Dynamical aspects of the generalized Schrödinger problem via Otto calculus –: A heuristic point of view
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Ivan Gentil [1] ; Christian Léonard [2] ; Luigia Ripani [1]
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[1] Claude Bernard University Lyon 1
Claude Bernard University Lyon 1
Arrondissement de Lyon, Francia
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[2] Paris West University Nanterre La Défense
Paris West University Nanterre La Défense
París, Francia
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- Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 36, Nº 4, 2020, págs. 1071-1112
- Idioma: inglés
- DOI: 10.4171/rmi/1159
- Enlaces
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Resumen
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.
