# Revista Matemática Iberoamericana

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**Published online first: 2020-01-10**

**DOI: 10.4171/rmi/1159**

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

Ivan Gentil^{[1]}, Christian Léonard

^{[2]}and Luigia Ripani

^{[3]}(1) Université Claude Bernard Lyon 1, Villeurbanne, France

(2) Université Paris Nanterre, France

(3) Université Claude Bernard Lyon 1, Villeurbanne, France

The defining equation
$$(\ast)\qquad \dot \omega_t=-F'(\omega_t)$$
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 *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 the Schrödinger problem, with a general function on the Wasserstein space. The corresponding minimal cost approaches the squared Wasserstein distance when the fluctuation parameter $\varepsilon$ 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.

*Keywords: *Schrödinger problem, Wasserstein distance, Otto calculus, Newton equation, curvature- dimension conditions

Gentil Ivan, Léonard Christian, Ripani Luigia: Dynamical aspects of the generalized Schrödinger problem via Otto calculus – A heuristic point of view. *Rev. Mat. Iberoam.* Electronically published on January 10, 2020. doi: 10.4171/rmi/1159 (to appear in print)