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

Abstract : The defining equation $(\ast):\ \dot \omega_t=-F'(\omega_t),$ 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 \emph{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 Schrödinger problem, with a general function of the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when some fluctuation parameter tends to zero. We show heuristically that the solutions satisfy a 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 inequality under 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.
Type de document :
Pré-publication, Document de travail
2018
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-01806572
Contributeur : Ivan Gentil <>
Soumis le : mercredi 6 juin 2018 - 10:39:44
Dernière modification le : mercredi 4 juillet 2018 - 23:14:02
Document(s) archivé(s) le : vendredi 7 septembre 2018 - 12:57:13

Fichiers

Gentil-Leonard-Ripani-HAL-2.pd...
Fichiers produits par l'(les) auteur(s)

Identifiants

  • HAL Id : hal-01806572, version 2
  • ARXIV : 1806.01553

Citation

Ivan Gentil, Christian Léonard, Luigia Ripani. Dynamical aspects of generalized Schrödinger problem via Otto calculus -- A heuristic point of view. 2018. 〈hal-01806572v2〉

Partager

Métriques

Consultations de la notice

92

Téléchargements de fichiers

32