The strong solution of the Monge-Ampère equation on the Wiener space for log-concave densities
type: Item
instance of: scholarly article
title: The strong solution of the Monge-Ampère equation on the Wiener space for log-concave densities
zbMATH Open document ID: 1051.60070
author: Denis Feyel
author: A. S. Üstünel
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2004-08-20T00:00:00Z
review text: The explicit form of the strong solution of the Monge-Ampère equation on the Wiener space \((W, H, \mu)\) is given. More precisely, let \(\nu\) be a measure of the form \(d\nu = \frac{e^{-f}}{c} d\mu \), \(f\) is lower bounded, \(H\)-convex. It is proved that the potential of transport \(\mu\) to \(\nu\), solving Monge-Kantorovich transportation problem [see the authors, Probab. Theory Relat. Fields 128, 347--385 (2004; Zbl 1055.60052)] belongs to \(\mathbb{D}_{2,2}\). This enables to prove that the solution of the Monge problem of transporting \(\mu\) to \(\nu\) realizing the Wasserstein distance between \(\mu\) and \(\nu\) is the strong solution of the Monge-Ampère equation on \(W\). As a corollary the explicit formula for the Wasserstein distance is obtained.
reviewed by: Jacek Jakubowski
Mathematics Subject Classification ID: 60H30
Mathematics Subject Classification ID: 60H07
Mathematics Subject Classification ID: 46N30
Mathematics Subject Classification ID: 46F25
zbMATH DE Number: 2094018
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: Wiener space
zbMATH Keywords: Monge-Kantorovich transportation problem
zbMATH Keywords: Wasserstein distance
MaRDI profile type: Publication
full work available at URL: j.crma.2004.04.013
OpenAlex ID: W2002878397
cites work: Q3658220
cites work: Polar factorization and monotone rearrangement of vector‐valued functions
cites work: The Regularity of Mappings with a Convex Potential
cites work: Monotonicity properties of optimal transportation and the FKG and related inequalities
cites work: Q5336643
cites work: Gaussian capacities
cites work: The notion of convexity and concavity on Wiener space
cites work: Measure transport on Wiener space and the Girsanov theorem.
cites work: Monge-Kantorovitch measure transportation and Monge-Ampère equation on Wiener space
cites work: Q4337939
cites work: An introduction to analysis on Wiener space
DOI: 10.1016/J.CRMA.2004.04.013
description: scientific article; zbMATH DE number 2094018