Invariant integrated convexity profiles for Hamilton-Jacobi-Bellman equations and applications

Conforti Giovanni, IPParis, École Polytechnique

It is well known that log-concavity is preserved along the heat flow. Equivalently, this can be phrased as the fact that if the terminal condition is convex, then the solution of the Hamilton-Jacobi Bellman (HJB) equation stays convex at all times. Quite surprisingly, this basic geometric invariance principle has found in recent and not-so-recent times several striking applications. For example, it was used back in 1976 by Brascamp and Lieb to establish convexity of the ground state for Schrödinger operators. Furthermore, over the last couple of years it has been employed to generalise Caffarelli's theorem on the existence of Lipschitz transport maps, to show the exponential convergence of scaling algorithms for entropic optimal transport such as Sinkhorn's algorithm, and as a tool in the renormalisation approach to functional inequalities. In this talk, I will show how some probabilistic constructions based on coupling ideas reveal the existence of families of functions that are invariant for Hamilton-Jacobi-equations but that are not necessarily convex. More precisely, they are only asymptotically convex in the sense that the integral of the second directional derivative over a long segment is non-negative. If time allows, some applications of this result will be presented.

Area: CS34 - Advances in stochastic optimal transport and applications (Luca Tamanini)

Keywords: couplings, convexity Schrödinger problem

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