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Article Dans Une Revue Revista Matemática Iberoamericana Année : 2020

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

Résumé

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 et 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. Revista Matemática Iberoamericana, 2020, 36 (4), pp.1071-1112. ⟨10.4171/RMI/1159⟩. ⟨hal-01806572v3⟩
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