The strong solution of the Monge-Ampère equation on the Wiener space for log-concave densities
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.
Monge-Kantorovich transportation problem
scientific article; zbMATH DE number 2094018