Bounds on optimal transport maps onto log-concave measures

TL;DR

This paper investigates the properties of optimal transport maps from Gaussian measures to strictly log-concave measures with degenerate bounds, establishing local Lipschitz continuity and controlled eigenvalue growth.

Contribution

It proves the local Lipschitz continuity of optimal transport maps and bounds the eigenvalues of their Jacobians for measures with degenerate bounds at infinity.

Findings

Optimal transport maps are locally Lipschitz.

Eigenvalues of Jacobians have controlled growth at infinity.

Results apply to measures with degenerate bounds at infinity.

Abstract

We consider strictly log-concave measures, whose bounds degenerate at infinity. We prove that the optimal transport map from the Gaussian onto such a measure is locally Lipschitz, and that the eigenvalues of its Jacobian have controlled growth at infinity.