A Dacorogna-Moser construction of transport maps on $\mathbb{R}^d$ with application to geodesics on the space of couplings

Abstract

A seminal work by Dacorogna and Moser introduced a way of constructing regular transport maps from a probability distribution on a bounded domain to another one. In this work, we extend this construction to the whole $\mathbb{R}^d$ for strictly asymptotically log-concave measures, a wide class of distributions that encompasses Lipschitz-perturbations of log-concave measures. We then leverage this construction to study geodesics in the space of probability measures on a product set with imposed marginal laws (couplings), for which we derive optimality conditions, answering an open question in a recent work by Conforti, Lacker and Pal. Taking inspiration from Brenier's variational model for incompressible fluids and its regularization, we further introduce an entropic regularization of the geodesic problem, which can be seen as the Schr\"odinger bridge problem on the space of couplings, for which we also derive optimality conditions. We eventually study convergence of minimizers as the regularization vanishes, and we prove convergence of the related Lagrange multipliers. Our approach involves proving uniform-in-time global regularity estimates on elliptic and parabolic equations on $\mathbb{R}^d$, by exploiting the structure of asymptotically log-concave measures using the probabilistic notion of reflection coupling.