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