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We investigate the quadratic Schr\"odinger bridge problem, a.k.a. EntropicOptimal Transport problem, and obtain weak semiconvexity and semiconcavitybounds on Schr\"odinger potentials under mild assumptions on the marginals thatare substantially weaker than log-concavity. We deduce from these estimatesthat Schr\"odinger bridges satisfy a logarithmic Sobolev inequality on theproduct space. Our proof strategy is based on a second order analysis ofcoupling by reflection on the characteristics of the Hamilton-Jacobi-Bellmanequation that reveals the existence of new classes of invariant functions forthe corresponding flow.

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