Civita connection in this setting However this cannot be done The reason is that we don t have a notion of smoothness for vector fields therefore not only we don t know how
104. e Riemannian submersion Pf from BM to 2 R is the push forward map Pf BM gt A R T gt Tx 85 We claim that Pf is a Riemannian submersion and that the fiber Pf p is isometric to the manifold Arn p We start considering the fibers Fix p Po R7 Observe that Pf p r EBM Typ p and that the tangent space Tanr Pf p is the set of vector fields u such that 4 ght tu zp 0 so that from Id tuoT 4 Typ d T tu 4p Id tuoT up V uoT p d dtlt 0 Fle o d dt lt 0 we have TanrPf p vector fields u on R such that V uo T p o and the scalar product between two vector fields in Tany Pf mA p is the one inherited by the one in BM i e is the scalar product in L p Now choose a distinguished map T Pf p and notice that the right composition with T provides a natural bijective map from Arn p into Pf because Sgp p SoT up p We claim that this right composition also provides an isometry between the Riemannian manifolds Arn p and Pf p indeed if v TangArn p then the perturbed maps S tv are sent to S o T tv o T which means that the perturbation v of S is sent to the perturbation u v o T of S o T by the differential of the right composition The conclusion follows from the change of variable formula which gives lePde f lupan Clearly the kernel of the differential dPf of Pf at T is given by TanrPf
105. e cost distance squared is the celebrated result of Y Brenier who also observed that it implies the polar factorization result 1 28 18 19 Brenier s ideas have been generalized in many directions One of the most notable one is R Mc Cann s theorem 1 33 concerning optimal maps in Riemannian manifolds for the case cost squared distance 64 R McCann also noticed that the original hypothesis in Brenier s theorem which was u amp Lt can be relaxed into pu gives 0 mass to Lipschitz hypersurfaces In 42 W Gangbo and R McCann pointed out that to get existence of optimal maps in R with c x y x y 2 it is sufficient to ask to the measure u to be regular in the sense of the Definition 1 25 The sharp version of Brenier s and McCann s theorems presented here where the necessity of the regularity of u is also proved comes from a paper of the second author of these notes 46 Other extensions of Brenier s result are e Infinite dimensional Hilbert spaces the authors and Savar 6 e cost functions induced by Lagrangians Bernard Buffoni 13 namely 1 dary int f E e 0 4 0 dts 0 2 901 0 e Carnot groups and sub Riemannian manifolds c dc 2 the first author and S Rigot 10 A Figalli and L Rifford 39 e cost functions induced by sub Riemannian Lagrangians A Agrachev and P Lee 1 e Wiener spaces E H y D Feyel A S st nel 36 Here F is a Banach space y Z
106. e isoperimetric inequality via a change of variable argument is due to Gro mov 65 in Gromov s proof it is not used the optimal transport map but the so called Knothe s map Such a map has the property that its gradient has non negative eigenvalues at every point and the reader can easily check that this is all we used of Brenier s map in our proof so that the argument of Gromov is the same we used here The use of Brenier s map instead of Knothe s one makes the difference when studying the quantitative version of the isoperimetric problem Figalli Maggi and Pratelli in 38 using tools coming from optimal transport proved the sharp quantitative isoperi metric inequality in R endowed with any norm the sharp quantitative isoperimetric inequality for the Euclidean norm was proved earlier by Fusco Maggi and Pratelli in 40 by completely different means The approach used here to prove the Sobolev inequality has been generalized by Cordero Erasquin Nazaret and Villani in 30 to provide a new proof of the sharp Gagliardo Nirenberg Sobolev inequality together with the identification of the functions realizing the equality 79 5 Variants of the Wasserstein distance In this chapter we make a quick overview of some variants of the Wasserstein distance W2 together with their applications No proofs will be reported our goal here is only to show that concepts coming from the transport theory can be adapted to cover a broader range of
107. e obtain from 7 16 that v is also left continuous Thus it is continuous and in particular the volume of the spheres y d y x r is 0 for any r gt 0 In particular m y 0 for any y X and the proof is concluded An interesting geometric consequence of the Brunn Minkowski inequality in conjunction with the non branching hypothesis is the fact that the cut locus is negligible Proposition 7 16 Negligible cut locus Assume that X d m is a CD 0 N space and that it is non branching Then for every x supp m the set of y s such that there is more than one geodesic from x to y is m negligible In particular form x m a e x y there exists only one geodesic y from z to y and the map X 3 x y gt q Geod X is measurable Proof Fix x supp m R gt 0 and consider the sets A x Br x z Fix t lt Land y A We claim that there is only one geodesic connecting it to x By definition we know that there is some z Br ax and a geodesic y from z to x such that y y Now argue by contradiction and assume that there are 2 geodesics 71 from y to x Then starting from z following y for time 1 t and 118 then following each of y1 7 for the rest of the time we find 2 different geodesics from z to z which agree on the non trivial interval 0 1 t This contradicts the non branching hypothesis Clearly A C As C Br x fort lt s thus t gt m A is non decreasing B
108. econd part of the proof of Theorem 2 10 shows that if X d is Polish and W X W2 is geodesic then X d is geodesic as well Indeed given x y X and a geodesic u connecting 5 to y we can build a measure H P Geod X satisfying 2 7 Then every y supp js is a geodesic connecting x toy W Definition 2 15 Non branching spaces A geodesic space X d is said non branching if for any t 0 1 a constant speed geodesic y is uniquely determined by its initial point yo and by the point y In other words X d is non branching if the map Geod X gt y gt 0 X is injective for any t 0 1 34 Non branching spaces are interesting from the optimal transport point of view because for such spaces the behavior of geodesics in A X is particularly nice optimal transport plan from inter mediate measures to other measures along the geodesic are unique and induced by maps it is quite surprising that such a statement is true in this generality compare the assumption of the proposition below with the ones of Theorems 1 26 1 33 Examples of non branching spaces are Riemannian manifolds Banach spaces with strictly convex norms and Alexandrov spaces with curvature bounded below Examples of branching spaces are Banach spaces with non strictly convex norms Proposition 2 16 Non branching and interior regularity Let X d be a Polish geodesic non branching space Then P2 X W2 is non branching as well Furt
109. ect all the smoothness issues Fix d p and observe that without loss of generality we can assume f gt 0 and f f 1 so that our aim is to prove that 1 p vsr gt C 4 1 78 for some constant C not depending on f Fix once and for all a smooth non negative function g R gt R satisfying f g 1 define the probability measures map eo vig and let T be the optimal transport map from p to v w r t the cost given by the distance squared The change of variable formula gives g T x P Yz R fo fote forrtp fawn tert As for the case of the isoperimetric inequality we know that T is the gradient of a convex function thus VT x is a symmetric matrix with non negative eigenvalues and the arithmetic geometric mean inequality gives det VT x lt Tia Thus we get forte fy F 1 2 fTT Vf where gt 1 Finally by H lder inequality we have fo aeii f mime a 0 3 a wuta Jory Since g was a fixed given function 4 1 is proved Hence we have 4 4 Bibliographical notes The possibility of proving Brunn Minkowski inequality via a change of variable is classical It has been McCann in his PhD thesis 62 to notice that the use of optimal transport leads to a natural choice of reparametrization It is interesting to notice that this approach can be generalized to curved and non smooth spaces having Ricci curvature bounded below see Proposition 7 14 The idea of proving th
110. ed balls in X are compact the argument simplifies Indeed from the uniform bound on the second moments and the inequality R u X Br ao lt lee d o du we get the tightness of the sequence Hence up to pass to a subsequence we can assume that 4n narrowly converges to a limit measure u and then using the lower semicontinuity of W2 w r t narrow convergence we can conclude lim W2 H Hn S lim limn W2 um Hn 0 30 measures Hn 1 En ndx Where n is chosen such that end T n r To bound from above W j1 Hn leave fixed 1 n u move n p to T and then move dz into 6 this gives WE n lt en f a2 e 2 dule enD so that lim W2 1 Hn lt r Conclude observing that lim d x dun lim 1 en fe x T du End n T J Paadu n oo n co thus the second moments do not converge Since clearly Hn weakly converges to u we proved that there is no local compactness a 2 2 X geodesic space In this section we prove that if the base space X d is geodesic then the same is true also for P2 X W2 and we will analyze the properties of this latter space Let us recall that a curve y 0 1 X is called constant speed geodesic provided d Y Ys lt sld 70 11 Vt s 0 1 2 5 or equivalently if lt always holds Definition 2 9 Geodesic space A metric space X d is called geodesic if for every x y X there exists a constant speed geodesic
111. ed by CGN Gai The operator u O u is the adjoint of O i e it is defined by O7 u w u Oy w Vw L p u It is clear that the operator norm of O and O is bounded by Lip v Observe that in writing O u O u we are losing the reference to the base measure u which certainly plays a role in the definition this simplifies the notation and hopefully should create no confusion as the measure we are referring to should always be clear from the context Notice that if v C3 R4 R these operators read as O u a By Vo u O u Vv P u The introduction of the operators and O allows to give a precise meaning to formula 6 23 for general regular curves Theorem 6 20 Covariant derivative of P uz Let u be a regular curve v its velocity vec tor field and let u be an absolutely continuous vector field along it Then P u is absolutely continuous as well and for a e t it holds D d ree uz Pu Su O w 6 25 101 Proof The fact that P uz is absolutely continuous has been proved with inequality 6 21 To get the thesis start from equation 6 22 and conclude noticing that for a e t it holds Lip v z lt co and thus Pi V ve Na VY v4 Npa vt Ve Ov Vo Corollary 6 21 Total derivatives of P u and Pee uz Let u be a regular curve let v be its velocity vector field and let u be a
112. ena s t r which is precisely the EVI 3 30 written in integral form Uniqueness and contractivity It remains to prove that the solution to the EVI is unique and the contractivity 3 31 The heuristic argument is the following pick x and y solutions of the EVI starting from 7 y respectively Choose y y in the EVI for x to get ld 2 3 delet das ye zf 1 y E x lt Ely Symmetrically we have ld 2 2 3 dg l t Ys zf t yt Ely lt E z 66 Adding up these two inequalities we get L P e Yt lt 2dd x4 Yt a e t The rigorous proof follows this line and uses a doubling of variables argument 4 la Kruzkhov Uniqueness and contraction then follow by the Gronwall lemma 3 3 Applications to the Wasserstein case The aim of this section is to apply the abstract theory developed in the previous one to the case of functionals on A2 R W2 As we will see various diffusion equations may be interpreted as Gradient Flows of appropriate energy functionals w r t to the Wasserstein distance and quantitive analytic properties of the solutions can be derived by this interpretation Most of what we are going to discuss here is valid in the more general contexts of Riemannian manifolds and Hilbert spaces but the differences between these latter cases and the Euclidean one are mainly technical thus we keep the discussion at a level of R to avoid complications that would just
113. ere exists a set A C 0 1 of full Lebesgue measure such that t gt f pdp is differentiable at t A for every y D we can also assume that the metric derivative 1 exists for every t E A Also by 2 25 we know that for to A the linear functional Li Vy yp E D gt R given by d Vo gt Li Vy Sli ee 47 satisfies Leo VEP lt Vellu Ato and thus it can be uniquely extended to a linear and bounded functional on Tan 22 R By the Riesz representation theorem there exists a vector field v Tan Ve R2 such that d ai is T pdu Li Ve VY Uto dtp YED 2 26 and whose norm in L ut is bounded above by the metric derivative 1 at t to It remains to prove that the continuity equation is satisfied in the sense of distributions This is a consequence of 2 26 see Theorem 8 3 1 of 6 for the technical details Part B Up to a time reparametrization argument we can assume that v 72 lt L for some L R for a e t Fix a Gaussian family of mollifiers p and define Hi Mt p EA My It is clear that d EE et V vim 0 Moreover from Jensen inequality applied to the map X 2 gt 2 X z X z X vm it follows that Ive llz2quzy lt Neele lt E 2 27 This bound together with the smoothness of vf implies that there exists a unique locally Lipschitz map T 0 1 x R R4 t 0 1 satisfying d E He a vf T t x
114. es and requiring geodesic convexity on the whole 2 X would lead to a notion not invariant under isomorphism of metric measure spaces Also for the CD 0 N condition one requires the geodesic convexity of all amp y to ensure the following compatibility condition if X is a CD 0 N space then it is also a CD 0 N space for any N gt N Using Proposition 2 16 it is not hard to see that such compatibility condition is automatically satisfied on non branching spaces a Remark 7 9 How to adapt the definitions to general bounds on curvature the dimension It is pretty natural to guess that the notion of bound from below on the Ricci curvature by K R and bound from above on the dimension by N can be given by requiring the functional amp y to be K geodesically convex on A X W2 However this is wrong because such condition is not compatible with the Riemannian case The hearth of the definition of CD K N spaces still con cerns the properties of y but a different and more complicated notion of convexity is involved a 112 Let us now check that the definitions given have the qualitative properties that we discussed in the introduction of this chapter Intrinsicness This property is clear from the definition Compatibility To give the answer we need to do some computations on Riemannian manifolds Lemma 7 10 Second derivative of the internal energy Let M be a compact and smooth Rieman nian manifold m its normalized
115. es e outgoing from z edges e incoming in z Then for a 0 1 one minimizes w e length e edges e of G among all admissible graphs G Observe that for a 0 this problem reduces to the classical Steiner problem while for a 1 it reduces to the classical optimal transport problem for cost distance It is not hard to show the existence of a minimizer for this problem What is interesting is that a continuous formulation is possible as well which allows to discuss the minimization problem for general initial and final measure in A R Definition 5 1 Admissible continuous dynamical transfer Let p v A R An admissible continuous dynamical transfer from u to v is given by a countably H rectifiable set T an orientation on itr T gt S anda weight function w T 0 00 such that the R valued measure Jv 7r w defined by ID rw I wrt 80 satisfies ys IT rw V Hp which is the natural generalization of the Kirchoff rule Given a 0 1 the cost function associated to I 7 w is defined as Eal Jr rw wid r Theorem 5 2 Existence Let p v A IR with compact support Then for all 0 1 there exists a minimizer of the cost in the set of admissible continuous dynamical transfers connecting u to v If u 6 and v L o ye the minimal cost is finite if and only if a gt 1 1 d The fact that 1 1 d is a limit value to get a finite cost can be heuristically
116. ets of a given metric space A bad property is that it is not suitable to study convergence of metric measure spaces which are endowed with infinite reference measures well the definition can easily be adapted but it would lead to a too strict notion of convergence very much like the Gromov Hausdorff distance which is not used to discuss convergence of non compact metric spaces The only notion of convergence of Polish spaces endowed with o finite measures that we are aware of is the one discussed by Villani in Chapter 27 of 80 Definition 27 30 It is interesting to remark that this notion of convergence does not guarantee uniqueness of the limit which can be though of as a negative point of the theory yet bounds from below on the Ricci curvature are stable w r t such convergence which in turn is a positive point as it tells that these bounds are even more stable The discussion on the local Poincar inequality and on Lemma 7 19 is extracted from 57 There is much more to say about the structure and the properties of spaces with Ricci curvature bounded below This is an extremely fast evolving research area and to give a complete discussion on the topic one would probably need a book nowadays Two things are worth to be quickly mentioned The first one is the most important open problem on the subject is the property of being a CD K N space a local notion That is suppose we have a metric measure space X d m
117. etting Tin T and defining Lin 41 asa minimizer of T 2 x x7 F x a 54 It is immediate to verify that a minimum exists and that it is unique thus the sequence n gt Tin is well defined The Euler Lagrange equation of Lin 1 is Tin41 Tin T O F Tia41 which is a time discretization of 3 2 It is then natural to introduce the rescaled curve t gt x7 by Te e e where denotes the integer part and to ask whether the curves t gt x7 converge in some sense to a limit curve x which solves 3 2 as T 0 This is the case and this procedure is actually the heart of the proof of Theorem 3 1 What is important for the discussion we are making now is that the minimization procedure just described can be naturally posed in a metric setting for a general functional EF X RU 00 it is sufficient to pick E lt co T gt 0 define Tio T and then recursively P T x7 Tin 41 E argmin 4 x gt E x 3 11 We this give the following definition Definition 3 7 Discrete solution Let X d be a metric space E X R U 00 lower semi continuous T E E lt co and T gt 0 A discrete solution is a map 0 00 gt t gt af defined by y E Cers satisfies 3 11 where Tio T and Tin Clearly in a metric context it is part of the job the identification of suitable assumptions that ensure that the minimization problem 3 11 admits at least a m
118. from T in the EVI sense w rt A Then equation 3 9 holds Proof First we assume that x is locally Lipschitz The claim will be proved if we show that t gt E a is locally Lipschitz and it holds d 1 1 gE zll 5 VEP ae a e t gt 0 Let us start observing that the triangle inequality implies dg zg a1 y gt tild xt y Vy X a e t gt 0 thus plugging this bound into the EVI we get x4 d axe y 3T at E x lt Ely Vy X a e t gt 0 which implies Ele Ey d xt y Fix an interval a b C 0 00 let L be the Lipschitz constant of x in a b and observe that for any y X it holds lt e a e t gt 0 3 10 VEe Tim d ae ow gt a d a y gt Ld 2 y a e t a b Plugging this bound in the EVI we get Ld z y T zE ey F E x lt E y a e t E a b 53 and by the lower semicontinuity of t E x the inequality holds for every t a b Taking y x and then exchanging the roles of s we deduce E xz E xs lt Ld a4 s Plen a lt Lt s z rie s Wt s a b thus the map t gt E z is locally Lipschitz It is then obvious that it holds d a Elti E tsn _ E t E t4n d tttn tt Paa E m h m d Tt h Xt h 1 i lt VE z ti lt zV EL 2 zlil a e t Thus to conclude we need only to prove the opposite inequality Integrate the EVI from t to t h
119. g process it happens that the volume measures of the approximating manifolds are not converging to the volume measure of the limit one Another important fact to keep in mind is the following if we want to derive useful ana lytic geometric consequences from a weak definition of Ricci curvature bound we should also known what is the dimension of the metric measure space we are working with consider for instance the Brunn Minkowski and the Bishop Gromov inequalities above both make sense if we know the di mension of M and not just that its Ricci curvature is bounded from below This tells that the natural notion of bound on the Ricci curvature should be a notion speaking both about the curvature and the dimension of the space Such a notion exists and is called CD K N condition K being the bound from below on the Ricci curvature and NV the bound from above on the dimension Let us tell in advance that we will focus only on two particular cases the curvature dimension condition CD K oo where no upper bound on the dimension is specified and the curvature dimension con dition C_D 0 N where the Ricci curvature is bounded below by 0 Indeed the general case is much more complicated and there are still some delicate issues to solve before we can say that the theory is complete and fully satisfactory Before giving the definition let us highlight which are the qualitative properties that we expect from a weak notion of curvature dimension bound
120. get d T 1 lt d T 7 The fact that r H E x is non increasing now follows from d Tro T 271 a xy T 271 d Bros E n a 271 lt E x lt E t For the second part of the statement observe that from E a bate lt E y ae Vy X we get E x Ely _ d y T d 27 7 _ d y z d x d x Z d y z d zr y 7 2rd x y 2rd x y Z d x 2 d y Z 27 i Taking the limsup as y x we get the thesis 56 By Theorem 3 9 and Lemma 3 10 it is natural to introduce the following variational interpo lation in the Minimizing Movements scheme as opposed to the classical piecewise constant affine interpolations used in other contexts Definition 3 11 Variational interpolation Let X E be satisfying Assumption 3 8 D E and 0 lt T lt T We define the map 0 00 gt t gt xf in the following way e 2 T bd nT XG Tin41 r is chosen among the minimizers of 3 12 with amp replaced by x e x witht nt n 1 r is chosen among the minimizers of 3 12 with and T replaced by w x and t nt respectively For a7 defined in this way we define the discrete speed Dsp 0 00 0 00 and the Discrete slope Dsl 0 00 0 00 by a Ane Eingir pape geile T 3 15 d x 27 7 TE tE nt n 1 r Dsl Although the object Dsl does not look like a slope we chose this name because from 3
121. hat o is a geodesic and that the K convexity inequality is satisfied along it To check that it is a geodesic just notice that for any partition of 0 1 we have Wo uo p lim Wo oh of lim Wlot oF D lim Walog ot 2 gt Walt Tny l 7 X n oo Passing to the limit in 7 12 recalling Proposition 7 5 to get that u gt ui i 0 1 and that lim l Hr gt limno amp 0 o7 gt 0 we conclude To deal with general jug u1 we start recalling that the sublevels of amp are tight indeed using first the bound z log z gt t and then Jensen s inequality we get 1 C gt m X E e e Exo u 2 erostoyam gt u E log 4 115 for any u pm such that u lt C and any Borel E C X This bound gives that if m E gt 0 then u En 0 uniformly on the set of ws such that 6 u4 lt C This fact together with the tightness of m gives the claimed tightness of the sublevels of Now the conclusion follows by a simple truncation argument using the narrow compactness of the sublevels of and the lower semicontinuity of o w r t narrow convergence For the stability of the C D 0 N condition the argument is the following we first deal with the case of uo y with bounded densities with exactly the same ideas used for Then to pass to the general case we use the fact that if X d m is a CD 0 N space then supp m d m is a doubling space Proposition 7 15 below
122. hat a statement stronger than the one of Remark 1 15 holds namely under the assumptions of Theorems 1 13 and 1 17 for any c concave couple of functions p p maximizing the dual problem and any optimal plan y it holds supp y C Oy Indeed we already know that for some c concave y we have y L 1 p L v and supp y C O for any optimal y Now pick another maximizing couple y for the dual problem 1 16 and notice that G x y lt c a y for any x y implies lt g and therefore G 6 is a maximizing couple as well The fact that L v follows as in the proof of Theorem 1 17 Conclude noticing that for any optimal plan it holds ear ora odut ford f ole o wera f deyar z f edu f grav so that the inequality must be an equality E 13 Definition 1 19 Kantorovich potential A c concave function such that p yt is a maximizing pair for the dual problem 1 16 is called a c concave Kantorovich potential or simply Kantorovich potential for the couple u v A c convex function is called c convex Kantorovich potential if p is a c concave Kantorovich potential Observe that c concave Kantorovich potentials are related to the transport problem in the follow ing two different but clearly related ways e as c concave functions whose superdifferential contains the support of optimal plans accord ing to Theorem 1 13 e as maximizing functions together with
123. hat exp OT y x C O za a From this proposition and following the same ideas used in the Euclidean case we give the following definition Definition 1 32 Regular measures in Y M We say that u E Y M is regular provided it van ishes on the set of points of non differentiability of y for any semiconvex function yy M gt R The set of points of non differentiability of a semiconvex function on M can be described as in the Euclidean case by using local coordinates For most applications it is sufficient to keep in mind that absolutely continuous measures w r t the volume measure and even measures vanishing on Lipschitz hypersurfaces are regular By Proposition 1 30 we can derive a result about existence and characterization of optimal trans port maps in manifolds which closely resembles Theorem 1 26 Theorem 1 33 McCann Let M be a smooth compact Riemannian manifold without boundary and u E Y M Then the following are equivalent i for every v E Y M there exists only one transport plan from pn to v and this plan is induced by a map T ii uis regular If either i or ii hold the optimal map T can be written as x gt exp V x for some c concave function y M gt R 19 Proof ii gt i and the last statement Pick v Y M and observe that since d 2 is uniformly bounded condition 1 4 surely holds Thus from Theorem 1 13 and Remark 1 15 we get that any optimal plan y
124. he following definition Definition 1 23 c c hypersurfaces A set E C R is called c c hypersurface if in a suitable system of coordinates it is the graph of the difference of two real valued convex functions i e if there exists convex functions f g R R such that B y t ER ye R teER t f y 9 y here c c stands for convex minus convex and has nothing to do with the c we used to indicate the cost function 15 Then it holds the following theorem which we state without proof Theorem 1 24 Structure of sets of non differentiability of convex functions Let A C R Then there exists a convex function Y R R such that A is contained in the set of points of non differentiability of if and only if A can be covered by countably many c c hypersurfaces We give the following definition Definition 1 25 Regular measures on R A measure p A IR is called regular provided u E 0 for any c c hypersurface E C R Observe that absolutely continuous measures and measures which give 0 mass to Lipschitz hy persurfaces are automatically regular because convex functions are locally Lipschitz thus a c c hypersurface is a locally Lipschitz hypersurface Now we can state the result concerning existence and uniqueness of optimal maps Theorem 1 26 Brenier Let u A IR be such that f x dy x is finite Then the following are equivalent i for every v A R with
125. he whole Fea so that we need to make some assump tions on wz w to be sure that NV ue we is well defined and absolutely continuous Indeed 102 observe that from a purely formal point of view we expect that the total derivative of N uz we is something like d d dt t d N p Ue Wt Ng Suw N a u Wt as the derivative of M applied to the couple u w some tensor which we may think di d i Forget about the last object and look at the first two addends given that the domain of definition of N is not the whole Ee ls in order for the above formula to make sense we should ask that in each of the couples tu w and ue fut at least one vector is Lipschitz Under the assumption that Ee Lip uz dt lt oo and fo Lip 4u dt lt 00 it is possible to prove the following theorem whose proof we omit Theorem 6 22 Let u be an absolutely continuous curve let v be its velocity vector field and let u we be two absolutely continuous vector fields along it Assume that J Lip uz dt lt co and J Lip u dt lt 00 Then Nq ut w is absolutely continuous and it holds d d d gNr Ue We N u Suw Nu u Sw Ov Nu tes te Ppa O3 Mae tes we 6 27 Corollary 6 23 Let u be a regular curve and assume that its velocity vector field v satisfies i d Lip 5r dt lt 6 28 0 dt Then for every absolutely continuous vector field u both Oy u and
126. her information because its value on R is given by v 2 I pL J Wel gga p and the object on the right hand side is called Fisher information on R It is possible to prove that a formula like the above one is writable and true on general CD K 00 spaces see 7 but we won t discuss this topic Proposition 7 18 HWI inequality Let X d m be a metric measure space satisfying the condi tion CD K 00 Then K In particular choosing v m it holds K y2 Finally if K gt 0 the log Sobolev inequality with constant K holds I lt T 2K Proof Clearly to prove 7 18 it is sufficient to deal with the case v amp o H lt 00 Let u be a constant speed geodesic from pu to v such that Eoo 7 20 Elh lt t Bo tot A W2 u 0 Then from V I p gt limao H Goo ut W2 u ut we get the thesis Equation 7 20 now follows from 7 19 and the trivial inequality valid for any a b gt 0 The log Sobolev inequality is a notion of global Sobolev type inequality and it is known that it implies a global Poincar inequality we omit the proof of this fact When working on metric measure spaces however it is often important to have at disposal a local Poincar inequality see e g the analysis done by Cheeger in 29 Our final goal is to show that in non branching C D 0 N spaces a local Poincar inequality holds The importance of the non branch
127. hermore if m C PA2 X isa constant speed geodesic then for every t 0 1 there exists only one optimal plan in Opt U0 ut and this plan is induced by a map from u Finally the measure p E A Geod X associated to u via 2 7 is unique Proof Let m C Ao X be a constant speed geodesic and fix to 0 1 Pick yt Opt uo Mt and y Opt 1z H1 We want to prove that both yt and y are induced by maps from u To this aim use the gluing lemma to find a 3 plan 2 X such that 1 2 _ 21 Ty a 23 2 my a and observe that since u is a geodesic it holds lal II z2 e lt ld a n d m w 22 ex lt ldir n llada a l ld r 0 Il e2cqry llar 7 II ca q2 Wa Ho Heo W2 tto 1 Wa Ho p1 so that mt 73 za Opt uo u1 Also since the first inequality is actually an equality we have that d x y d y z d x z for a ae x y z which means that x y z lie along a geodesic Furthermore since the second inequality is an equality the functions x y z gt d x y and x y z d y z are each a positive multiple of the other in supp q It is then immediate to verify that for every x y z E supp q it holds d x y 1 to d z z d y z tod a z We now claim that for x y z 2 y 2 E supp q it holds x y z a y z if and only if y y Indeed pick x y z x y 2 supp a and assume for inst
128. hich contradicts 1 8 a 1 5 Bibliographical notes G Monge s original formulation of the transport problem 66 was concerned with the case X Y R and c z y x y and L V Kantorovich s formulation appeared first in 49 The equality 1 2 saying that the infimum of the Monge problem equals the minimum of Kan torovich one has been proved by W Gangbo Appendix A of 41 and the first author Theorem 2 1 in 4 in particular cases and then generalized by A Pratelli 68 22 In 50 L V Kantorovich introduced the dual problem and later L V Kantorovich and G S Rubinstein 51 further investigated this duality for the case c x y d x y The fact that the study of the dual problem can lead to important informations for the transport problem has been investigated by several authors among others M Knott and C S Smith 52 and S T Rachev and L Riischendorf 69 71 The notions of cyclical monotonicity and its relation with subdifferential of convex function have been developed by Rockafellar in 70 The generalization to c cyclical monotonicity and to c sub super differential of c convex concave functions has been studied among others by Riischendorf 71 The characterization of the set of non differentiability of convex functions is due to Zaj ek 83 see also the paper by G Alberti 2 and the one by G Alberti and the first author 3 Theorem 1 26 on existence of optimal maps in R for th
129. hich proves the torsion free identity for the covariant derivative Example 6 9 The velocity vector field of a geodesic Let u be the restriction to 0 1 of a geodesic defined in some larger interval e 1 and let v be its velocity vector field Then we know by Proposition 6 3 that u is regular Also from formula 6 5 it is easy to see that it holds Us T t s Ut Vt s 0 1 and thus v is absolutely continuous and satisfies fu 0 and a fortiori Pu 0 Thus as expected the velocity vector field of a geodesic has zero convariant derivative in analogy with the standard Riemannian case Actually it is interesting to observe that not only the covariant derivative is 0 in this case but also the total one Now we pass to the question of parallel transport The definition comes naturally Definition 6 10 Parallel transport Let u be a regular curve A tangent vector field u along it is a parallel transport if it is absolutely continuous and u 0 a e t dt 93 It is immediate to verify that the scalar product of two parallel transports is preserved in time indeed the compatibility with the metric 6 9 yields diy 9 3 1 2 D J uz uz TA Ut Ut u ut 0 a e t dt 4 dt ne dt T for any couple of parallel transports In particular this fact and the linearity of the notion of parallel transport give uniqueness of the parallel transport itself in the sense that for any
130. holds fix to 0 1 and let y be the unique optimal plan in Opt ju0 Hto The thesis will be proved if we show that u depends only on y Observe that from Theorem 2 10 and its proof we know that 0 eto Opt Ho Hito and thus o ez Y By the non branching hypothesis we know that eo cto Geod X gt X is injective Thus it it invertible on its image letting F the inverse map we get w Fey and the thesis is proved Theorem 2 10 tells us not only that geodesics exists but provides also a natural way to interpo late optimal plans once we have the measure u Y Geod X satisfying 2 7 an optimal plan from 4 to us is simply given by e es Now we know that the transport problem has a natural dual problem which is solved by the Kantorovich potential It is then natural to ask how to inter polate potentials In other words if p p are c conjugate Kantorovich potentials for uo 41 is there a simple way to find out a couple of Kantorovich potentials associated to the couple Ht Hs The answer is yes and it is given shortly said by the solution of an Hamilton Jacobi equation To see this we first define the Hopf Lax evolution semigroup H which in R4 produces the viscosity solution of the Hamilton Jacobi equation via the following formula d x ng GaP tye ites Aj p x 4 2 ift s 2 12 2 apa Oe ift gt s yex s t To fully appreciate the mechanisms behi
131. i 1 N Choosing N 1 and v1 y1 ZY we get p z lt 0 Conversely from the c cyclical monotonicity of IT we have p X gt 0 Thus Y T 0 Also it is clear from the definition that y is c concave Choosing again N 1 and 21 41 Z 7 using 1 3 we get p z lt x 9 e z y lt a x bY e z y which together with the fact that a L pu yields max y 0 L u Thus we need only to prove that 0 y contains I To this aim choose Z T let x1 y1 J and observe that by definition of y x we have pla lt ele g e 9 inf e ye 2 42 elen 3 9 c x 9 9 2 By the characterization 1 3 this inequality shows that Z 9 0 y as desired iii i Let y Adm u v be any transport plan We need to prove that f cdy lt f cd Recall that we have I Na x y V x y E supp y x y Va X y EY p x p y p x p y Cc C IA 10 and therefore ole y dy x 9 I ple 6 y dy e y J vle du e f oe y dr y J ple e y d x 4 lt J ola y d x 9 Remark 1 14 Condition 1 4 is natural in some but not all problems For instance problems with constraints or in Wiener spaces infinite dimensional Gaussian spaces include oo valued costs with a large set of points where the cost is not finite We won t discuss these topics An important
132. ian gles in which Q is divided by its diagonals Let po xqY and define the function v Q gt R as the gradient of the convex map max z y as in the figure Set also w v the rotation by 1 2 of v in Q and w 0 out of Q Notice that V wno 0 Set u Id tv guo and observe that for positive t the support Q of u is made of 4 connected components each one the translation of one of the sets T and that u xQ 2 os Ho H It is immediate to check that u is a geodesic in 0 00 so that from 6 3 we know that the restriction of u to any interval e 1 with gt 0 is regular Fix gt 0 and note that by construction the flow maps of u in e 1 are given by T t s Id sv o Id tv Vt s e 1 Now set ws w o T t 0 and notice that w is tangent at u because w is constant in the connected components of the support of u so we can define a CXC function to be affine on each connected component and with gradient given by w and then use the space between the components themselves to rearrange smoothly the function Since w 4 0 T t t h w we have fw 0 and a fortiori 2 w 0 Thus w is a parallel transport in e 1 Furthermore since V wHo 0 we have wo w Tan 2 R Therefore there is no way to extend w to a continuous tangent vector field on the whole 0 1 In particular there is no way to extend the parallel transport up to t 0
133. ic measure spaces with Ricci curvature bounded from below Contents 1 The optimal transport problem 4 1 1 Monge and Kantorovich formulations of the optimal transport problem 4 1 2 Necessary and sufficient optimality conditions 6 1 3 The dual problem s sancs 6 4 de oP AEE OE Pigs el be BD ee eee 11 1 4 Existence of optimal maps 0 2 00 eee ee ee ee 14 1 5 Bibliographical notes 2 0 0 0 0 20 0000 002000000 22 2 The Wasserstein distance W3 24 Del XQ POWSH Space yese ey tut sb amp wade etna eo te Dp el See 20 6 Ben ls 24 2 2 A geodesic Space be 2a bes ee a ey ee Ra EU et oO had 31 2 3 X Riemannian manifold 0 0 0 0 20 00000000 40 2 3 1 Regularity of interpolated potentials and consequences 40 2 3 2 The weak Riemannian structure of P3 M W2 0 43 2 4 Bibliographical notes 2 2 2 2 ee 49 3 Gradient flows 49 3 1 Hilbertian theory of gradient flows oaoa a 49 3 2 The theory of Gradient Flows in a metric setting o ooo e 51 3201s he framework s aay cps aA ad E E E as ee et Ra is 51 3 2 2 General l s c functionals and EDI aaa a 55 3 2 3 The geodesically convex case EDE and regularizing effects 59 3 2 4 The compatibility of Energy and distance EVI and error estimates 63 3 3 Applications to the Wasserstein case 2 2 2 0 2 ee ee 67 3 3 1 Elements of subdifferential calculus in P3 R W2
134. ical measurable selection theorems we know that there exists a Borel map GeodSel X Geod X such that for any x y X the curve GeodSel z y is a constant speed geodesic connecting x to y Define the Borel probability measure u E A Geod X by H GeodSelyy and the measures u E P X by p er eM We claim that t ju is a constant speed geodesic connecting u to ut Consider indeed the map e9 1 Geod X X and observe that from eo e1 GeodSel a x y we get eo e1 u 7 2 8 In particular po eo u TY p and similarly p1 pt so that the curve t p connects u to 1 The facts that the measures u have finite second moments and u is a constant speed geodesic follow from 2 7 2 1 AN 2 d e4 e6 7 dye QD iy _ 5 J eol e1 7 duly D U 9 f Ple w dr e E PWR 0 The fact that 27 implies i follows from the same kind of argument just used So we turn to i gt ii For n gt 0 we use iteratively the gluing Lemma 2 1 and the Borel map GeodSel to build a measure u P C 0 1 X such that Cian e i41 2 pH Opt Hijo H i 1 2 Vi 0 2 1 and u a e yis a geodesic in the intervals i 2 i 1 2 i 0 2 1 Fix n and observe that for any 0 lt j lt k lt 2 it holds 9 k 1 k 1 l e e5 2 x 2 ll zaqumy S De aleiz ecn las lt Mele ees 2 llzer i j tJ k 1
135. ification in the action functional on curves arising in the Benamou Brenier formula this leads to many different optimal transportation distances maybe more difficult to describe from the Lagrangian viepoint but still with quite useful implications in evolution PDE s and functional inequalities The last one deals with transportation distance between measures with unequal mass a variant useful in the modeling problems with Dirichlet boundary conditions Chapter 6 deals with a more detailed analysis of the differentiable structure of 2 R besides the analytic tangent space arising from the Benamou Brenier formula also the geometric tangent space based on constant speed geodesics emanating from a given base point is introduced We also present Otto s viewpoint on the duality between Wasserstein space and Arnold s manifolds of measure preserving diffeomorphisms A large part of the chapter is also devoted to the second order differentiable properties involving curvature The notions of parallel transport along sufficiently regular geodesics and Levi Civita connection in the Wasserstein space are discussed in detail Finally Chapter 7 is devoted to an introduction to the synthetic notions of Ricci lower bounds for metric measure spaces introduced by Lott amp Villani and Sturm in recent papers This notion is based on suitable convexity properties of a dimension dependent internal energy along Wasserstein geodesics Synthetic Ricci bound
136. inequalities None of the results proven here are new in the sense that they all were well known before the proofs coming from optimal transport appeared Still it is interesting to observe how the tools described in the previous sections allow to produce proofs which are occasionally simpler and in any case providing new informations when compared to the standard ones 4 1 Brunn Minkowski inequality Recall that the Brunn Minkowski inequality in R is 22 ALE 2 etay ety and is valid for any couple of compact sets A B C R To prove it let A B C Rf be compact sets and notice that without loss of generality we can assume that 4 A 4 B gt 0 Define 1 lo r L d 1 ZA lA d H1 ZB B and let u be the unique geodesic in 2 IR7 W2 connecting them Recall from 3 47 that for u z d z171 4 z the functional p f u p dL4 is geodesi cally convex in 22 R1 W2 Also simple calculations show that E ju d 2 A 4 1 E u1 d 4 B 1 Hence we have Elya lt S t 24B a Now notice that Theorem 2 10 see also Remark 2 13 ensures that 4 2 is concentrated on oes 2 thus letting 1 2 2 A B 2 14 I and applying Jensen s inequality to the convex function u we get A B 2 E u1 2 Eli d 55 1 which concludes the proof TT 4 2 Isoperimetric inequality On R the isoperimetric inequality can be written as expt lt dZ4
137. ing assumption is due to the following lemma Lemma 7 19 Let X d m be anon branching CD 0 N space B C X a closed ball of positive measure and 2B the closed ball with same center and double radius Define the measures u m B m and u yg u x u A Geod X where x y y is the map which associates to each x y the unique geodesic connecting them such a map is well defined form x m a e x y by Proposition 7 16 Then 2N er M lt m B 2B vt 0 1 120 Proof Fix x B t 0 1 and consider the homothopy map B 3 y gt Hom y y7 By Proposition 7 16 we know that this map is well defined for m a e y and that using the characteriza tion of geodesics given in Theorem 2 10 t u Hom is the unique geodesic connecting dx to u We have m Hom 1 E EC X Borel mB 5 VEC ore ui E n Hom E The non branching assumption ensures that Homy is invertible therefore from the fact that Hz Hom E Hom Hom 1 E E the Brunn Minkowski inequality and the fact that m x 0 we get m E gt t m Hom B and therefore u E lt ae Given that E was arbitrary we deduce T m Ht S iNm B 7 21 Notice that the expression on the right hand side is independent on z Now pick p as in the hypothesis and define u e The equalities ea I aoe p y du y ply du x du y ede f ply duly x Xx valid for any y Cy X show that
138. ing lemma which has its own interest Lemma 2 1 Gluing Let X Y Z be three Polish spaces and let y A X xY yY P Y x Z be such that my mY Then there exists a measure y E P X x Y x Z such that XY a Ty F YZ n2 Ty y Proof Let p thy m7 and use the disintegration theorem to write dyt x y du y dy x and dy y z du y dyz z Conclude defining y by dy x y 2 dulyjd y x 5 2 2 Theorem 2 2 W gt is a distance W gt is a distance on Po X Proof It is obvious that W2 u 0 and that W2 u v W2 v u To prove that W2 u v 0 implies u v just pick an optimal plan y Opt u v and observe that f d x y dy x y 0 implies that y is concentrated on the diagonal of X x X which means that the two maps 71 and 7 coincide y a e and therefore mhy T3 Y For the triangle inequality we use the gluing lemma to compose two optimal plans Let M1 H2 H3 E P X and let y Opt t1 u2 Y3 Opt u2 H3 By the gluing lemma we know that there exists y Y2 X such that 12 2 Ty V it 2 3 me Y V2 Since THY u and mY u3 we have mY Adm u u3 and therefore from the triangle 25 inequality in L y it holds Wo H1 u3 S Penzan renzo yI d x1 amp 3 dY 21 2 3 lt yJ d a1 2 d z2 3 dy z1 2 3 lt Peneirenenes 4 Penerten aay yI d x1 2 dy7 21 2 JI d x2 x3 dy3 2 3 W2 p1
139. inimum so that discrete solutions exist We now divide the discussion into three parts to see under which conditions on the functional and the metric space X it is possible to prove existence of Gradient Flows in the EDI EDE and EVI formulation 3 2 2 General l s c functionals and EDI In this section we will make minimal assumptions on the functional and show how it is possible starting from them to prove existence of Gradient Flows in the EDI sense Basically there are two independent sets of assumptions that we need those which ensure the existence of discrete solutions and those needed to pass to the limit To better highlight the structure of the theory we first introduce the hypotheses we need to guarantee the existence of discrete solution and see which properties the discrete solutions have Then later on we introduce the assumptions needed to pass to the limit We will denote by D E C X the domain of F i e D E E lt co Assumption 3 8 Hypothesis for existence of discrete solutions X d is a Polish space and E X RU 00 be a l s c functional bounded from below Also we assume that there exists T gt 0 such that for every 0 lt T lt Fand T D E there exists at least a minimum of 2 Pa r gt Eee 3 12 2T 55 Thanks to our assumptions we know that discrete solutions exist for every starting point 7 for T sufficiently small The big problem we have to face now is to show tha
140. iqueness contraction and regularizing effects On the other hand this formulation depends on a compatibility condition between energy and distance this condition is fulfilled in Non Positively Curved spaces in the sense of Alexandrov if the energy is convex along geodesics Luckily enough the compatibility condition holds even for some important model functionals in 2 R sum of the so called internal potential and interaction energies even though the space is Positively Curved in the sense of Alexandrov In Chapter 4 we illustrate the power of optimal transportation techniques in the proof of some classical functional geometric inequalities the Brunn Minkowski inequality the isoperimetric in equality and the Sobolev inequality Recent works in this area have also shown the possibility to prove by optimal transportation methods optimal effective versions of these inequalities for instance we can quantify the closedness of E to a ball with the same volume in terms of the vicinity of the isoperimetric ratio of E to the optimal one Chapter 5 is devoted to the presentation of three recent variants of the optimal transport problem which lead to different notions of Wasserstein distance the first one deals with variational problems giving rise to branched transportation structures with a Y shaped path opposed to the V shaped one typical of the mass splitting occurring in standard optimal transport problems The second one involves mod
141. is an injective isometry where on the source space we put the L distance w rt u Thus Ly always extends to a natural isometric embedding of Tan Y2 R into Tan P2 R2 Furthermore the following statements are equivalent i the space Tan P2 R D is an Hilbert space ii the map t Tany A2 R gt Tan A2 R2 is surjective iii the measure n is regular definition 1 25 We comment on the second part of the theorem The first thing to notice is that the space of di rections Tan 72 R can be strictly larger than the space of gradients Tan 72 R This is actually not surprising if one thinks to the case in which u is a Dirac mass Indeed in this situ ation the space Tan 72 R D coincides with the space 2 R W2 this can be checked 87 directly from the definition however the space Tan 2 R is actually isometric to R itself and is therefore much smaller The reason is that geodesics are not always induced by maps that is they are not always of the form t Id tu for some vector field u cL To some extent here we are facing the same problem we had to face when starting the study of the optimal transport problem maps are typically not sufficient to produce optimal transports From this perspective it is not surprising that if the measure we are considering is regular that is if for any v Ao IR there exists a unique optimal plan and this plan is induced
142. is convergence in L on each coordinate and sup Lip v lt co n for at least 3 indexes i Thus in order for the curvature tensor to be well defined we need at least 3 of the 4 vector fields involved to be Lipschitz However for some related notion of curvature the situation simplifies Of particular relevance is the case of sectional curvature Example 6 28 The sectional curvature If we evaluate the curvature tensor R on a 4 ple of vectors of the kind u v u v and we recall the antisymmetry of M we obtain R u v u v 3M u ol Thanks to the simplification of the formula the value of R u v u v is well defined as soon as either u or v is Lipschitz That is R u v u v is well defined for u v LNL In analogy with the Riemannian case we can therefore define the sectional curvature K u v at the measure p along the directions u v by Kee Riu v u Uy 3 Na u ol UG ent O lol woa Mellel us oy 2 This expression confirms the fact that the sectional curvatures of IR are positive coherently with Theorem 2 20 and provides a rigorous proof of the analogous formula already appeared in 67 and formally computed using O Neill formula 6 4 Bibliographical notes The idea of looking at the Wasserstein space as a sort of infinite dimensional Riemannian manifold is due to F Otto and given in his seminal paper 67 The whole discussion in Section 6 1 is directly taken from there The fac
143. is convex and lower semicontinuous In this case y O p x if and only if y oT p z Proof Observe that ote int ETHE _ yig g a int EE e y BE ey ole EL inf BE v0 which proves the first claim For the second observe that y amp ty x gt pln lal e plz lt z yP 2 o y Vz eR iy p x x 2 x y yl 2 y y p z 2 2 lt z y lyl 2 p y Vz ER S p z z 2 lt p x z 2 z 2 y Yz ER S y p 17 22 Syed Pa Therefore in this situation being concentrated on the c superdifferential of a c concave map means being concentrated on the graph of the subdifferential of a convex function Remark 1 22 Perturbations of the identity via smooth gradients are optimal An immediate consequence of the above proposition is the fact that if y C IR then there exists gt 0 such that Id eV is an optimal map for any e lt Z Indeed it is sufficient to take such that Id lt EV lt Id With this choice the map x a 2 e z is convex for any e lt Z and thus its gradient is an optimal map Proposition 1 21 reduced the problem of understanding when there exists optimal maps reduces to the problem of convex analysis of understanding how the set of non differentiability points of a convex function is made This latter problem has a known answer in order to state it we need t
144. ization of random variables with minimum L distance J Multivariate Anal 32 1990 pp 48 54 G SAVARE Gradient flows and diffusion semigroups in metric spaces under lower curvature bounds C R Math Acad Sci Paris 345 2007 pp 151 154 G SAVARE Gradient flows and evolution variational inequalities in metric spaces In prepa ration 2010 K T STURM On the geometry of metric measure spaces I Acta Math 196 2006 pp 65 131 On the geometry of metric measure spaces II Acta Math 196 2006 pp 133 177 K T STURM AND M K VON RENESSE Transport inequalities gradient estimates entropy and Ricci curvature Comm Pure Appl Math 58 2005 pp 923 940 V N SUDAKOV Geometric problems in the theory of infinite dimensional probability distri butions Proc Steklov Inst Math 1979 pp i v 1 178 Cover to cover translation of Trudy Mat Inst Steklov 141 1976 N S TRUDINGER AND X J WANG On the Monge mass transfer problem Calc Var Partial Differential Equations 13 2001 pp 19 31 C VILLANI Topics in optimal transportation vol 58 of Graduate Studies in Mathematics American Mathematical Society Providence RI 2003 127 80 Optimal transport old and new Springer Verlag 2008 81 Q XIA Optimal paths related to transport problems Commun Contemp Math 5 2003 pp 251 279 82 Interior regularity of optimal transport paths Calc Var Parti
145. locally absolutely continuous curve xo z and IA 1 fF Le af E as 5J t dr gt VE x dr E x YO lt t lt s 3 9 t t 52 Definition 3 5 Evolution Variation Inequality definition of GF EVI Let E X gt RU co T E E lt o andX E R We say that 0 00 3 t gt a X is a Gradient Flow in the EVI sense with respect to A starting at T provided it is a locally absolutely continuous curve in 0 00 x T as t 0 and d E x P zy 3P Eny lt E y Vy X a e t gt Q 1 2 dt There are two basic and fundamental things that one needs understand when studying the problem of Gradient Flows in a metric setting 1 Although the formulations EDI EDE and EVI are equivalent for convex functionals on Hilbert spaces they are not equivalent in a metric setting Shortly said it holds EVI gt EDE gt EDI and typically none of the converse implication holds see Examples 3 15 and 3 23 below Here the second implication is clear for the proof of the first one see Proposition 3 6 below 2 Whatever definition of Gradient Flow in a metric setting we use the main problem is to show existence The main ingredient in almost all existence proofs is the Minimizing Movements scheme which we describe after Proposition 3 6 Proposition 3 6 EVI implies EDE Let E X RU 00 be a lower semicontinuous func tional T X a given point A R and assume that x is a Gradient Flow for E starting
146. m the definition by taking x yo To prove the opposite one observe that since e9 e1 4 E Opt uo u1 and y is a c convex Kantorovich potential for Ho H1 we have from Theorem 1 13 that o 71 90 11 p0 thus 0 1 Co 0 1 Cx p x sup c x y p y e 2 71 M1 yEX t x y 90 11 p0 Plugging this inequality in the definition of y we get ae 0 s Ps Ys inf c 7 z ys p x gt inf x ys c x y 2 70 71 200 gt c ys y1 y0 71 P Y Y0 Ys p0 Step 3 We know that an optimal transport plan from p to jus is given by ez s 44 thus to conclude the proof we need to show that Ps lY Ps E 1 E e Ys Yy supplu t s where y is the c conjugate of the c concave function ys The inequality lt follows from the definition of c conjugate To prove opposite inequality start observing that ps y inf x y ply lt 90 4 ol lt yo Y l y elo and conclude by chs pst inf 4 y Ys y C 90 w plo c9 y0 Ys E Ye Ys plo 2 13 _ ChS Yt Ys Ys Ys 38 We conclude the section studying some curvature properties of 22 X W2 We will focus on spaces positively non positively curved in the sense of Alexandrov which are the non smooth analogous of Riemannian manifolds having sectional curvature bounded from below above by 0 Definition
147. m the one of 10 p1 33 Plugging v pp ps and recalling that Wz Toei en fe eiyan Jeu Oas m m for every n N we get that Jeze m m 1 m eS m x Hin WAHM lt eg f Poona f Gorda m m gt oo 203 1 gaa W2 Ho p1 Letting n we get that u C PY_ Geod X is a Cauchy sequence and the conclusion Lemma 2 11 The multivalued map from G X Geod X which associates to each pair x y the set G x y of constant speed geodesics connecting x to y has closed graph Proof Straightforward Lemma 2 12 A variant of gluing Let Y Z be Polish spaces v v P Y and f g Y gt Z be two Borel maps Let y Adm fv gg Then there exists a plan B P Y such that ee 1H D fom gon 4B y Proof Let vz vz be the disintegrations of v w rt f g respectively Then define B v X Vz dy z Z Z2 Remark 2 13 The Hilbert case If X is an Hilbert space then for every x y X there exists only one constant speed geodesic connecting them the curve t gt 1 t a ty Thus Theorem 2 10 reads as t gt pu is a constant speed geodesic if and only if there exists an optimal plan y Opt ju9 p1 such that Lit a t r tr ga If y is induced by a map T the formula further simplifies to pi 1 t Id tT Ho 2 11 a Remark 2 14 A slight modification of the arguments presented in the s
148. mannian manifold that no assumptions on the measures were given and that both the existence Theorem 6 15 for the parallel transport along a regular curve and counterexamples to its general existence the Example 6 16 were provided These results have been published by the authors of these notes in 5 Later on after having beed aware of Lott s results the second author generalized the construction to the case of Wasserstein space built over a manifold in 44 Not all the results have been reported here we mention that it is possible to push the analysis up show the differentiability properties of the exponential map and the existence of Jacobi fields 7 Ricci curvature bounds Let us start recalling what is the Ricci curvature for a Riemannian manifold M which we will always consider smooth and complete Let R be the Riemann curvature tensor on M x M and u v TyM Then the Ricci curvature Ric u v R is defined as Ric u v 5 R u ei v i where e is any orthonormal basis of T M An immediate consequence of the definition and the symmetries of R is the fact that Ric u v Ric v u Another more geometric characterization of the Ricci curvature is the following Pick z M a small ball B around the origin in TyM and let u be the Lebesgue measure on B The exponential map exp B gt M is injective and smooth thus the measure exp has a smooth density w r t the volume measure Vol on M For any u
149. map T i e p 1 t Id tT pHo Then a simple calculation shows that u satisfies the continuity equation d at V vim 0 43 with ve T Id o 1 t Id tT for every t in the sense of distributions Indeed for CV R it holds foam 5 f o H1aH17 du vo a pia er 7 14 duo Jovan Now the continuity equation describes the link between the motion of the continuum p and the instantaneous velocity v R R of every atom of p It is therefore natural to think at the vector field v as the infinitesimal variation of the continuum p From this perspective one might expect that the set of smooth curves on 9 IR and more generally on M is somehow linked to the set of solutions of the continuity equation This is actually the case as we are going to discuss now In order to state the rigorous result we need to recall the definition of absolutely continuous curve on a metric space Definition 2 28 Absolutely continuous curve Let Y d be a metric space and let 0 1 gt t gt yt Y be a curve Then y is said absolutely continuous if there exists a function f L 0 1 such that A t Ys x if f r dr Yt lt s 0 1 2 18 t We recall that if y is absolutely continuous then for a e t the metric derivative y exists given by d Yt h Yt lim _ 2 1 el D Ih 9 and that Z 0 1 and is the smallest L function
150. micontinuous functional and p P R be a regular measure such that E u lt co The set OW E u C Tan A2 R is the set of vector fields v L u R such that A E u if Ty Id v du ZW2 mv lt EU Wwe A R where here and in the following T will denote the optimal transport map from the regular measure Lt to v whose existence and uniqueness is guaranteed by Theorem 1 26 Observe that the subdifferential of a geodesically convex functional F has the following mono tonicity property which closely resembles the analogous valid for convex functionals on an Hilbert space J v Ty Id du j w TH Id dv lt AW2 u v 3 42 for every couple of regular measures ju v in the domain of E and v OW E u w OW E v To prove 3 42 just observe that from the definition of subdifferential we have and add up these inequalities The definition of subdifferential leads naturally to the definition of Gradient Flow it is sufficient to transpose the definition given with the system 3 2 Definition 3 27 Subdifferential formulation of Gradient Flow Let E be a geodesically con vex functional on Pz R and p A R Then p is a Gradient Flow for E starting from p provided it is a locally absolutely continuous curve ju ast 0 w r t the distance Wa pu is regular for t gt 0 and it holds v OW E u a e t where v is the vector field uniquely identified by the cur
151. n D d FP ut ve geye Ep uz Vp i Ut p gt Pus ut Pre Vo i v ay t bt d ge VE Phil Pk Oe 6 22 This formula characterizes the scalar product of 2P u with any Vy when varies on C3 R Since the set Vp is dense in Tan 2 R for any t 0 1 the formula actually identifies PP a uz However from this expression it is unclear what is the value of ZP p ut w for a general w Tan A2 R because some regularity of Vy seems required to compute V 4 v In order to better understand what the value of 2P us is fix t 0 1 and assume for a moment that vi CX R2 Then compute the gradient of x gt V x ve x to obtain V Vy v V p vs Vu Vo and consider this expression as an equality between vector fields in L Taking the projection onto the Normal space we derive PI V7y v P Vu Ve 0 99 Plugging the expression for P3 V7 v into the formula for the covariant derivative we get D d ak i t FP vo aimee Pi ut Pr Vu Ve a d 1 Hue ve 3 Vor Poa ur VP which identifies 2 P uz as D d ae u Pur Su Vu PE w 6 23 We found this expression assuming that v was a smooth vector field but given that we know that PPu uz exists for a e t it is realistic to believe that the expression makes sense also for general Lipschitz v s The problem is that the object Vv may very well be
152. n Y and for the c convexity Definition 1 10 c superdifferential and c subdifferential Let y X RU co bea c concave function The c superdifferential p C X x Y is defined as ap f r EX XY ola oy ela y The c superdifferential 0 p x at x X is the set of y Y such that x y O py A symmetric definition is given for c concave functions Y RU oo The definition of c subdifferential O of a c convex function p X 00 is analogous d p f z EX XY ola o y e y Analogous definitions hold for c concave and c convex functions on Y Remark 1 11 The base case c x y x y Let X Y R and c x y x y Then a direct application of the definitions show that e a set is c cyclically monotone if and only if it is cyclically monotone e a function is c convex resp c concave if and only if it is convex and lower semicontinuous resp concave and upper semicontinuous e the c subdifferential of the c convex resp c superdifferential of the c concave function is the classical subdifferential resp superdifferential e the c_ transform is the Legendre transform Thus in this situation these new definitions become the classical basic definitions of convex analysis a Remark 1 12 For most applications c concavity is sufficient There are several trivial relations between c convexity c concavity and related notions For instance y is c concave if and
153. n absolutely continuous vector field along it Then P uz is absolutely continuous and it holds d d SP ltt Pu Lue Pu OR te Ou Puet 6 26 d d aTe ut Pr Su Pu Oo ut On Pas uz Proof The absolute continuity of P4 u follows from the fact that both u and P u are absolutely continuous Similarly the second formula in 6 26 follows immediately from the first one noticing that u P u Ps ue yields Zu 4P u apt uz Thus we have only to prove the first equality in 6 26 To this aim let w be an arbitrary absolutely continuous vector field along u and observe that it holds d d d ag Pu ue We FP ue we a a ue a Pp te DP d D Po ts Pae We SEP te P t Ht Ht Since the left hand sides of these expression are equal the right hand sides are equal as well thus we get d D d D SP ut raat uz us aa ae Pu ut a E rae w Ai Pr ut Pu Sw Pp o 6 25 _ P n uz O wz Oy Pig ut we Ht Ht Ht Ht so that the arbitrariness of w gives d D ae ut z ae uz agi On Pu uz and the conclusion follows from 6 25 Along the same lines the total derivative of M uz w for given absolutely continuous vector fields u w along the same regular curve u can be calculated The only thing the we must take care of is the fact that M is not defined on t
154. nd the theory it is better to introduce the rescaled costs cs defined by x y Yt lt s zt yE X s t ch ay Observe that fort lt r lt s Cire ey Sere VoL E X and equality holds if and only if there is a constant speed geodesic y t s X such that z Vt Y Yr Z Ys The notions of er and c transforms convexity concavity and sub super differential are defined as in Section 1 2 Definitions 1 8 1 9 and 1 10 The basic properties of the Hopf Lax formula are collected in the following proposition 36 Proposition 2 17 Basic properties of the Hopf Lax formula We have the following three prop erties i For any t s 0 1 the map H is order preserving that is lt y gt H7 lt Hf vy ii For any t lt s 0 1 it holds iii For any t s 0 1 it holds H o Ht o H H Proof The order preserving property is a straightforward consequence of the definition To prove property ii observe that H H 9 2 sup inf o e c a y ey t s which gives the equality Ht HF pE in particular choosing x x we get the claim the proof of the other equation is similar For the last property assume t lt s the other case is similar and observe that by i we have He o Ht oH gt H gt Id and H o Ht o H lt H eS lt Id The fact that Kantorovich potentials evolve according to the Hopf Lax formula is expre
155. ng way given three vector fields p Vyf Tan P2 R2 i 1 3 the curvature tensor R calculated on them at the measure p is defined as R V gp Ven Ven Vve Vvo Ven Voor Vye Von Vivo Yo V Pn where the objects like Vyy VY are heuristically speaking the covariant derivative of the vector field u Vy along the vector field u gt Vyp However in order to give a precise meaning to the above formula we should be sure at least that the derivatives we are taking exist Such an approach is possible but heavy indeed consider that we should define what are Ct and C vector fields and in doing so we cannot just consider derivatives along curves Indeed we would need to be sure that the partial derivatives have the right symmetries otherwise there won t be those cancellations which let the above operator be a tensor Instead we adopt the following strategy e First we calculate the curvature tensor for some very specific kind of vector fields for which we are able to do and justify the calculations Specifically we will consider vector fields of the kind u gt Vy where the function y C M does not depend on the measure p e Then we prove that the object found is actually a tensor i e that its value depends only on the js a e value of the considered vector fields and not on the fact that we obtained the formula assuming that the functions y s were independent on the measure 104
156. notonicity We say that T C X x Y is ccyclically monotone if ziyi ET 1 lt i lt N implies N 5 c zi yi lt 5 c i Yo forall permutations o of 1 N i 1 i l Definition 1 8 c transforms Let Yy Y RU 00 be any function Its c transform wo X RU oo is defined as ye a inf c a y Py yeY Similarly given p X RU 00 its c transform is the function p Y RU 00 defined by p y inf ce z y p x The c_ transform w X RU 00 ofa function w on Y is given by ye x sup e x y Yy yeY and analogously for c_ transforms of functions on X Definition 1 9 c concavity and c convexity We say that p X RU oo is c concave if there exists Y Y RU oo such that p w Similarly Y Y RU o0 is c concave if there exists p Y RU o0 such that y yr Symmetrically p X RU 00 is c convex if there exists y Y RU 00 such that g 4 andy Y gt RU 00 is c convex if there exists p Y RU co such that p er Observe that y X RU oo is c concave if and only if y vy This is a consequence of the fact that for any function Y Y gt R U 00 it holds Y w indeed C4 C4 C4 f f x ieee iss Y x ae ee c x J 2 9 c Z y vy and choosing x we get gt w while choosing y y we get Stor lt We Similarly for functions o
157. notonicity inequality v w z y gt Alx y Vu OF x w O F y We will denote by V F x the element of minimal norm in F x which exists and is unique as soon as OT F x 0 because O F x is closed and convex For convex functions a natural generalization of Definition 3 1 of Gradient Flow is possible we say that x is a Gradient Flow for F starting from z H if it is a locally absolutely continuous curve in 0 00 such that lima T 3 2 t 0 i x O F x for a e t gt 0 We now summarize without proof the main existence and uniqueness results in this context Theorem 3 1 Gradient Flows in Hilbert spaces Brezis Pazy If F H RU 00 is convex and lower semicontinuous then the following statements hold i Existence and uniqueness for all z D F 3 2 has a unique solution x ii Minimal selection and Regularizing effects t holds lir V F x for every t gt 0 that is the right derivative of x always exists and realizes the element of minimal norm in o7 F zx and TEF o a t V F x t for every t gt 0 Also 1 F x lt inf 4 F v v z OEM RORE i 1 VF a lt inf 4 VF v 5w al y VRE lt int VFO gle a iii Energy Dissipation Equality VF 2 L2 0 00 F a AC oc 0 00 and loc the following Energy Dissipation Equality holds 1 fe 1 fF F x F a F VF a dr F z dr 0 lt t lt
158. nt to prove that t E x is absolutely continuous as then the inequality lm E t n E t lt lm E x E tetn RTO h RTO h 7 BE a1 E tttn dee Cen lt lt E i d Lt tph ma h AERE valid for any t 0 1 gives 3 23 Define the functions f g 0 1 R by f t E z t f s t O w sop FO F s4t s tl Let D be the diameter of the compact set ejo 1 use the fact that x is 1 Lipschitz formula 3 21 and the trivial inequality at lt a b b7 valid for any a b R to get E a1 E ws T t lt su a E A dls Lt 2 Therefore the thesis will be proved if we show that geL gt f s FOI lt i g r dr Yt lt s 3 24 t Fix M gt Oand define f min f M Now fix gt 0 pick a smooth mollifier pe R R with support in e and define f g e 1 R by ee JE F pe t n aip F 4 SAE sft s t Since f is smooth and g gt fM y it holds Me OLS obra 3 25 t From the trivial bound f h lt ht we get SCFM t 1 f s r tpe r dr S F r f s 1 pe rar lt sup s gz t lt sup s s t s t r f s r t sup SED SED pelr ar lt f alt rypelr dr g pelt 3 26 Thus the family of functions g e is dominated in L 0 1 From 3 25 and 3 26 it follows that the family of functions f uniformly converge
159. ntial of W at u Therefore if u is a Gradient Flow of W made of regular measures it solves the non local evolution equation d ait V VW ps ue in the sense of distributions in R x 0 00 Sketch of the Proof Fix p CS R let p Id eV 4 and observe that 1 1 wi 5 J W a y dp x du y 5 W x y e Vo a Vely du a duly 5 we y du ax du y zJ VW x y V x Veo y du x du y o e Now observe that Owe Vote anodni f f VW vydno Vole aut J VW na Vola du z and similarly J VW 2 y Voly dule du y i VW u y Volu duly J VW 2 Vo dul Thus the conclusion follows by applying the equivalence 3 50 Proposition 3 38 Subdifferential of Let u 0 20 R be convex C on 0 00 bounded from below and satisfying conditions 3 44 and 3 45 Let pL P2 R be an absolutely continuous measure with smooth density Then V u p is the unique element in OW E u Therefore if u is a Gradient Flow for E and p is absolutely continuous with smooth density p for every t gt 0 then t p solves the equation L p V pV p 75 Note this statement is not perfectly accurate because we are neglecting the integrability issues Indeed a priori we don t know that V u p belongs to L p Sketch of the Proof Fix p CYX R and define p Id eVy 4p For
160. o check the convexity of the functionals Let 1 be an interpolating curve with base the regular measure u and To T the optimal trans port maps from u to vo and v respectively The only if part of i follows simply considering interpolation of deltas For the if observe that Youn V nta V U HTE 21 e dul 2 V To dyu t VT odua i t To e T 2 dula lt 1 t V r tV v a tW vo v1 3 46 For ii we start claiming that W2 u x u v x v 2W2 u v for any u v P2 R To prove this it is enough to check that if y Opt u v then nt nt n 1 ny Opt u x p v x v To see this let y R RU 00 be a convex function such that supp y C 37 y and define the convex function on R 4 by z y y x y y Itis immediate to verify that supp 7 C 87 9 so that is optimal as well This argument also shows that if v is an interpolating curve with base u then t gt 4 X 1 is an interpolating curve from vo x vo to v x v with base u x u Also 1 2 W x1 x2 is convex if W is The conclusion now follows from case i We pass to iii We will make the simplifying assumption that u lt and that Ty and T are smooth and satisfy det VTo x 0 det VT x 0 for every x supp u up to an approximation argument it is possible to reduce to this case we omit the details Then writing u pL4 from the change of varia
161. o the set B There are several advantages in the Kantorovich formulation of the transport problem e Adm v is always not empty it contains u x v e the set Adm s 1 is convex and compact w r t the narrow topology in A X x Y see below for the definition of narrow topology and Theorem 1 5 and y gt f cdy is linear e minima always exist under mild assumptions on c Theorem 1 5 e transport plans include transport maps since Tyu v implies that y Id x T gu belongs to Adm 1 v In order to prove existence of minimizers of Kantorovich s problem we recall some basic notions concerning analysis over a Polish space We say that a sequence un C P X narrowly converges to u provided foim 9 peu Vy X Cp X being the space of continuous and bounded functions on X It can be shown that the topology of narrow convergence is metrizable A set C C A X is called tight provided for every gt 0 there exists a compact set Ke C X such that WX Ke lt e Wek It holds the following important result Theorem 1 3 Prokhorov Let X d be a Polish space Then a family K C P X is relatively compact w r t the narrow topology if and only if it is tight Notice that if K contains only one measure one recovers Ulam s theorem any Borel probability measure on a Polish space is concentrated on a o compact set Remark 1 4 The inequality Y X x Y Ky x Ko lt U X Ki Y Ko 1 1 valid for any y Adm p
162. o u v P X respectively Pick y Opt iUn Vn and use Remark 1 4 and Prokhorov theorem to get that Yn admits a subsequence not relabeled narrowly converging to some y Y X7 It is clear that THY pu and Tay vy thus it holds W2 u v js fe x yj dy z y lt lim z y dy y lim W2 un Vn n oo n oo Now we pass to the second part of the statement that is we need to prove that with the same notation just used it holds y Opt u v Choose a x b x d x xo for some zo X in the bound 1 4 and observe that since u v A X Theorem 1 13 applies and thus optimality is equivalent to c cyclical monotonicity of the support The same for the plans y Fix N N and pick x y supp y i 1 N From the fact that 7 narrowly converges to y it is not hard to infer the existence of 7 Y supp 7 such that lim alai z alai 0 Vi 1 N n gt co Thus the conclusion follows from the c cyclical monotonicity of supp 7 and the continuity of the cost function Now we are going to prove that 2 X W2 is a Polish space In order to enable some construc tions we will use a version of Kolmogorov s theorem which we recall without proof see e g 31 51 28 Theorem 2 6 Kolmogorov Let X be a Polish space and E A X n N be a sequence of measures such that 1 n 1 Ty vt Hn Pn 1 gt vn 2 2 Then there exists a measure
163. obscure the main ideas The secton is split in two subsections in the first one we discuss the definition of subdifferential of a geodesicaly convex functional on Y2 R which is based on the interpretation of P2 IR as a sort of Riemannian manifold as discussed in Subsection 2 3 2 In the second one we discuss three by now classical applications for which the full power of the abstract theory can be used i e we will have Gradient Flows in the EVI formulation Before developing this program we want to informally discuss a fundamental example Let us consider the Entropy functional E P2 R7 R U 00 defined by d d by J etos oyac if u pLl 00 otherwise We claim that the Gradient Flow of the Entropy in P2 R W2 produces a solution of the Heat equation This can be proved rigorously see Subsection 3 3 2 but for the moment we want to keep the discussion at the heuristic level By what discussed in the previous section we know that the Minimizing Movements scheme produces Gradient Flows Let us apply the scheme to this setting Fix an absolutely continuous measure p here we will make no distinction between an absolutely continuous measure and its density fix 7 gt 0 and minimize gt ad 2T 3 37 It is not hard to see that the minimum is attained at some absolutely continuous measure p actually the minimum is unique but this has no importance Our claim will be proved if we show that for
164. odesically convex on A2 M W2 if and only if M has non negative Ricci curvature and dim M lt N Sketch of the Proof We will give only a formal proof neglecting all the issues which arise due to the potential non regularity of the objects involved We start with i Assume that Ric v v gt K v for any v Pick a geodesic ppm C P2 M and assume that p E C for any t 0 1 By Theorem 1 33 we know that there exists a function p M gt R differentiable ppm a e such that exp Vy is the optimal transport map from pom to pim and pm exp Vy 4 pom Assume that y is C Then by Lemma 7 10 with u us we know that d Szlem V Ric Ty Ve podm gt K Vel poam 114 Since f Vy p0dm W3 po p1 the claim is proved The converse implication follows by an explicit construction if Ric v v lt K v for some x M andv TyM then fore lt 6 lt 1 define pio com z ey co being the normalizing constant and pz T Ho where T y exp t6Vy y and p C is such that V x v and V y x 0 Using Lemma 7 10 again and the hypothesis Ric v v lt K v it is not hard to prove that is not A geodesically convex along u We omit the details Now we turn to ii Let pm and y as in the first part of the argument above Assume that M has non negative Ricci curvature and that dim M lt N Observe that for u uy Lemma 7 10 gives d 1 ai DEJ B oss r ON 08 ii 1 x PR
165. of the internal energy kind on P3 Xn W2 Mosco converge to the corresponding functional on 24 X W2 Thus fix a con vex and continuous function u 0 00 gt R define u z 1 and for every compact metric space X d define the functional amp A X RU 00 by B ulv ulp av w c0 p X 7 6 where u pv p is the decomposition of u in absolutely continuous py and singular part u w r t tov Lemma 7 4 decreases under y Let X dx mx and Y dy my be two metric measure space and d a coupling between them Then it holds E ygeulmy lt ulmx Wwe P X E yy v mx lt vimy Vu P Y 110 Proof Clearly it is sufficient to prove the first inequality Let u pm x and yyy nmy with n given by 7 5 By Jensen s inequality we have Sergulmy f undam fu i ple c die lt f udr ledn f ulole ar e 9 f w ole amx 2 E umx Proposition 7 5 Mosco convergence of internal energy functionals Let Xn dn Mn 3 X d m and dn Yn E Opt dn Mn d m Then the following two are true Weak T lim For any sequence n un E P3 Xn such that n gt Yn Hn narrowly converges to some u P X it holds lim amp in tMn gt 8 ulm n Co Strong T lim For any p Y amp X with bounded density there exists a sequence n gt un PS Xn such that W2 Yn Hn H gt 0 and lim un Mn lt ulm n oo Note
166. okl Akad Nauk USSR 28 1940 pp 212 215 On the translocation of masses Dokl Akad Nauk USSR 37 1942 pp 199 201 English translation in J Math Sci 133 4 2006 1381D1382 L V KANTOROVICH AND G S RUBINSHTEIN On a space of totally additive functions Vestn Leningrad Univ 13 7 1958 pp 52 59 M KNOTT AND C S SMITH On the optimal mapping of distributions J Optim Theory Appl 43 1984 pp 39 49 K KUWADA N GIGLI AND S I OHTA Heat flow on alexandrov spaces preprint 2010 S LISINI Characterization of absolutely continuous curves in Wasserstein spaces Calc Var Partial Differential Equations 28 2007 pp 85 120 G LOEPER On the regularity of solutions of optimal transportation problems Acta Math 202 2009 pp 241 283 J LOTT Some geometric calculations on Wasserstein space Comm Math Phys 277 2008 pp 423 437 J LOTT AND C VILLANI Weak curvature conditions and functional inequalities J Funct Anal 2007 pp 311 333 J LOTT AND C VILLANI Ricci curvature for metric measure spaces via optimal transport Ann of Math 2 169 2009 pp 903 991 126 59 X N MA N S TRUDINGER AND X J WANG Regularity of potential functions of the optimal transportation problem Arch Ration Mech Anal 177 2005 pp 151 183 F MADDALENA AND S SOLIMINI Transport distances and irrigation models J Convex Anal 16 2009 pp 121 152 F MADDALEN
167. on of a geodesic we know that L SUPz 0 1 Lip v lt oo and that fu 0 recall Example 6 9 In particular condition 6 28 is fulfilled Fix to 0 1 u La and define u u o T t to so that fu 0 From equations 6 26 and 6 29 and by induction it follows that P uz is C Also Pu uz is the sum of addends each of which is the composition of projections onto the tangent or normal space and up to n operators O and O applied to the vector u Since the operator norm of O and O is bounded by L we deduce that d Gat me ue lt Weller E eller E Wn EN tE 0 1 Ht Defining the curve t gt U P u o T to t ae the above bound can be written as _U lt U dt lt to Lto lus L Yn EN tE 0 1 which implies that the curve t U Di is analytic This means that for close to to it holds t to d Pu uz 2 T to t a D ay gg lito Pre us n gt 0 Now notice that equations 6 26 and 6 29 and the fact that ERTA 0 ensure that a Py ue An u where An Li Lj is bounded Thus the thesis follows by dt t t6 the arbitrariness of u Dic Now we have all the technical tools we need in order to study the curvature tensor of the mani fold P R Following the analogy with the Riemannian case we are lead to define the curvature tensor in the followi
168. onal The study of gradient flows in the Wasserstein space began in the seminal paper by R Jordan D Kinderlehrer and F Otto 47 where it was proved that the minimizing movements procedure for the functional pee Joos p Vpd on the space 2 IR W2 produce solutions of the Fokker Planck equation Later F Otto in 67 showed that the same discretization applied to pee aL a 1 with the usual meaning for measures with a singular part produce solutions of the porous medium equation The impact of Otto s work on the community of optimal transport has been huge not only he was able to provide concrete consequences in terms of new estimates for the rate of convergence 76 of solutions of the porous medium equation out of optimal transport theory but he also clearly described what is now called the weak Riemannian structure of 2 R W2 see also Chapter 6 and Subsection 2 3 2 Otto s intuitions have been studied and extended by many authors The rigorous description of many of the objects introduced by Otto as well as a general discussion about gradient flows of A geodesically convex functionals on 2 R W2 has been done in the second part of 6 the discussion made here is taken from this latter reference 4 Geometric and functional inequalities In this short Chapter we show how techniques coming from optimal transport can lead to simple proofs of some important geometric and functional
169. only if y is c convex y y and 0 y O y Therefore roughly said every statement concerning c concave functions can be restated in a statement for c convex ones Thus choosing to work with c concave or c convex functions is actually a matter of taste Our choice is to work with c concave functions Thus all the statements from now on will deal only with these functions There is only one important part of the theory where the distinction between c concavity and c convexity is useful in the study of geodesics in the Wasserstein space see Section 2 2 and in particular Theorem 2 18 and its consequence Corollary 2 24 We also point out that the notation used here is different from the one in 80 where a less symmetric notion but better fitting the study of geodesics of c concavity and c convexity has been preferred An equivalent characterization of the c superdifferential is the following y 0 y x if and only if it holds or equivalently if p z e z y gt elz clz y V2 EX 1 3 A direct consequence of the definition is that the c superdifferential of a c concave function is always a c cyclically monotone set indeed if x y O y it holds X cleny by p z Mi Dd P 2i Yo S 2 C Xi Yo i t for any permutation o of the indexes What is important to know is that actually under mild assumptions on c every c cyclically mono tone set can be obtained as the c
170. onvergence plus convergence of length We omit the technical proof of this fact and focus instead on the important case of geodesics Proposition 6 3 Regular geodesics Let u be a constant speed geodesic on 0 1 Then its re striction to any interval e 1 with gt 0 is regular In general however the whole curve ut may be not regular on 0 1 Proof To prove that u may be not regular just consider the case of po dz and py 5 dy dy it is immediate to verify that for the velocity vector field v it holds Lip v t7 For the other part recall from Remark 2 25 see also Proposition 2 16 that for t 0 1 and s 0 1 there exists a unique optimal map T from ju to us It is immediate to verify from formula 2 11 that these maps satisfy Ti Id T Id s t si t Vt 0 1 s 0 1 Thus thanks to Proposition 2 32 we have that v is given by S _ 70 v lim 2 i ae 6 5 sot g t t Now recall that Remark 2 25 gives Lip T lt 1 t to obtain Lipo SENU 9 1 Thus t Lip v is integrable on any interval of the kind e 1 gt 0 Definition 6 4 Vector fields along a curve A vector field along a curve u is a Borel map t u x such that us L for a e t It will be denoted by ut Observe that we are considering also non tangent vector fields that is we are not requiring uz Tan Yo R for ae
171. ooth density that each L function is C and so on Let us recall the definition of Riemannian submersion Let M N be Riemannian manifolds and let f M N a smooth map f is a submersion provided the map df Kert df x TN 84 is a surjective isometry for any x E M A trivial example of submersion is given in the case M N x L for some Riemannian manifold L with M endowed with the product metric and f M gt N is the natural projection More generally if f is a Riemannian submersion for each y N the set f y C M isa smooth Riemannian submanifold The duality between the Wasserstein and the Arnold Manifolds consists in the fact that there exists a Big Manifold BM which is flat and a natural Riemannian submersion from BM to 2 R whose fibers are precisely the Arnold Manifolds Let us define the objects we are dealing with Fix once and for all a reference measure p P gt R recall that we are assuming that all the measures are absolutely continuous with smooth densities so that we will use the same notation for both the measure and its density The Big Manifold BM is the space L p of maps from R to R which are L w r t the reference measure p The tangent space at some map T BM is naturally given by the set of vector fields belonging to L p where the perturbation of T in the direction of the vector field u is given by t gt T tu The target manifold of the submersion is
172. ory we just give an example showing the difficulty that can arise in a curved setting The example will show a smooth compact manifold and two measures absolutely continuous with positive and smooth densities such that the optimal transport map is discontinuous We remark that similar behaviors occur as soon as M has one point and one sectional curvature at that point which is strictly negative Also even if one assumes that the manifold has non negative sectional curvature everywhere this is not enough to guarantee continuity of the optimal map what comes into play in this setting is the Ma Trudinger Wang tensor an object which we will not study Example 1 36 Let M C R be a smooth surface which has the following properties e M is symmetric w r t the x axis and the y axis e M crosses the line x y 0 0 at two points namely O O 21 e the curvature of M at O is negative These assumptions ensure that we can find a b gt 0 such that for some Za zp the points A a 0 2a A a 0 Za B 0 b z B 0 b z belong to M and d A B gt d A O amp O B d being the intrinsic distance on M By continuity and symmetry we can find gt 0 such that P x y gt d x 0 d 0 y Ye Be A UB A y Be B U Be B 1 8 Now let f resp g be a smooth probability density everywhere positive and symmetric w r t the x y axes such that JecauB an f dvol gt 4 resp Je BU
173. p2 W2 u2 u3 Finally we need to prove that W3 is real valued Here we use the fact that we restricted the analysis to the space 2 X from the triangle inequality we have Wa u v lt Wa u d2 Wav bao yJ x zo du x jay fe x zo dv x lt A trivial yet very useful inequality is W2 fem gpu lt I f x g 2 du e 2 1 valid for any couple of metric spaces X Y any y A X and any couple of Borel maps f g X gt Y This inequality follows from the fact that f 9 is an admissible plan for the measures f b 941 and its cost is given by the right hand side of 2 1 Observe that there is a natural isometric immersion of X d into 2 X W2 namely the map LH Op Now we want to study the topological properties of A X W2 To this aim we introduce the notion of 2 uniform integrability K C P X is 2 uniformly integrable provided for any gt 0 and xo X there exists Re gt 0 such that sup f d x zo du lt HEK J X Bpr xo Remark 2 3 Let X dx Y dy be Polish and endow X x Y with the product distance d x1 y1 x2 y2 d amp x1 2 d y1 y2 Then the inequality lezie f dano erey Ble sodva Br z0 x Br yo Br x0 xY Br xo0 x Br yo lt dlato du c f Ray x y Br z0 X x Br yo lt dlxto du c Biwi Br z0 Br yo 26 valid for any y E Adm u v and the analogous one with the integral of d
174. pex on a vector tensor field stands for covariant differentia tion so that in particular we have Jo Id 7 8 Je V 113 The fact that Jacobi fields are the differential of the exponential map reads in our case as VTi x v I x v therefore we have D det J 7 9 Also Jacobi fields satisfy the Jacobi equation which we write as Jt Ard 0 7 10 where A x Texp Ygl M gt Texp tvy x M is the map given by Alx v RW v Yt where 7 exp tV x Recalling the rule det B det B tr B B valid for a smooth curve of linear operators we obtain from 7 9 the validity of Di Ditr JIJ 7 11 Evaluating this identity at t 0 and using 7 8 we get the first of 7 7 Recalling the rule B7 B B B7 valid for a smooth curve of linear operators and differentiating in time equation 7 11 we obtain Di D HI N D R Ie Di HIT tr AI I having used the Jacobi equation 7 10 Evaluate this expression at t 0 use 7 8 and observe that tr Ao tr v gt R V u v Ric Vy Ve to get the second of 7 7 Theorem 7 11 Compatibility of weak Ricci curvature bounds Let M be a compact Riemannian manifold d its Riemannian distance and m its normalized volume measure Then i the functional x is K geodesically convex on P M W2 if and only if M has Ricci curvature uniformly bounded from below by K ii the functional Ey is ge
175. pick v A R let T be the optimal transport map from ju to v and recall that T is the gradient of a convex function Assume that is smooth and define v x o ax x 2 The geodesic u from u to v can then be written as m L tld 7 pu 1 td V0 pu Id tV o yh From the A convexity hypothesis we know that d Fv gt F u Jilt 07 Me F z W2 H v therefore since we know that TE ov Ht J v Vp dp from the arbitrariness of v we deduce v OW F p Proposition 3 36 Subdifferential of V Let V R gt R be convex and Ct let V be as in Definition 3 30 and let yp P R be regular and satisfying V u lt Then OW V u is non empty if and only if VV L p and in this case VV is the only element in the subdifferential of V at u 74 Therefore if u is a Gradient Flow of V made of regular measures it solves d m V VV qi VV ne in the sense of distributions in R x 0 00 Sketch of the Proof Fix p CX R and observe that I is V UId eV 4M V p im f Vo UId eVy V e 0 E e 0 E da f VV Vo dp Conclude using the equivalence 3 50 Proposition 3 37 Subdifferential of W Let W R R be A convex even and Ct let W be defined by 3 31 and u be regular and satisfying W u lt co Then OV W p 0 if and only if VW x u belongs to L u and in this case VW x u is the only element in the subdiffere
176. pport in Q and f d x 0Q dpn gt f x 0Q dp as n ov e Finally a plan y Adm v is optimal i e it attains the minimum cost among admissible plans if and only there exists a c concave function p which is identically 0 on OQ such that supp y C 0 here c x y x yl Observe that M2 Q Wb is always a geodesic space while from Theorem 2 10 and Remark 2 14 we know that P Q W2 is geodesic if and only if Q is that is if and only if Q is convex It makes perfectly sense to extend the entropy functional to the whole M2 Q the formula is still E u f plog p for u pla and E u oo for measures not absolutely continuous The Gradient Flow of the entropy w r t Wb2 produces solutions of the Heat equation with Dirichlet boundary conditions in the following sense Theorem 5 6 Let u E M2 Q be such that E u lt Then e for every T gt 0 there exists a unique discrete solution p starting from and constructed via the Minimizing Movements scheme as in Definition 3 7 e Ast 0 the measures p converge to a unique measure pi in M2 Q Wb2 for any t gt 0 e The map x t gt p x is a solution of the Heat equation so Apt in Q x 0 00 pt gt H weakly as t 0 subject to the Dirichlet boundary condition p x e in OQ for every t gt 0 that is pp e7 belongs to Ht Q for every t gt 0 83 The fact that the boundary value is given by e can be heuristically guessed by the fact th
177. ric space and let E X RU 00 be satisfying the Assumptions 3 8 and 3 13 Also let x D E and for0 lt T lt T define the discrete solution via the variational interpolation as in Definition 3 11 Then it holds 57 e the set of curves x7 is relatively compact in the set of curves in X w rt local uniform convergence e any limit curve x is a Gradient Flow in the EDI formulation Definition 3 3 Sketch of the Proof Compactness By Corollary 3 12 we have T 2 T d a7 z lt Depar lt Ef Dsp dr lt 2T E inf E Vex T 0 0 for any T nt Therefore for any T gt 0 the set x7 lt r is uniformly bounded in 7 As this set is also contained in FE lt E Z it is relatively compact The fact that there is relative compactness w r t local uniform convergence follows by an Ascoli Arzela type argument based on the inequality 2 d x 27 f Dsp ar lt 2 s t E T inf E Vt nt s mT7 n lt meNn 3 17 Passage to the limit Let 7 0 be such that x converges to a limit curve 2 locally uniformly Then by standard arguments based on inequality 3 17 it is possible to check that t gt x is abso lutely continuous and satisfies dr lt lim F Dsp dr WO lt t lt s 3 18 n oo By the lower semicontinuity of V E and 3 14 we get n o0 n o0 thus Fatou s lemma ensures that for any t lt s it holds i VE a dr lt J lim VE x7 dr lt lim Dsi
178. rom the lower semicontinuity again and the bounded compactness of the sublevels of we immediately get that the minimization problem 3 12 admits a solution if r lt 1 A7 The lower semicontinuity of the slope is a direct consequence of 3 21 and of the lower semi continuity of E Thus to conclude we need only to show that Ln gt x sup VE rn E an lt co gt lim E n lt E x 3 22 n n o0 From 3 21 with x y replaced by n x respectively we get E x gt E an VE n d n P e wn and the conclusion follows by letting n oo Thus Theorem 3 14 applies directly also to this case and we get existence of Gradient Flows in the EDI formulation To get existence in the stronger EDE formulation we need the following result which may be thought as a sort of weak chain rule observe that the validity of the proposition below rules out behaviors like the one described in Example 3 15 Proposition 3 19 Let E be a geodesically convex and l s c functional Then for every absolutely continuous curve x C X such that E x lt for every t it holds E xs E a lt VE x dr Yt lt s 3 23 t Proof We may assume that the right hand side of 3 23 is finite for any t s 0 1 and by a reparametrization argument we may also assume that 2 1 for a e t in particular x is 60 1 Lipschitz so that t VE az is an L function Notice that it is sufficie
179. rties of D Proposition 7 3 Properties of D The inf in 7 4 is realized and a coupling realizing it will be called optimal Also let X be the set of isomorphism classes of metric measure spaces satisfying Assumption 7 1 Then D is a distance on X and in particular D is 0 only on couples of isomorphic metric measure spaces Finally the space X D is complete separable and geodesic Proof See Section 3 1 of 74 We will denote by Opt dx mx dy my the set of optimal couplings between X dx mx and Y dy my i e the set of couplings where the inf in 7 4 is realized Given a metric measure space X d m we will denote by Y3 X C A X the set of measures which are absolutely continuous w r t m To any coupling d y of two metric measure spaces X dx mx and Y dy my it is natu rally associated a map yy Y3 X P9 Y defined as follows LL pmx Yeu nmy where nis defined by n y TEOG 7 5 where 7 is the disintegration of y w r t the projection on Y Similarly there is a natural map Ye PSY P X given by v nmy yg v pmx where pis defined by p x f naiaretu where obviously 7 is the disintegration of y w r t the projection on X Notice that yymx my and Yy my m y and that in general Ve Ygl u Also if y is induced by a map T X gt Y i e if y Id T mx then yau Typ for any p P X Our goal now is to show that if Xn dn Mn D X d m
180. s are completely consistent with the smooth Riemannian case and stable under measured Gromov Hausdorff limits For this reason these bounds and their analytic implications are a useful tool in the description of measured GH limits of Riemannian manifolds Acknowledgement Work partially supported by a MIUR PRIN2008 grant 1 The optimal transport problem 1 1 Monge and Kantorovich formulations of the optimal transport problem Given a Polish space X d i e a complete and separable metric space we will denote by A X the set of Borel probability measures on X By support supp j of a measure u A X we intend the smallest closed set on which u is concentrated If X Y are two Polish spaces T X Y is a Borel map and u A X a measure the measure Tyu P Y called the push forward of u through T is defined by Tyu E T E VECY Borel The push forward is characterized by the fact that J fatan f toTay for every Borel function f Y R U 00 where the above identity has to be understood in the following sense one of the integrals exists possibly attaining the value oc if and only if the other one exists and in this case the values are equal Now fix a Borel cost function c X x Y RU 00 The Monge version of the transport problem is the following Problem 1 1 Monge s optimal transport problem Let u P X v P Y Minimize T gt f c x T x du x x among all transport maps
181. s choice will be important in discussing the stability issue Definition 7 2 Coupling between metric measure spaces Given two metric measure spaces X dx mx Y dy my we consider the product space X x Y Dxy where Dxy is the distance defined by Dxy 1 91 2 y2 1 22 d y1 y2 We say that a couple d y is an admissible coupling between X dx mx and Y dy my we write d y Adm dx mx dy my if e d is a pseudo distance on suppmx U suppmy i e it may be zero on two different points which coincides with dx resp dy when restricted to supp mx X supp my resp supp My X supp My e a Borel w rt the Polish structure given by Dxyy measure y on supp Mx X supp My such that my mx and T Y my It is not hard to see that the set of admissible couplings is always non empty The cost C d y of a coupling is given by Clan J SE Pedy supp Uscriptsizem xsupp Uscriptsizem The distance D X dx mx Y dy my is then defined as D X dx mx Y dy my inf VC d Y 7 4 the infimum being taken among all couplings d y of X dx mx and Y dy my A trivial consequence of the definition is that if X dx mx and X dg m resp Y dy my and Y dy my are isomorphic then D X dx mx Y dy my D X dg my dp my 109 so that D is actually defined on isomorphism classes of metric measure spaces In the next proposition we collect without proof the main prope
182. same lines taking into account that for a general of the form m A m it holds En u N L m A and that as before if m Ao m A1 gt 0 it cannot be m Ao Ai z 0 or we would violate the convexity inequality A consequence of Brunn Minkowski is the Bishop Gromov inequality Proposition 7 15 Bishop Gromov Let X d m be a CD 0 N space Then it holds me Be ae Var supp m 7 15 gt m Br 2 R In particular supp m d m is a doubling space Proof Pick x supp m and assume that m x 0 Let v r m B a Fix R gt 0 and apply the Brunn Minkowski inequality to Ag x Ai Br x observing that Ao Ai C B r x to get vV N tR gt m Ao Aili gt twv R VO lt t lt 1 Now let r tR and use the arbitrariness of R t to get the conclusion It remains to deal with the case m x 4 0 We can also assume supp m x otherwise the thesis would be trivial under this assumption we will prove that m x 0 for any x X A simple consequence of the geodesic convexity of y tested with delta measures is that supp m is a geodesically convex set therefore it is uncountable Then there must exist some x supp m such that m z 0 Apply the previous argument with x in place of x to get that gt Z we lt re lt r 7 16 where now v r is the volume of the closed ball of radius r around x By definition v is right continuous letting r R w
183. se spaces are the subclass of CD K N spaces where the heat flow studied in 45 53 7 is linear References 1 10 11 12 tairi A AGRACHEV AND P LEE Optimal transportation under nonholonomic constraints Trans Amer Math Soc 361 2009 pp 6019 6047 G ALBERTI On the structure of singular sets of convex functions Calc Var and Part Diff Eq 2 1994 pp 17 27 G ALBERTI AND L AMBROSIO A geometrical approach to monotone functions in R Math Z 230 1999 pp 259 316 L AMBROSIO Lecture notes on optimal transport problem in Mathematical aspects of evolv ing interfaces CIME summer school in Madeira Pt P Colli and J Rodrigues eds vol 1812 Springer 2003 pp 1 52 L AMBROSIO AND N GIGLI Construction of the parallel transport in the Wasserstein space Methods Appl Anal 15 2008 pp 1 29 L AMBROSIO N GIGLI AND G SAVARE Gradient flows in metric spaces and in the space of probability measures Lectures in Mathematics ETH Ziirich Birkhauser Verlag Basel sec ond ed 2008 Calculus and heat flows in metric measure spaces with ricci curvature bounded below preprint 2011 Spaces with riemannian ricci curvature bounded below preprint 2011 L AMBROSIO B KIRCHHEIM AND A PRATELLI Existence of optimal transport maps for crystalline norms Duke Mathematical Journal 125 2004 pp 207 241 L AMBROSIO AND S RIGOT Optimal mass transportation
184. sequence n Np Un Un Tan P2 R is a Cauchy sequence Fix such a sequence un Vn let L sup min Lip un Lip vn Z C N be the set of indexes n such that Lip u lt L and fix two smooth vectors amp 3 CV R1 R2 100 Notice that for n m J it holds My un vn Nu lUm Vm lly lt My Un Un 7 Mla oI Mu un Um p T N Um z Um lly lt Lon llu Lip Un umlly Lilom Oly and thus o im Miu Un Un Nu lum vm ly lt 2L v Olu n mel this expression being vacuum if J is finite If n I and m I we have Lip vm lt L and Ma uns Un N plum Vm lla lt N un vn lla Np un llu NG Um NG um Um Ile lt Ltn 0 Lip llun ll Lip 0 vmllu Llum ely which gives T Wylttns Yn Ny lum vm lly lt Ello llu Ellu ally nEI m I Exchanging the roles of the u s and the v s in these inequalities for the case in which n I we can conclude a Tia Multia n Ny tens Yn Mle lt 2E lly 2Lllu iy n Since are arbitrary we can let u gt u and gt v in r and conclude that n gt Np n Un isa Cauchy sequence as requested The other claims follow trivially by the sequential continuity Definition 6 19 The operators O and O Let u Pa R and v L with Lip v lt oo Then the operator u O u is defin
185. sportation discussed in the previous section Indeed in this problem the mass tends to spread out along an action minimizing curve rather than to glue together 5 3 An extension to measures with unequal mass Let us come back to the Heat equation seen as Gradient Flow of the entropy functional E p J plog p with respect to the Wasserstein distance W2 as discussed at the beginning of Section 3 3 and in Subsection 3 3 2 We discussed the topic for arbitrary probability measures in R4 but actually everything could have been done for probability measures concentrated on some open bounded set Q C R with smooth boundary that is consider the metric space P Q W2 and the entropy functional E p f plog p for absolutely continuous measures and E u 00 for measures with a singular part Now use the Minimizing Movements scheme to build up a family of discrete solutions p starting from some given measure p Q It is then possible to see that these discrete families converge as 7 0 to the solution of the Heat equation with Neumann boundary condition 4p Apt inQ x 0 00 pt gt P weakly as t 0 Vor n 0 in OQ x 0 00 where 77 is the outward pointing unit vector on OQ The fact that the boundary condition is the Neumann s one can be heuristically guessed by the fact that working in Z Q enforces the mass to be constant with no flow of the mass through the boundary It is then natural to ask whether it is possible
186. ssed in the following theorem We remark that in the statement below one must deal at the same time with c concave and c convex potentials Theorem 2 18 Interpolation of potentials Let X d be a Polish geodesic space m C Po X a constant speed geodesic in P X W2 and yp ac c t convex Kantorovich potential for the couple uo p1 Then the function ps H y is a c 5 concave Kantorovich potential for the couple us u for any t lt s Similarly if is a c concave Kantorovich potential for m po then Ht is a ch convex Kantorovich potential for u 4s for any t lt s Observe that that for t 0 s 1 the theorem reduces to the fact that Hi p y is a c concave Kantorovich potential for 41 Ho a fact that was already clear by the symmetry of the dual problem discussed in Section 1 3 Proof We will prove only the first part of the statement as the second is analogous Step 1 We prove that H is a c concave function for any t lt s and any Y X gt RU 00 This is a consequence of the equality x y inf z y c 2 37 from which it follows HE i f 0 s inf t s inf 0 t o x ee x y VY inf c 2 2 ink o z y Ply Step 2 Let u Y Geod X be a measure associated to the geodesic u via equation 2 7 We claim that for every y supp jz and s 0 1 it holds s Ys 70 Y0 Ys 2 13 Indeed the inequality lt comes directly fro
187. structive and is based on the identity Va Pt u 0 Yu TyoM 6 13 l 0 7 which tells that the vectors Pf u are a first order approximation at t 0 of the parallel transport Taking 6 11 into account 6 13 is equivalent to P tro u Pi u o t we Tyo M 6 14 Equation 6 14 follows by applying inequalities 6 12 note that tr u Pj u T TM P trolu P5 u lt Ct tr u Po u lt C7t ul Now let be the direct set of all the partitions of 0 1 where for P Q W P gt Qif Pisa refinement of Q For P 0 to lt t lt lt ty 1 Pand u Ta M define P u Ty M as tn P u PY Pans Po u Our first goal is to prove that the limit P w for P exists This will naturally define a curve t u T M by taking partitions of 0 t instead of 0 1 the final goal is to show that this curve is actually the parallel transport of u along the curve y The proof is based on the following lemma Lemma 6 11 Let 0 lt s lt s2 lt s3 lt 1 be given numbers Then it holds P35 u PoP u lt C ul sy S2 S2 s3 Vue Ty M Proof From P33 u Py t133 u Py tres tr u we get PS u Pop Psp u Pep trs u P3 u Since u T M and trf u P u Trz M the proof follows applying inequalities 6 12 From this lemma an easy induction shows that for any 0 lt
188. sufficiently small u is absolutely continuous and its density p satisfies by the change of variable formula the identity 2 p z det Id eV y x p x eVy x Using the fact that o detUd eV y zx Ay x we have d de e d d 2 d x p qe leno H j Elo ul y dy leo zm ray det Id V y x dx z i E E Vou o u o Ve J V u 0 Ve b and the conclusion follows by the equivalence 3 50 As an example let u z zlog x and let V be a convex smooth function on R Since u z log z 1 we have pV u p Ap thus a gradient flow p of F E V solves the Fokker Plank equation d et Api V VV pi Also the contraction property 3 31 in Theorem 3 25 gives that for two gradient flows pz p it holds the contractivity estimate W2 pt Pt lt e Wa po po 3 4 Bibliographical notes The content of Section 3 2 is taken from the first part of 6 we refer to this book for a detailed bibliographical references on the topic of gradient flows in metric spaces with the only exception of Proposition 3 6 whose proof has been communicated to us by Savar see also 72 73 The study of geodesically convex functionals in A2 IR W2 has been introduced by R Mc Cann in 63 who also proved that conditions 3 44 and 3 45 were sufficient to deduce the geodesic convexity called by him displacement convexity of the internal energy functi
189. sume that Assumption 3 17 Geodesic convexity hypothesis X d is a Polish geodesic space E X RU 00 is lower semicontinuous geodesically convex for some A R Also we assume that the sublevels of E are boundedly compact i e the set E lt c N B ax is compact for any c R r gt 0c2E xX What we want to prove is that for X E satisfying these assumptions there is existence of Gradient Flows in the formulation EDE Definition 3 4 Our first goal is to show that in this setting it is possible to recover the results of the previous section We start claiming that it holds cap EO BW Nar at V E a sup dy 5a a 3 21 so that the lim in the definition of the slope can be replaced by a sup Indeed we know that x hm t x lt su t T To prove the opposite inequality fix y x and a constant speed geodesic y connecting x to y for which 3 20 holds Then observe that i 2 a pde eee Riles Using this representation formula we can show that all the assumptions 3 8 and 3 13 hold Proposition 3 18 Suppose that Assumption 3 17 holds Then Assumptions 3 8 and 3 13 hold as well Sketch of the Proof From the A geodesic convexity and the lower semicontinuity assumption it is possible to deduce we omit the details that has at most quadratic decay at infinity i e there exists T X a b gt 0 such that E x gt a bd x A d x 2 Vae x Therefore f
190. t To define the time smoothness of a vector field u defined along a regular curve u we will make an essential use of the flow maps notice that the main problem in considering the smoothness of uz is that for different times the vectors belong to different spaces To overcome this obstruction we will define the smoothness of t gt uz L in terms of the smoothness of t gt uz o T to t Ee Hto Definition 6 5 Smoothness of vector fields Let u be a regular curve T t s its flow maps and uz a vector field defined along it We say that u is absolutely continuous or Ct or C or C or analytic provided the map t u o T to t Li is absolutely continuous or Ct or C or C or analytic for every ty 0 1 Since uso T t t uro T to t o T t1 to and the composition with T t1 to provides an isometry from Liro to Lia it is sufficient to check the regularity of t gt ut o T to t for some to 0 1 to be sure that the same regularity holds for every to 90 Definition 6 6 Total derivative With the same notation as above assume that uz is an absolutely continuous vector field Its total derivative is defined as phe laa Ut h O T t t h Ut dt ho h where the limit is intended in Ey Observe that we are not requiring the vector field to be tangent and that the total derivative is in general a non tangent vector field even
191. t z e Jimla lt pa IF F0 dma dmly Pe m B JBxB 2 fo f duly Geod X where p is defined as in the statement of Lemma 7 19 Observe that for any geodesic y the map t f y is Lipschitz and its derivative is bounded above by d yo 71 V f 7 for a e t Hence since any geodesic y whose endpoints are in B satisfies d 7yo 1 lt 2r we have Fc Om Foal lt 2 Foo Moo amo f f iwsateoasat By Lemma 7 19 we obtain 2 f ideana lt Zi f m By the Bishop Gromov inequality we know that m 2B lt 2 m B and thus QN 1 lt Zr AE Jas S10 S aE JF which is the conclusion 7 3 Bibliographical notes The content of this chapter is taken from the works of Lott and Villani on one side 58 57 and of Sturm 74 75 on the other The first link between K geodesic convexity of the relative entropy functional in 22 M W2 and the bound from below on the Ricci curvature is has been given by Sturm and von Renesse n 76 The works 74 75 and 58 have been developed independently The main difference between them is that Sturm provides the general definition of CD K N bound which we didn t speak about with the exception of the quick citation in Remark 7 9 while Lott and Villani focused on the cases C_D K oo and CD 0 N Apart from this the works are strictly related and the differences are mostly on the technical side We mention only one of these In giving the definition of C D 0
192. t that the tangent space made of gradients Tan 2 R was not sufficient to study all the aspects of the Riemannian geometry of 2 IR W2 has been understood in 6 in con nection with the definition of subdifferential of a geodesically convex functional in particular con cerning the issue of having a closed subdifferential In the appendix of 6 the concept of Geometric Tangent space discussed in Section 6 2 has been introduced Further studies on the properties of Tan 2 M have been made in 43 Theorem 6 1 has been proved in 46 The first work in which a description of the covariant derivative and the curvature tensor of Po M W2 M being a compact Riemannian manifold has been given beside the formal calculus of the sectional curvature via O Neill formula done already in 67 is the paper of J Lott 56 rigorous formulas are derived for the computation of such objects on the submanifold Ao M 106 made of absolutely continuous measures with density C and bounded away from 0 In the same paper Lott shows that if M has a Poisson structure then the same is true for Yow M a topic which has not been addressed in these notes Independently on Lott s work the second author built the parallel transport on 2 IR7 W2 in his PhD thesis 43 along the same lines provided in Section 6 3 The differences with Lott s work are the fact that the analysis was carried out on R rather than on a compact Rie
193. t the discrete solutions satisfy a discretized version of the EDI suitable to pass to the limit The key enabler to do this is the following result due to de Giorgi Theorem 3 9 Properties of the variational interpolation Let X E be satisfying the Assumption 3 8 Fix amp X and for any 0 lt T lt T choose x among the minimizers of 3 12 Then the map TH E a elas is locally Lipschitz in 0 7 and it holds d d x d x L Elz i DAEA e 7 1 a7 F 572 a e T 0 7 3 13 d 70 d x7 Proof Observe that from E x x2 lt E x 32 we deduce a Er X E x En SR tp 52 1 1 Ti To E x EES d x R z7 2 2 Een lt E o g Pena Deena Arguing symmetrically we see that d Er 2To d 7 T s a E r F E x T Wri 271 2T0T1 The last two inequalities show that r gt E x eae 3 13 holds is locally Lipschitz and that equation Lemma 3 10 With the same notation and assumptions as in the previous theorem T gt d T x is non decreasing and T gt E x is non increasing Also it holds d a E vaags w 3 14 T Proof Pick 0 lt To lt T lt 7 From the minimality of and we get d 75 2 d m T E O 0 lt E j 19 En HERE lt Elen HT a2 d r T E TE 1 lt E nE 0 x i 2T x 5 2T Adding up and using the fact that gt 0 we
194. tely continuous measures Let M be a Riemannian manifold u C P2 M a geodesic and assume that uo is absolutely continuous w r t the volume measure resp gives O mass to Lipschitz hypersurfaces of codimension 1 Then p is absolutely continuous w rt the volume measure resp gives O mass to Lipschitz hypersurfaces of codimension 1 for every t lt 1 In particular the set of absolutely continuous measures is geodesi cally convex and the same for measures giving O mass to Lipschitz hypersurfaces of codimension 1 Proof Assume that po is absolutely continuous let A C M be of 0 volume measure t 0 1 and let T be the optimal transport map from u to uo Then for every Borel set A C M it holds T T A D A and thus H A lt p T Ti A Ho Ti A The claims follow from the fact that T is locally Lipschitz Remark 2 27 The set of regular measures is not geodesically convex It is natural to ask whether the same conclusion of the previous proposition holds for the set of regular measures Definitions 1 25 and 1 32 The answer is not there are examples of regular measures fo p1 in P R such that the middle point of the geodesic connecting them is not regular 2 3 2 The weak Riemannian structure of 2 W2 In order to introduce the weak differentiable structure of Y2 X W2 we start with some heuristic considerations Let X R and u be a constant speed geodesic on Y2 IR induced by some optimal
195. that 41 Ty Bn Yn for every n N The inequality Ya I d a r z2x 6 Ya alrt 2 n2 x2 4 gt Wo Mi Mit1 lt n 1 shows that n gt 1 XN i X is a Cauchy sequence in L 3 X i e the space of maps f XN X such that J d f nae lt co for some and thus every ro X endowed with the distance d f g a d f y dB y Since X is complete L7 3 X is complete as well and therefore there exists a ve map 7 of the Cauchy sequence 7 Define p m3 B and notice that by 2 1 we have RT lt f a n dB 0 so that u is the limit of the Cauchy sequence un in W X W2 The fact that W X W2 is separable follows from 2 4 by considering the set of finite convex combinations of Dirac masses centered at points in a dense countable set in X with rational coefficients The last claim now follows Remark 2 8 On compactness properties of 72 X An immediate consequence of the above theorem is the fact that if X is compact then Y2 X W2 is compact as well indeed in this case the equivalence 2 4 tells that convergence in 2 X is equivalent to weak convergence It is also interesting to notice that if X is unbounded then X is not locally compact Actu ally for any measure u X and any r gt 0 the closed ball of radius r around ju is not compact To see this fix X and find a sequence xn C X such that d n oo Now define the 3again if clos
196. the time evolution of Kantorovich potentials and the Hopf Lax semigroup are discussed in detail Also when looking at geodesics in this space and in particular when the underlying metric space X is a Riemannian manifold M one is naturally lead to the so called time dependent optimal transport problem where geodesics are singled out by an action minimization principle This is the so called Benamou Brenier formula which is the first step in the interpretation of Z M as an infinite dimensional Riemannian manifold with W2 as Riemannian distance We then further exploit this viewpoint following Otto s seminal work 67 In Chapter 3 we make a quite detailed introduction to the theory of gradient flows borrowing almost all material from 6 First we present the classical theory for A convex functionals in Hilbert spaces Then we present some equivalent formulations that involve only the distance and therefore are applicable at least in principle to general metric space They involve the derivative of the distance from a point the EVI formulation or the rate of dissipation of the energy the EDE and EDI formulations For all these formulations there is a corresponding discrete version of the gradient flow formulation given by the implicit Euler scheme We will then show that there is convergence of the scheme to the continuous solution as the time discretization parameter tends to 0 The EVI formulation is the stronger one in terms of un
197. their c_ tranforms in the dual problem 1 4 Existence of optimal maps The problem of existence of optimal transport maps consists in looking for optimal plan y which are induced by a map T X gt Y i e plans y which are equal to Id T 4 for u THY and some measurable map T As we discussed in the first section in general this problem has no answer as it may very well be the case when for given u A X v P Y there is no transport map at all from yz to v Still since we know that 1 2 holds when u has no atom it is possible that under some additional assumptions on the starting measure u and on the cost function c optimal transport maps exist To formulate the question differently given u v and the cost function c is that true that at least one optimal plan is induced by a map Let us start observing that thanks to Theorem 1 13 the answer to this question relies in a natural way on the analysis of the properties of c monotone sets to see how far are they from being graphs Indeed Lemma 1 20 Let y Adm p v Then y is induced by a map if and only if there exists a y measurable set T C X x Y where y is concentrated such that for u a e x there exists only one y T x Y such that x y EI In this case y is induced by the map T Proof The if part is obvious For the only if let I be as in the statement of the lemma Possibly removing from I a product N x Y with N p negligible we can assume that I is a
198. tion of tangent space at u The idea is to think the tangent space at u as the space of directions or which is the same as the set of constant speed geodesics emanating from u More precisely let the set Geod be defined by constant speed geodesics starting from u Ix Geet and defined on some interval of the kind 0 T i where we say that u u4 provided they coincide on some right neighborhood of 0 The natural distance D on Geod is aare Wolini D u u4 Tm aA 6 1 The Geometric Tangent space Tan 2 R is then defined as the completion of Geod u Wit the distance D The natural question here is what is the relation between the space of gradients Tan 2 R and the space of directions Tan R Recall that from Remark 1 22 we know that given y CZ R1 the map t gt Id tVp 4u is a constant speed geodesic on a right neighborhood of 0 This means that there is a natural map t from the set Vy y CZ into Geod and therefore into Tan 72 R which sends Vy into the equivalence class of the geodesic t Id tVyp 4p The main properties of the Geometric Tangent space and of this map are collected in the following theorem which we state without proof Theorem 6 1 The tangent space Let u Y2 R Then e the lim in 6 1 is always a limit e the metric space Tan A2 R D is complete and separable e the map t Ve gt Tan P2 R
199. tion 3 32 Internal energy Let u 0 00 R U 00 be a convex function bounded from below such that u 0 0 and lim ule gt co for some a gt oe 44 230 2 d 2 Pen let u 00 lim _ u z z The internal energy functional E associated to u is E u f WPL o R where p pL u is the decomposition of u in absolutely continuous and singular parts w r t the Lebesgue measure 71 Condition 3 44 ensures that the negative part of u p is integrable for u A2 R so that is well defined possibly 00 Indeed from 3 44 we have u z lt az bz for some a lt 1 satisfying 2a 1 a gt d and it holds eow f EAA a fal Pde lt o enact fa ee e Under appropriate assumptions on V W and e the above defined functionals are compatible with the distance W2 As said before we will use as interpolating curves those given in Definition 3 29 Proposition 3 33 Let gt 0 The following holds i The functional V is convex along interpolating curves in P2 R2 W2 if and only if V is A convex ii The functional W is convex along interpolating curves P2 R W2 if W is convex iii The functional E is convex along interpolating curves P2 R W2 provided u satisfies z m 2tu 2z 4 is convex and non increasing on 0 00 3 45 Proof Since the second inequality in 3 29 is satisfied by the interpolating curves that we are con sidering inequality 3 43 we need only t
200. tive of the squared distance Let u and v as above and v P R Then for a e t 0 1 it holds d W3 m v 2 f vi Ti Id dpe dt where T is the unique optimal transport map from u to v which exists and is unique by Theo rem 1 26 due to our assumptions on p We conclude the section with a sketch of the proof of Theorem 2 29 Sketch of the Proof of Theorem 2 29 Reduction to the Euclidean case Suppose we already know the result for the case R and we want to prove it for a compact and smooth manifold M Use the Nash embedding theorem to get the existence of a smooth map i M R whose differential provides an isometry of T M and its image for any x M Now notice that the inequality i x i y lt d x y valid for any x y E M ensures that Wa ipu ipv lt W2 u v for any u v P2 M Hence given an absolutely continuous curve u C P2 M the curve igt C Ao R is absolutely continuous as well and there exists a family vector fields v such that 2 20 is fulfilled with 71 in place of u 46 and vzl r2 44 S ipe lt pis for a e t Testing the continuity equation with functions constant on i M we get that for a e t the vector field v is tangent to i M for i4 4 a e point Thus the v s are the isometric image of vector fields on M and part A is proved Viceversa let 4 C A M be a curve and the vs vector fields in M such that h ve z2 dt lt and assume that
201. to get 2 32 t h t h d ea d xt y i f Ev ds f Sa ws y ds lt hE y t t Let y x to obtain d ttih e lt i E a E a ds Aran nf E a E amp tsne dr IEE t 0 Now let A C 0 00 be the set of points of differentiability of t 4 E x and where t exists choose t AN a b divide by h the above inequality let h 0 and use the dominated convergence theorem to get 1g wa fF Ela Fea d 1d git S tim f h a Sre f memea geet Recalling 3 10 we conclude with Bfe sire sll SIVEP 20 mersi Finally we see how the local Lipschitz property of 2 can be achieved It is immediate to verify that the curve t x 4 is a Gradient Flow in the EVI sense starting from x for all h gt 0 We now use the fact that the distance between curves satisfying the EVI is contractive up to an exponential factor see the last part of the proof of Theorem 3 25 for a sketch of the argument and Corollary 4 3 3 of 6 for the rigorous proof We have d s Carn lt e 8 9 d T tin Vs gt t Dividing by h letting h 0 and calling B C 0 co the set where the metric derivative of x exists we obtain lts lt Jile Vs teB s gt t This implies that the curve z is locally Lipschitz in 0 00 Let us come back to the case of a convex and lower semicontinuous functional F on an Hilbert space Pick z D F fix 7 gt 0 and define the sequence n gt Tin recursively by s
202. to avoid technicalities and just focus on the main concepts For an interpolating curve as in the definition it holds W3 u lt 1 t W3 u vo tW2 u 1 t t W3 vo vi 3 43 Indeed the map 1 t To tT is optimal from p to 14 because we know that To and T are the gradients of convex functions yo p respectively thus 1 t T gt tT is the gradient of the convex function 1 t yo ty1 and thus is optimal and we know by inequality 2 1 that W2 vo v1 lt Zo Ti 72 gt thus it holds W3 m 0 t To tT lieg 1 t To Tdl 72q HIT IdllZ2q t AIT Ti lleg lt 1 t W3 u vo tW3 u v1 t 1 t W3 vo 1 We now pass to the description of the three functionals we want to study Definition 3 30 Potential energy Let V R RU 00 be lower semicontinuous and bounded from below The potential energy functional V P2 IR R U 00 associated to V is defined by V u fva Definition 3 31 Interaction energy Let W R RU 00 be lower semicontinuous even and bounded from below The interaction energy functional W P2 R R U 00 associated to W is defined by W t 5 We x2 du X x1 2 Observe that the definition makes sense also for not even functions W however replacing if neces sary the function W x with W x W 2 2 we get an even function leaving the value of the functional unchanged Defini
203. to covariantly differentiate vector fields but we don t know either which are the vector fields regular enough to be differentiated In a purely Riemannian setting this problem does not appear as a Riemannian man ifold borns as smooth manifold on which we define a scalar product on each tangent space but the space 9 IR does not have a smooth structure there is no diffeomorphism of a small ball around the origin in Tan A2 R onto a neighborhood of in Y2 IR Thus we have to proceed in a different way which we describe now Regular curves first of all we drop the idea of defining a smooth vector field on the whole mani fold We will rather concentrate on finding an appropriate definition of smoothness for vector fields defined along curves We will see that to do this we will need to work with a particular kind of curves which we call regular see Definition 6 2 Smoothness of vector fields We will then be able to define the smoothness of vector fields defined along regular curves Definition 6 5 Among others a notion of smoothness of particular relevance is that of absolutely continuous vector fields for this kind of vector fields we have a natural notion of total derivative not to be confused with the covariant one see Definition 6 6 Levi Civita connection At this point we have all the ingredients we need to define the covariant derivative and to prove that it is the Levi Civita connection on Y2 R Definiton 6 8
204. to modify the transportation distance in order to take into account measures with unequal masses and such that the Gradient Flow of the entropy 82 functional produces solutions of the Heat equation in Q with Dirichlet boundary conditions This is actually doable as we briefly discuss now Let Q C R be open and bounded Consider the set M2 Q defined by Mo Q measures u on Q such that J Paana lt and for any u v Mo Q define the set of admissible transfer plans Adm p v by y Adm u v if and only if y is a measure on Q such that THY g Hs THI q U Notice the difference w r t the classical definition of transfer plan here we are requiring the first respectively second marginal to coincide with u respectively v only inside the open set Q This means that in transferring the mass from js to v we are free to take put as much mass as we want from to the boundary Then one defines the cost C of a plan y by Cy i le yPay e and then the distance W bz by Wbo p v inf C y where the infimum is taken among all y Admp pu v The distance Wb shares many properties with the Wasserstein distance W2 Theorem 5 5 Main properties of Wb2 The following hold e Wb is a distance on M2 Q and the metric space M2 Q Wb2 is Polish and geodesic e A sequence un C Mo Q converges to u wrt Wo if and only if un converges weakly to u in duality with continuous functions with compact su
205. tric properties of the base space X d For this reason we split the foregoing discussion into three sections on which we deal with the cases in which X is a general Polish space a geodesic space and a Riemannian manifold A word on the notation when considering product spaces like X with 7 X X we intend the natural projection onto the i th coordinate i 1 n Thus for instance for u v A X and y Adm u v we have myy p and T3 Yy v Similarly with 1 X X we intend the projection onto the i th and j th coordinates And similarly for multiple projections 2 1 X Polish space Let X d be a complete and separable metric space The distance Wz is defined as Wo u v eit J Panare uw j I P x y dy e y YY Ope m v The natural space to endow with the Wasserstein distance W3 is the space 2 X of Borel 24 Probability measures with finite second moment P X u P X P zodno lt oo for some and thus any o x Notice that if either u or v is a Dirac delta say v 6 then there exists only one plan y in Adm u v the plan u x s which therefore is optimal In particular it holds d x x0 du x W3 1 dro that is the second moment is nothing but the squared Wasserstein distance from the corresponding Dirac mass We start proving that W2 is actually a distance on 4 X In order to prove the triangle inequal ity we will use the follow
206. ts yields that the sequence n pin is tight Thus to prove narrow convergence it is sufficient to check that f fdjin gt f fdu for every f C X Since Lipschitz functions are dense in Ce X w r t uniform convergence it is sufficient to check the convergence of the integral only for Lipschitz f s This follows from the inequality f ran f tam fto fody ey lt f fle foldres lt Lip f f d x y dy 0 4 lt Lint f d x y dyn 2 y Lip f W2 1 ttn 29 Since the argument does not depend on the subsequence chosen the claim is proved We pass to the converse implication in 2 4 Pick y E Opt 1 fn and use Remark 1 4 to get that the sequence v7 is tight hence up to passing to a subsequence we can assume that it narrowly converges to some y By Proposition 2 5 we know that y Opt u u which forces J x y dy x y 0 By Proposition 2 4 and our assumption on un x we know that un is 2 uniformly integrable thus by Remark 2 3 again we know that y is 2 uniformly integrable as well Since the map x y gt d a y has quadratic growth in X it holds lim W3 lin 4 lim feeder f Pyare 0 Now we prove that P X W2 is complete Pick a Cauchy sequence un and assume with out loss of generality that 7 W2 Hn Hn 1 lt For every n N choose Opt Hn Hn 1 and use repeatedly the gluing lemma to find for every n N a measure 3 42 X such
207. tu1 C Y we know that for any geodesic y supp fz it holds yo 71 Y Since Y is totally convex this implies that y Y for any t and any y supp jt i e p ec 4 E P Y Therefore u is a geodesic connecting Ho to u1 in Y d Conclude noticing that for any u 2 Y it holds du du _ i du du E lo lt 4 dmy log m Y Tm log TA dm du 1 57 1 du 1 57 f a dmy mv a where we wrote my for m Y mj ii Fix a gt 0 and let d ad and W2 be the Wasserstein distance on A X induced by the 116 distance d It is clear that a plan y Adm u v is optimal for the distance W2 if and only if it is optimal for W3 thus W2 aW2 Now pick pio p E A X and let uz C A X be a constant speed geodesic connecting them such that Bo He lt 1 1 8 0 8 tu C tW uo m then it holds Bolu lt 1 1 8 uo tEn ztl HW 10 1 and the proof is complete A similar argument applies for the case C D 0 N For Ao A C X we define Ao Aij C X as Ao Ail TO yis a constant speed geodesic such that y 0 Ao y 1 Ay Observe that if Ao Ai are open resp compact Ao A is open resp compact hence Borel Proposition 7 14 Brunn Minkowski Let X d m be a metric measure space and Ag Ay C supp m compact subsets Then i if X d m isa CD K co space it holds log na Ao Ai e 1 log m 40 tlog m 41 t1 1 D 4o 41 7
208. u Tan A2 R there exists at most one parallel transport u along u satisfying up u Thus the problem is to show the existence There is an important analogy which helps under standing the proof that we want to point out we already know that the space A2 IR W2 looks like a Riemannian manifold but actually it has also stronger similarities with a Riemannian manifold M embedded in some bigger space say on some Euclidean space RP indeed in both cases e we have a natural presence of non tangent vectors elements of L Tan 2 R for P R and vectors in R non tangent to the manifold for the embedded case e The scalar product in the tangent space can be naturally defined also for non tangent vectors scalar product in r for the space IR and the scalar product in R for the embedded case This means in particular that there are natural orthogonal projections from the set of tangent and non tangent vectors onto the set of tangent vectors P L gt Tan 22 R9 for P2 R and P RP T M for the embedded case e The Covariant derivative of a tangent vector field is given by projecting the time derivative onto the tangent space Indeed for the space Yz IR we know that the covariant derivative is given by formula 6 8 while for the embedded manifold it holds d Vaut Py Zu l 6 11 where t gt y is a smooth curve and t gt u T M is a smooth tangent vector field
209. v shows that if K C A X and K2 C A Y are tight then so is the set y P X xY THEY EK TY Ka Existence of minimizers for Kantorovich s formulation of the transport problem now comes from a standard lower semicontinuity and compactness argument Theorem 1 5 Assume that c is lower semicontinuous and bounded from below Then there exists a minimizer for Problem 1 2 Proof Compactness Remark 1 4 and Ulam s theorem show that the set Adm y v is tight in A X x Y and hence relatively compact by Prokhorov theorem To get the narrow compactness pick a sequence 7 C Adm u v and assume that y gt Y narrowly we want to prove that y Adm u v as well Let y be any function in C X and notice that x y p x is continuous and bounded in X x Y hence we have eaky vadra tim o e ay e y im f odr i f edu so that by the arbitrariness of y C X we get THY u Similarly we can prove my which gives y Adm u v as desired Lower semicontinuity We claim that the functional y gt f cdy is l s c with respect to narrow convergence This is true because our assumptions on c guarantee that there exists an increasing sequence of functions cn X x Y R continuous an bounded such that c x y sup Cn y so that by monotone convergence it holds fedty sup f endy Since by construction y f cn dy is narrowly continuous the proof is complete We will denote by Opt u v the
210. ve u via d at V veu 0 v Tanp Po R a e t 69 recall Theorem 2 29 and Definition 2 31 Thus we have a total of 4 different formulations of Gradient Flows of A geodesically convex func tionals on Y2 IR based respectively on the Energy Dissipation Inequality the Energy Dissipation Equality the Evolution Variational Inequality and the notion of subdifferential The important point is that these 4 formulations are equivalent for A geodesically convex func tionals Proposition 3 28 Equivalence of the various formulation of GF in the Wasserstein space Let E be a geodesically convex functional on P2 R and u a curve made of regular measures Then for u the 4 definitions of Gradient Flow for E EDI EDE EVI and the Subdifferential one are equivalent Sketch of the Proof We prove only that the EVI formulation is equivalent to the Subdifferential one Recall that by Proposition 2 34 we know that ld y 5 g2 Hov f oT Id djn a e t where T is the optimal transport map from u to v Then we have u OW E u a e t T E u l v Th Id du 3 W2 ev lt E v Ww P2 R a e t gt ld r E u z g2 Hov 3 W2 wv lt E v Ww Pa R a e t 3 3 2 Three classical functionals We now pass to the analysis of 3 by now classical examples of Gradient Flows in the Wasserstein space Recall that in terms of strength the best theor
211. vepe 0 dt with p p and p ut A natural generalization of the distance W gt comes by considering a new action modified by putting a weight on the density that is given a smooth function A 0 co 0 00 we define 1 We p Lt pL int f J Phoa 5 1 0 81 where the infimum is taken among all the distributional solutions of the non linear continuity equa tion d a V u h pz 0 5 2 with p p and p ft The key assumption that leads to the existence of an action minimizing curve is the concavity of h since this leads to the joint convexity of Giai 5 h p so that using this convexity with J vh p one can prove existence of minima of 5 1 Particularly important is the case given by h z z for lt 1 from which we can build the distance Wa defined by 1 1 a Wal P Zt p1 Z in ee e p eyae a at 5 3 0 the infimum being taken among all solutions of 5 2 with p p and p pt The following theorem holds Theorem 5 4 Let a gt 1 i Then the infimum in 5 3 is always reached and if it is finite the minimizer is unique Now fix a measure u Y R The set of measures v with Wa u v lt endowed with Wa is a complete metric space and bounded subsets are narrowly compact We remark that the behavior of action minimizing curves in this setting is in some very rough sense dual of the behavior of the branched optimal tran
212. vl p From this lemma and the inequality Lip T s BAS 1d lt elie Mip ar pg Vt 0 1 fh Tapa t whose simple proof we omit where C efo Lip vr ar Lip v dr t Lip v dr t These inequalities are perfectly analogous to the 6 12 well the only difference is that here the bound on the angle is L in t s while for the embedded case it was L but this does not really change anything Therefore the arguments presented before apply also to this case and we can derive the existence of the parallel transport along regular curves 1 it is immediate to verify that it holds luo T s t P ullas lt Cllulla u Tany 2P2 R 6 20 PEW llus lt Clellue we Tan P2 R 97 Theorem 6 15 Parallel transport along regular curves Let u be a regular curve and u Tanpo Z2 R2 Then there exists a parallel transport uz along u such that ug v Now we know that the parallel transport exists along regular curves and we know also that regular curves are dense it is therefore natural to try to construct the parallel transport along any absolutely continuous curve via some limiting argument However this cannot be done as the fol lowing counterexample shows Example 6 16 Non existence of parallel transport along a non regular geodesic Let Q 0 1 x 0 1 be the unit square in R and let T i 1 2 3 4 be the four open tr
213. volume measure u 0 00 be convex continuous and C on 0 00 with u 0 0 and define the pressure p 0 00 gt R by p z zu z u z Vz gt 0 and p 0 0 Also let u pm E YS M with p C M pick p CS M and define T M gt M by T x exp tVp x Then it holds Salif Tan PO 0 Be p o Ag Vo Ric Vo Ve dm where by V2y 2 we mean the trace of the linear map Vp x TM gt T M in coordi nates this reads as 0ij9 Proof Computation of the second derivative Let D x det VTi yt Ti u prVol By compactness for t sufficiently small T is invertible with smooth inverse so that D p E C M For small t the change of variable formula gives p t p pe Te x det VT x D x Thus we have all the integrals being w r t m d d p p pD p f p aes Lps l D 4 D D zJ lt faf i f 5 2 DTU D t PAD O lao feo Feo P 2 D Joo p p Do having used the fact that Dp 1 Evaluation of Dj and Dj We want to prove that D x Ay c Di x Ay 2 V p x Ric Ve e Vo 2 For t gt 0 and x M let J a be the operator from Ty M to Tox evy x M given by 7 7 J a v the value at s t of the Jacobi field js along the geodesic s exp sVy z having the initial conditions jo v jj V7y 0 where here and in the following the a
214. which are a metric analogous of Riemannian manifolds with bounded either from above or from below sectional curvature What became clear over time is that the correct non smooth object where one could try to give a notion of Ricci curvature bound is not a metric space but rather a metric measure space i e a metric space where a reference non negative measure is also given When looking to the Riemannian case this fact is somehow hidden as a natural reference measure is given by the volume measure which is a function of the distance There are several viewpoints from which one can see the necessity of a reference measure which can certainly be the Hausdorff measure of appropriate dimension if available A first cheap one is the fact that in most of identities inequalities where the Ricci curvature appears also the reference measures appears e g equations 7 1 7 2 and 7 3 above A more subtle point of view comes from studying stability issues consider a sequence Mn gn of Riemannian manifolds and assume that it converges to a smooth Riemannian manifold M g in the Gromov Hausdorff sense Assume that the Ricci curvature of Mn gn is uniformly bounded below by some K R Can we deduce that the Ricci curvature of M g is bounded below by K The answer is no while the same question with sectional curvature in place of Ricci one has affirmative answer It is possible to see that when Ricci bounds are not preserved in the limitin
215. y 7 14 and the fact that m x 0 proved in Proposition 7 15 we know that lim m A m Br z2 which means that m a e point in Br x is connected to x by a unique geodesic Since R and x are arbitrary uniqueness is proved The measurability of the map x y gt y is then a consequence of uniqueness of Lemma 2 11 and classical measurable selection results which ensure the existence of a measurable selection of geodesics in our case there is m x m almost surely no choice so the unique geodesic selection is measurable Corollary 7 17 Compactness Let N D lt co Then the family X N D of isomorphism classes of metric measure spaces X d m satisfying the condition CD 0 N with diameter bounded above by D is compact w rt the topology induced by D Sketch of the Proof Using the Bishop Gromov inequality with R D we get that m Bz 2 gt ew Y X d m X N D supp mx 7 17 Thus there exists n N D which does not depend on X X N D such that we can find at most n N D disjoint balls of radius in X Thus supp m x can be covered by at most n N D balls of radius 2e This means that the family N D is uniformly totally bounded and thus it is compact w r t Gromov Hausdorff convergence see e g Theorem 7 4 5 of 20 Pick a sequence Xn dn Mn X N D By what we just proved up to pass to a subsequence not relabeled we may assume that supp m dn
216. y with n large to get that the infimum is oo Thus we proved that imt f e esydr e y sup padula warty YE Adm p v yw where the supremum is taken among continuous and bounded functions y y satisfying 1 5 We now give the rigorous statement and a proof independent of the min max principle Theorem 1 17 Duality Let u A X v A Y andc X xY gt Ra continuous and bounded from below cost function Assume that 1 4 holds Then the minimum of the Kantorovich problem 1 2 is equal to the supremum of the dual problem 1 16 Furthermore the supremum of the dual problem is attained and the maximizing couple p is of the form p p for some c concave function p Proof Let y Adm u v and observe that for any couple of functions y L u and Y L v satisfying 1 5 it holds J dendre gt f oe odre n f oleae f vend This shows that the minimum of the Kantorovich problem is gt than the supremum of the dual prob lem To prove the converse inequality pick y Opt p v and use Theorem 1 13 to find a c concave function such that supp y C 0 y max y 0 L u and max y 0 L1 v Then as in the proof of iii i of Theorem 1 13 we have eenerten f oote oden f ole dle f ow arty and f cdy R Thus y L y and y L v which shows that p y is an admissible couple in the dual problem and gives the thesis Remark 1 18 Notice t
217. y to use is the one of Subsection 3 2 4 be cause the compatibility in Energy and distance ensures strong properties both at the level of discrete solutions and for the limit curve obtained Once we will have a Gradient Flow the Subdifferential formulation will let us understand which is the PDE associated to it Let us recall Example 2 21 that the space 2 R7 W2 is not Non Positively Curved in the sense of Alexandrov this means that if we want to check whether a given functional is compatible with the distance or not we cannot use geodesics to interpolate between points because we would violate the second inequality in 3 29 A priori the choice of the interpolating curves may depend on the functional but actually in what comes next we will always use the ones defined by Definition 3 29 Interpolating curves Let u vo v E Y2 R and assume that u is regular Def inition 1 25 The interpolating curve 1 from vo to v with base u is defined as v 1 t To tT eu where To and T are the optimal transport maps from n to vo and v respectively Observe that if L v the interpolating curve reduces to the geodesic connecting it to v 70 Strictly speaking in order to apply the theory of Section 3 2 4 we should define interpolating curves having as base any measure u IR and not just regular ones This is actually possible and the foregoing discussion can be applied to the more general definition but we prefer

A user's guide to optimal transport

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