Weak semiconvexity estimates for Schr\"odinger potentials and logarithmic Sobolev inequality for Schr\"odinger bridges

TL;DR

This paper establishes weak semiconvexity and semiconcavity bounds for Schr"odinger potentials in entropic optimal transport, leading to a logarithmic Sobolev inequality for Schr"odinger bridges under mild assumptions.

Contribution

It introduces new weak semiconvexity estimates for Schr"odinger potentials and derives a logarithmic Sobolev inequality for Schr"odinger bridges without requiring log-concavity.

Findings

Weak semiconvexity bounds for Schr"odinger potentials

Logarithmic Sobolev inequality for Schr"odinger bridges

Applicable under mild marginal assumptions

Abstract

We investigate the quadratic Schr\"odinger bridge problem, a.k.a. Entropic Optimal Transport problem, and obtain weak semiconvexity and semiconcavity bounds on Schr\"odinger potentials under mild assumptions on the marginals that are substantially weaker than log-concavity. We deduce from these estimates that Schr\"odinger bridges satisfy a logarithmic Sobolev inequality on the product space. Our proof strategy is based on a second order analysis of coupling by reflection on the characteristics of the Hamilton-Jacobi-Bellman equation that reveals the existence of new classes of invariant functions for the corresponding flow.