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Preprints, Working Papers, ... Year : 2018

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

Ivan Gentil
Christian Léonard
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  • PersonId : 944163
Luigia Ripani
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  • PersonId : 972341

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.
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Dates and versions

hal-01806572 , version 1 (04-06-2018)
hal-01806572 , version 2 (06-06-2018)
hal-01806572 , version 3 (12-03-2019)

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Ivan Gentil, Christian Léonard, Luigia Ripani. Dynamical aspects of generalized Schrödinger problem via Otto calculus -- A heuristic point of view. 2018. ⟨hal-01806572v1⟩
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