Monge-Kantorovitch Measure Transportation and Monge-Ampère Equation on Wiener Space

Abstract

Let (W, µ, H ) be an abstract Wiener space assume two ν i , i = 1, 2 probabilities on (W, B(W )) 1 . We give some conditions for the Wasserstein distance between ν 1 and ν 2 with respect to the Cameron-Martin spaceto be finite, where the infimum is taken on the set of probability measures β on W × W whose first and second marginals are ν 1 and ν 2 . In this case we prove the existence of a unique (cyclically monotone) map T = I W + ξ , with ξ : W → H , such that T maps ν 1 to ν 2 . Moreover, if ν 2 µ 2 , then T is stochastically invertible, i.e., there exists S : W → W such that S • T = I W ν 1 a.s. and T • S = I W ν 2 a.s. If, in addition, ν 1 = µ, then there exists a 1-convex function φ in the Gaussian Sobolev space D 2,1 , such that ξ = ∇φ. These results imply that the quasi-invariant transformations of the Wiener space with finite Wasserstein distance from µ can be written as the composition of a transport map T and a rotation, i.e., a measure preserving map. We give also 1-convex sub-solutions and Ito-type solutions of the Monge-Ampère equation on W .

Existence and stability for Fokker–Planck equations with log-concave reference measure

“…We shall denote this optimal transport map by t ν µ . This is one of the infinite-dimensional generalizations (see also [15] for another result in Wiener spaces) of the finite-dimensional result ensuring that whenever µ ∈ P 2 (R k ) is absolutely continuous with respect to L k , then there exists a unique optimal transport map that is also the gradient of a convex function.…”

Sampling via Measure Transport: An Introduction

“…Of particular interest is the result of [15], later extended by [55], which shows that when c(x, T (x)) is quadratic and µ ref is atomless, the optimal transport map exists and is unique; moreover this map is the gradient of a convex function and thus is monotone. Generalizations of this result accounting for different cost functions and spaces can be found in [19,2,27,11]. For a thorough contemporary development of optimal transport we refer to [88,87].…”

“…Thus, z * can be computed recursively via a sequence of n one-dimensional root-finding problems. Monotonicity of the triangular maps guarantees that (27) has a unique real solution for each k and any given x * . Here, one can use any off-the-shelf rootfinding algorithm [65].…”

“…to clean up the approximation of the root z * obtained from the recursive algorithm (27). An alternative way to evaluate the direct transport is to build a parametric representation of T itself via standard regression techniques.…”

Integrability of Absolutely Continuous Transformations of Measures and Applications to Optimal Mass Transportation

“…In particular, fruitful relations between these problems, nonlinear transformations of measures, nonlinear differential equations and nonlinear functional inequalities have been revealed (see [1], [3], [4], [8], [9], [10], [11], [12], [13], [14], [15], [17], [18], [20], [21], [23], [24], [26], [28], [29], where one can find additional references). We recall that the general Monge problem deals with measurable mappings T from a given probability space (X, A, µ) to a probability space (Y, B, ν) that transform µ into ν and minimize the integrals In this work, we consider a more general problem when a mapping T = I + F , where I(x) = x, transforms a measure µ into a measure g · µ and one is interested in the integrability of functions of |F | under appropriate integrability assumptions on g. An interesting result in this direction is due to Fernique [16] who considered a Gaussian measure γ on a separable Fréchet space and a probability measure g·γ such that g ∈ L p (γ) for some p > 1.…”