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

Understand

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.