Resumen de Dynamical aspects of the generalized Schrödinger problem via Otto calculus –: A heuristic point of view
Ivan Gentil, Christian Léonard, Luigia Ripani
The defining equation (∗)ω˙t=−F′(ωt) of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation (∗) into the family of slowed down gradient flow equations: ω˙εt=−εF′(ωεt), where ε>0, and (ii) by considering the accelerations ω¨εt. We shall focus on Wasserstein gradient flows. Our approach is mainly heuristic. It relies on Otto calculus.
A special formulation of the Schrödinger problem consists in minimizing some action on the Wasserstein space of probability measures on a Riemannian manifold subject to fixed initial and final data. We extend this action minimization problem by replacing the usual entropy, underlying the Schrödinger problem, with a general function on the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when the fluctuation parameter ε tends to zero.
We show heuristically that the solutions satisfy some Newton equation, extending a recent result of Conforti. The connection with Wasserstein gradient flows is established and various inequalities, including evolutional variational inequalities and contraction inequalities under a curvature-dimension condition, are derived with a heuristic point of view. As a rigorous result we prove a new and general contraction inequality for the Schrödinger problem under a Ricci lower bound on a smooth and compact Riemannian manifold.
