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

  • Ivan Gentil

    Université Claude Bernard Lyon 1, Villeurbanne, France
  • Christian Léonard

    Université Paris Nanterre, France
  • Luigia Ripani

    Université Claude Bernard Lyon 1, Villeurbanne, France
Dynamical aspects of the generalized Schrödinger problem via Otto calculus – A heuristic point of view cover

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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 into the family of slowed down gradient flow equations: , where , and (ii) by considering the accelerations . 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 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.

Cite this article

Ivan Gentil, Christian Léonard, Luigia Ripani, Dynamical aspects of the generalized Schrödinger problem via Otto calculus – A heuristic point of view. Rev. Mat. Iberoam. 36 (2020), no. 4, pp. 1071–1112

DOI 10.4171/RMI/1159