Small Noise Limit and Convexity for Generalized Incompressible Flows, Schrödinger Problems, and Optimal Transport

Abstract

This paper is concerned with six variational problems and their mutual connections: the quadratic Monge–Kantorovich optimal transport, the Schrodinger problem, Brenier’s relaxed model for incompressible fluids, the so-called Brodinger problem recently introduced by Arnaudon et al. (An entropic interpolation problem for incompressible viscid fluids, arXiv preprint arXiv:1704.02126, 2017) the multiphase Brenier model, and the multiphase Brodinger 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 $$\nu \rightarrow 0$$ 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 Lavenant (Calc Var Partial Differ Equ 56(6):170, 2017) and Benamou et al. (Generalized incompressible flows, multi-marginal transport and Sinkhorn algorithm, arXiv preprint arXiv:1710.08234, 2017).