Geodesics for a class of distances in the space of probability measures

TL;DR

This paper characterizes geodesics for a specific class of probability measure distances, establishing existence, optimality conditions, and PDE systems similar to Mean-Field Games, advancing understanding of geometric structures in probability spaces.

Contribution

It introduces a new characterization of geodesics for a class of distances in probability measure spaces, including existence proofs and PDE-based optimality conditions.

Findings

Existence of a potential function for geodesics.

Necessary and sufficient optimality conditions via PDEs.

Equivalent formulation in probability measures over curves.

Abstract

In this paper, we study the characterization of geodesics for a class of distances between probability measures introduced by Dolbeault, Nazaret and Savar e. We first prove the existence of a potential function and then give necessary and suffi cient optimality conditions that take the form of a coupled system of PDEs somehow similar to the Mean-Field-Games system of Lasry and Lions. We also consider an equivalent formulation posed in a set of probability measures over curves.