Seminar Analysis and Mathematical Physics: Gabriele Bocchi (Università degli Studi di Roma Tor Vergata)

A mean field planning approach to regularizations of the optimal transport problem

We analyze an optimal transport problem with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures m_0 and 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 have densities, we prove that the optimal curve remains positive and locally bounded in time. We focus on the case that the transport problem is set on a compact Riemannian manifold with Ricci curvature bounded below.

The approach follows ideas introduced by P.L. Lions in the theory of mean-field games about optimization problems with penalizing congestion terms. Crucial steps of our strategy include displacement convexity properties in the Eulerian approach and the analysis of distributional subsolutions to Hamilton-Jacobi equations.

The result provides a smooth approximation of Wasserstein-2 geodesics.

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