Regularizing effects of the entropy functional in optimal transport and planning problems

TL;DR

This paper demonstrates that adding an entropy functional to optimal transport problems induces elliptic regularization, ensuring smoothness of solutions and enabling better analysis of mean-field control and planning problems.

Contribution

It proves that entropic regularization yields smooth optimal transport curves and potentials, extending regularity results to bounded convex domains and Gaussian measures.

Findings

Optimal transport curves remain smooth over time with entropy regularization.

The approach provides smooth approximations for minimizers in mean-field control problems.

New estimates are derived using displacement convexity in the Eulerian framework.

Abstract

We analyze optimal transport problems with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures $m_0,m_1$. The effect of the additional entropy functional results into an elliptic regularization for the (so-called) Kantorovich potentials of the dual problem. Assuming the initial and terminal measures to be positive and smooth, we prove that the optimal curve remains smooth for all time. We focus on the case that the transport problem is set on a convex bounded domain in the $d$-dimensional Euclidean space (with no-flux condition on the boundary), but we also mention the case of Gaussian-like measures in the whole space. The approach follows ideas introduced by P.-L. Lions in the theory of mean-field games \cite{L-college}. The result provides with a smooth approximation of minimizers in optimization problems with penalizing congestion terms, which appear in mean-field control or mean-field planning problems. This allows us to exploit new estimates for this kind of problems by using displacement convexity properties in the Eulerian approach.