Small noise limit and convexity for generalized incompressible flows, Schr\"odinger problems, and optimal transport

TL;DR

This paper explores the connections between various variational problems in optimal transport, Schr"odinger problems, and fluid dynamics, establishing convergence results and new convexity properties as viscosity tends to zero.

Contribution

It establishes Gamma-convergence and pressure convergence between six related variational problems, extending previous results and analyzing entropy convexity.

Findings

Gamma-convergence of problems as viscosity approaches zero

Convergence of pressures from incompressibility constraints

New results on entropy time-convexity in dynamical interpolations

Abstract

This paper is concerned with six variational problems and their mutual connections: The quadratic Monge-Kantorovich optimal transport, the Schr\"odinger problem, Brenier's relaxed model for incompressible fluids, the so-called Br\"odinger problem recently introduced by M. Arnaudon & al. [3], the multiphase Brenier model, and the multiphase Br\"odinger problem. All of them involve the minimization of a kinetic action and/or a relative entropy of some path measures with respect to the reversible Brownian motion. As the viscosity parameter $\nu\to 0$ we establish Gamma-convergence relations between the corresponding problems, and prove the convergence of the associated pressures arising from the incompressibility constraints. We also present new results on the time-convexity of the entropy for some of the dynamical interpolations. Along the way we extend previous results by H. Lavenant [30] and J-D. Benamou & al. [10].