Weak semiconvexity estimates for Schrödinger potentials and logarithmic Sobolev inequality for Schrödinger bridges

We investigate the quadratic Schrödinger bridge problem, a.k.a. Entropic Optimal Transport problem, and obtain weak semiconvexity and semiconcavity bounds on Schrödinger potentials under mild assumptions on the marginals that are substantially weaker than log-concavity. We deduce from these estimates that Schrödinger 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.