Monge problem in metric measure spaces with Riemannian curvature-dimension condition

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Abstract

We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X,d,m) enjoying the Riemannian curvature-dimension condition RCD(K,N), with N<. For the first marginal measure, we assume that μ0m. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge–Kantorovich problem, attain the same minimal value.

Moreover we prove a structure theorem for d-cyclically monotone sets: neglecting a set of zero m-measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.

Introduction

Let (X,d,m) be a metric measure space verifying the Riemannian curvature dimension condition RCD(K,N) for K,NR with N1. In this note we prove the existence of a solution for the following Monge problem: given μ0,μ1P(X), the space of Borel probability measures over X, solve the following minimization problem infTμ0=μ1Xd(x,T(x))μ0(dx), provided μ0m. In more detail, the minimization of the functional runs over the set of μ0-measurable maps T:XX such that Tμ0=μ1, that is μ0(T1(A))=μ1(A),AB(X), where B(X) denotes the σ-algebra of all Borel subsets of X.

On the way to the proof of the existence of an optimal map, we will also prove a structure theorem for branching structures inside d-cyclically monotone sets. Before giving the statements of the two main results of this note and an account on the strategies to prove them, we recall some of the (extensive) literature on the Monge minimization problem.

The first formulation for (1.1) (Monge in 1781) was addressed in Rn with the cost given by the Euclidean norm and the measures μ0,μ1Ln were supposed to be supported on two disjoint compact sets. The original problem remained unsolved for a long time. In 1978 Sudakov in  [1] proposed a solution for any distance cost induced by a norm, but an argument about disintegration of measures contained in his proof was not correct, see  [2] for details. Then the Euclidean case was correctly solved by Evans and Gangbo in  [3], under the assumptions that sptμ0sptμ1=, μ0,μ1Ln and their densities are Lipschitz functions with compact support. After that, many results reduced the assumptions on the supports of μ0,μ1, see  [4], [5]. The result on manifolds with geodesic cost is obtained in  [6]. The case of a general norm as cost function on Rn has been solved first in the particular case of crystalline norms in  [7], and then in full generality independently by L. Caravenna in  [8] and by T. Champion and L. De Pascale in [9].

The study of the geodesic metric space framework started with  [10], where the metric space was assumed to be also non-branching. There the existence of solutions to (1.1) was obtained for metric spaces verifying the measure-contraction property MCP(K,N) (for instance the Heisenberg group). An application of the results of  [10] to the Wiener space can be found in  [11]. Then in  [12] the problem was studied removing the non-branching assumption but obtaining existence of solutions only in a particular case.

Non-branching metric measure spaces enjoying CD(K,N) also verify MCP(K,N), see  [13]. Then from  [10] the Monge problem is solved also in that case. So with respect to the most general known case, we impose a stronger curvature information (namely RCD(K,N)) and we remove the non-branching assumption.

The nowadays classical strategy to show existence of optimal maps is to relax the integral functional to the larger class of transport plans Π(μ0,μ1):={πP(X×X):(P1)π=μ0,(P2)π=μ1}, over where the functional we want to minimize has now the following expression d(x,y)η(dxdy). For i=1,2, Pi:X×XX denotes the projection map on the ith component. Assuming that the functional is finite at least on one element of Π(μ0,μ1), we have the existence of ηoptΠ(μ0,μ1) so that d(x,y)ηopt(dxdy)=infηΠ(μ0,μ1)Xd(x,y)η(dx), by linearity in η and tightness of Π(μ0,μ1). Then the central question, whose positive answer would prove existence of a solution to Monge problem, is whether ηopt is supported on the graph of a m-measurable map T:XX.

A property of ηopt inside Π(μ0,μ1) is the fact that is concentrated on a d-cyclically monotone set. We shall build an optimal map starting from this monotonicity. But while the Riemannian curvature-dimension condition RCD(K,N) gives crucial information on d2-cyclically monotone sets (neglecting a set of measure zero, they are the graph of a measurable map, see Section  2 and references therein), nothing is known under this curvature assumption on the structure of d-cyclically monotone sets. In particular what we would like to exclude is the presence of branching structures. Note that the first result proving absence of branching geodesics assuming a curvature condition, in that case strong CD(K,), is contained in  [14]. The same type of result, but only for L2-Wasserstein geodesics with end point a Dirac delta, was already present in an earlier work of Rajala, see  [15].

The strategy we will follow is: prove that d-cyclically monotone sets do not have branching structures m-almost everywhere; then use the approach with Disintegration Theorem (see for instance  [10] and references therein) to reduce the Monge problem to a family of 1-dimensional Monge problem. There one can apply the 1-dimensional theory. Thanks to the curvature assumption we can prove a suitable property for the first marginal measures and obtain the existence of the 1-dimensional optimal maps, one for each 1-dimensional Monge problem. Then gluing together all the one-dimensional optimal maps, one gets an optimal map T:XX solving the Monge problem (1.1). A more precise program on the use of Disintegration Theorem in the Monge problem will be given in Section  3.

We conclude this introductory part stating the two main results we will prove. The first is about the structure of the d-cyclically monotone set associated to a Kantorovich potential φd for the problem (1.1).

Theorem 1.1

Let (X,d,m) be a metric measure space verifying RCD(K,N) for some K,NR, with N1 . Let moreover Γ be a d-cyclically monotone set as   (3.1)   and let Te be the set of all points moved by Γ as in   Definition  3.2. Then there exists TTethat we call the transport set such thatm(TeT)=0,and for all xT, the transport ray R(x) is formed by a single geodesic and for xy, both in T, either R(x)=R(y) or R(x)R(y) is contained in the set of initial points ab as defined in   Definition  3.2.

All the terminology used in Theorem 1.1 will be introduced in Section  3. Taking advantage of Theorem 1.1 we then obtain the following.

Theorem 1.2

Let (X,d,m) be a metric measure space verifying RCD(K,N) for N< . Let μ0,μ1P(X) with W1(μ0,μ1)< and μ0m . Then there exists a Borel map T:XX such that Tμ0=μ1 andXd(x,T(x))μ0(dx)=X×Xd(x,y)ηopt(dxdy).

In the previous theorem, W1 denotes the L1-Wasserstein distance on the space of probability measures on (X,d).

A straightforward corollary of Theorem 1.2 is that the relaxation to the set of transference plan Π(μ0,μ1) does not lower the value of the minimum: infTμ0=μ1Xd(x,T(x))μ0(dx)d(x,T(x))μ0(dx)=minηΠ(μ0,μ1)Xd(x,y)η(dx)infTμ0=μ1Xd(x,T(x))μ0(dx). Hence minTμ0=μ1Xd(x,T(x))μ0(dx)=minηΠ(μ0,μ1)Xd(x,y)η(dx).

As it will be clear from their proofs, the results contained in Theorem 1.1, Theorem 1.2 can be obtained omitting the RCD(K,N) condition and assuming instead the metric measure space to satisfy the strong CD(K,N) condition. Even if strong CD(K,N) is a more general condition than RCD(K,N), the latter is stable with respect to measured Gromov–Hausdorff convergence. Hence we have decided in its favor to state and prove the results contained in this note.

The author wishes to thank Tapio Rajala for a discussion on an early version of this note. He is also in debt with the anonymous referees whose suggestions improved the paper.

Section snippets

RCD(K,N) spaces

Here we briefly give some references for RCD(K,N) and state some of the main properties of metric measure spaces verifying it.

Few notations: we will denote with Geo(X)C([0,1],X) the space of geodesics endowed with uniform topology and for a Borel set FX×X, we will often use the notation F(x) for P2(F{x}×X).

For μ0,μ1P2(X), the space of probability measures with finite second moment, we consider the following set of optimal geodesics: OptGeo(μ0,μ1):={νP(Geo(X)):d(x,y)2(es,et)ν=(ts)2W22(μ0

d-geodesics and d2-geodesics

To avoid the trivial case we can assume that the two marginal measures have finite L1-Wasserstein distance, W1(μ0,μ1)<. Consequently we infer the existence of ηΠ(μ0,μ1), such that X×Xd(x,y)η(dxdy)=inf{X×Xd(x,y)π(dxdy):πΠ(μ0,μ1)}=W1(μ0,μ1), where Π(μ0,μ1) is the set of transport plans, Π(μ0,μ1):={πP(X×X):(P1)π=μ0,(P2)π=μ1}.

The set of optimal transport plans, i.e. plans realizing the previous identity, will be denoted with Πopt(μ0,μ1). Since the cost is finite, we can also assume the

The transport set

We now prove that the set of transport rays R is an equivalence relation on a subset of Te. In order to do so, we study the branching geodesics in Γ. The presence of branching structures inside Γ can be modeled by the existence of x,z,wTe such that (x,z),(x,w)Γ,(z,w)R. Actually the previous condition only describes branching in the direction given by Γ. Branching in the direction of Γ1 will be treated analogously.

In the next lemma, using Lemma 3.1, we prove that, once a branching happens,

Structure of d-monotone sets

Theorem 4.6 says that the right set to look at in order to perform a reduction of the Monge problem to a family of 1-dimensional Monge problem is T:=Te(A+A). We will refer to T as the transport set. From now on we will always assume (X,d,m) to satisfy RCD(K,N).

The next step is to show that each equivalence class of R is formed by a single geodesic.

Lemma 5.1

Fix any xT . Then for any z,wR(x)there exists γGGeo(X) such that{x,z,w}{γs:s[0,1]}.If γˆG enjoys the same property, then({γˆs:s[0,1]}{γs:

Regularity of disintegration

Now we show that for q-a.e. ySmy(g(y,))L1.

Property (6.1) is linked to the behavior in time of the measure of evolving subsets of T, where the “evolving subsets” has to be made precise.

Since in RCD(K,N)-spaces a concavity estimate for densities of L2-geodesics in P2(X,d,m) holds, it is natural to look for a definition of evolution inside the transport set where an L2-structure can come into play.

Lemma 6.1

For each CT and δRthe set(C×{φd=δ})Γ,is d2-cyclically monotone.

Proof

The proof follows from

Existence of solution to the Monge problem

Using Theorem 6.6 we prove the existence of an m-measurable map Tˆ:XX such that Xd(x,Tˆ(x))μ0(dx)=infTμ0=μ1Xd(x,T(x))μ0(dx), with Tˆμ0=μ1, provided μ0 is absolutely continuous with respect to m. So assume μ0=ϱ0m.

Justified by Lemma 3.4, extension (3.2) and Proposition 4.5, we assume that μ0(T)=μ1(Te)=1. Then (5.1) gives that μ0=ϱ0mT=Sϱ0mαq(dα)=Sμ0,yqμ0(dy), where μ0,y=c(y)ϱ0mα with c(y) normalizing constant, and qμ0=c(y)1q.

Since R is an equivalence relation only on T and a priori μ1(TeT

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