Venerdì 7 luglio 2023 alle ore 14:30, Aula M3 Dipartimento di Matematica e Fisica di Roma Tre, Largo San Leonardo Murialdo,1- 00146 Roma.
Speaker: Giovanni Conforti (CMAP Ecole Polytechnique, Paris)
Title: Stable convexity profiles for Hamilton Jacobi Bellman equations and applications to entropic optimal transport.
Abstract: It is well known that solutions of the Hamilton-Jacobi-Bellman (HJB) equation with a convex terminal condition remain convex at all times. In this sense, the set of convex functions is stable under the action of the HJB equation and this result can be derived as a simple consequence of the Prékopa-Leindler inequality. In this talk I will illustrate a probabilistic method based on the analysis of coupling by reflection on a system of forward-backward stochastic differential equations (FBSDEs) that prompts the existence of stable sets for the HJB equation composed of functions that are only "asymptotically convex", but not necessarily convex. If time allows, I will move on to illustrate two consequences of this result to the Schrödinger problem, a.k.a. entropic optimal transport problem. The first is that the optimal coupling satisfies the logarithmic Sobolev inequality and the second one is that Sinkhorn's algorithm converges exponentially fast in the number of iterations to the optimal coupling. Partially based on joint work with A.Durmus and G.Greco