A quantitative stability result for regularity of optimal transport on compact manifolds

TL;DR

This paper establishes a quantitative stability result for the regularity of optimal transport maps on compact manifolds, showing that under certain conditions, smoothness persists within a specific Wasserstein neighborhood despite curvature challenges.

Contribution

It introduces a novel maximum principle approach to prove stability of optimal transport regularity under bounded log densities on compact manifolds.

Findings

Optimal transport maps remain smooth within a quantifiable Wasserstein neighborhood.

The Hessian of the Kantorovich potential stays bounded under specified conditions.

The approach overcomes issues caused by unhelpful MTW curvature.

Abstract

We use a Korevaar-style maximum principle approach to show the following: Fixing a $C^{2}$ bound on the log densities of a set of smooth measures, there is a quantifiably-sized Wasserstein neighborhood over which all pairs of such measures will enjoy smooth optimal transport. \ We do this in spite of unhelpful MTW\ curvature, by showing that when the gradient of the Kantorovich potential is small enough, the Hessian ``bound" places the Hessian in one of two disconnected regions, one bounded and the other unbounded. \ Tracking the estimate along a continuity path which starts in the bounded region, we conclude the Hessian must stay bounded.