Monge problem in metric measure spaces with Riemannian curvature-dimension condition
Introduction
Let be a metric measure space verifying the Riemannian curvature dimension condition for with . In this note we prove the existence of a solution for the following Monge problem: given , the space of Borel probability measures over , solve the following minimization problem provided . In more detail, the minimization of the functional runs over the set of -measurable maps such that , that is where denotes the -algebra of all Borel subsets of .
On the way to the proof of the existence of an optimal map, we will also prove a structure theorem for branching structures inside -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 with the cost given by the Euclidean norm and the measures 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 , and their densities are Lipschitz functions with compact support. After that, many results reduced the assumptions on the supports of , see [4], [5]. The result on manifolds with geodesic cost is obtained in [6]. The case of a general norm as cost function on 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 (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 also verify , 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 ) 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 over where the functional we want to minimize has now the following expression For , denotes the projection map on the th component. Assuming that the functional is finite at least on one element of , we have the existence of so that by linearity in and tightness of . Then the central question, whose positive answer would prove existence of a solution to Monge problem, is whether is supported on the graph of a -measurable map .
A property of inside is the fact that is concentrated on a -cyclically monotone set. We shall build an optimal map starting from this monotonicity. But while the Riemannian curvature-dimension condition gives crucial information on -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 -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 , is contained in [14]. The same type of result, but only for -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 -cyclically monotone sets do not have branching structures -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 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 -cyclically monotone set associated to a Kantorovich potential for the problem (1.1).
Theorem 1.1 Let be a metric measure space verifying for some , with . Let moreover be a -cyclically monotone set as (3.1) and let be the set of all points moved by as in Definition 3.2. Then there exists that we call the transport set such thatand for all , the transport ray is formed by a single geodesic and for , both in , either or is contained in the set of initial points 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 be a metric measure space verifying for . Let with and . Then there exists a Borel map such that and
In the previous theorem, denotes the -Wasserstein distance on the space of probability measures on .
A straightforward corollary of Theorem 1.2 is that the relaxation to the set of transference plan does not lower the value of the minimum: Hence
As it will be clear from their proofs, the results contained in Theorem 1.1, Theorem 1.2 can be obtained omitting the condition and assuming instead the metric measure space to satisfy the strong condition. Even if strong is a more general condition than , 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
spaces
Here we briefly give some references for and state some of the main properties of metric measure spaces verifying it.
Few notations: we will denote with the space of geodesics endowed with uniform topology and for a Borel set , we will often use the notation for .
For , the space of probability measures with finite second moment, we consider the following set of optimal geodesics:
-geodesics and -geodesics
To avoid the trivial case we can assume that the two marginal measures have finite -Wasserstein distance, . Consequently we infer the existence of , such that where is the set of transport plans,
The set of optimal transport plans, i.e. plans realizing the previous identity, will be denoted with . Since the cost is finite, we can also assume the
The transport set
We now prove that the set of transport rays is an equivalence relation on a subset of . In order to do so, we study the branching geodesics in . The presence of branching structures inside can be modeled by the existence of such that Actually the previous condition only describes branching in the direction given by . Branching in the direction of will be treated analogously.
In the next lemma, using Lemma 3.1, we prove that, once a branching happens,
Structure of -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 We will refer to as the transport set. From now on we will always assume to satisfy .
The next step is to show that each equivalence class of is formed by a single geodesic.
Lemma 5.1 Fix any . Then for any there exists such thatIf enjoys the same property, then
Regularity of disintegration
Now we show that for -a.e.
Property (6.1) is linked to the behavior in time of the measure of evolving subsets of , where the “evolving subsets” has to be made precise.
Since in -spaces a concavity estimate for densities of -geodesics in holds, it is natural to look for a definition of evolution inside the transport set where an -structure can come into play.
Lemma 6.1 For each and the setis -cyclically monotone.
Proof The proof follows from
Existence of solution to the Monge problem
Using Theorem 6.6 we prove the existence of an -measurable map such that with , provided is absolutely continuous with respect to . So assume .
Justified by Lemma 3.4, extension (3.2) and Proposition 4.5, we assume that . Then (5.1) gives that where with normalizing constant, and .
Since is an equivalence relation only on and a priori
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