Dynamical aspects of the generalized Schrödinger problem via Otto calculus – A heuristic point of view

Abstract

The defining equation

of a gradient flow is kinetic in essence. This article explores some dynamical (rather than kinetic) features of gradient flows (i) by embedding equation $(\ast )$ into the family of slowed down gradient flow equations: ${\dot{\omega }}_t^{\epsilon }=-\epsilon F^{\prime }({\omega }_t^{\epsilon })$, where $\epsilon >0$, and (ii) by considering the accelerations ${\ddot{\omega }}_t^{\epsilon }$. 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 $\epsilon$ 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.