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

Abstract

The defining equation $$ (\ast)\qquad \dot \omega\_t=-F'(\omega\_t) $$ 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: $\dot \omega^{\varepsilon}\_t=- \varepsilon F'( \omega ^{ \varepsilon}\_t)$, where $\varepsilon > 0$, and (ii) by considering the accelerations $\ddot \omega ^{ \varepsilon}\_t$. 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 $\varepsilon$ 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.