The Schrödinger problem is a stochastic mass transport problem that consists in minimising relative entropy under marginal constrains and provides with a model for an old question of Schrödinger about the most likely evolution of Brownian particles conditional to expectation. Optimal dual variables, known as Schrödinger potentials, are characterised through a coupled non linear PDE system (Schrödinger system) and are the entropic analogous of the Brenier potentials, arising as optimal dual variables in the Monge Kantorovich problem. In this talk, we address the problem of finding concavity/convexity estimates for Schrödinger potentials. Whereas recent progresses on this problem require log concavity assumptions on at least one the marginal inputs, we show how a probabilistic method drawing inspiration from coupling techniques allows to considerably relax this assumption. The main theoretical ingredient upon which proofs rely is the construction of novel sets of approximately convex functions that is invariant under the flow generated by the Hamilton Jacobi Bellman equation. This result may be seen as a partial generalisation of Prekopa-Leindler inequality in a non convex setting.
