Carath\'eodory Theory and A Priori Estimates for Continuity Inclusions in the Space of Probability Measures

TL;DR

This paper extends the theory of differential inclusions to Wasserstein spaces of probability measures, proving a new existence result under Carathéodory conditions and improving solution estimates using a set-valued Euler scheme.

Contribution

It introduces a novel existence theorem for continuity inclusions in Wasserstein spaces under minimal regularity assumptions, utilizing a set-valued Euler scheme adapted from Filippov's method.

Findings

Established a Peano-type existence result for continuity inclusions.

Developed new estimates and compactness properties for solutions.

Enhanced the understanding of solution set relaxation in the Wasserstein space.

Abstract

In this article, we extend the foundations of the theory of differential inclusions in the space of compactly supported probability measures, introduced recently by the authors, to the setting of general Wasserstein spaces. In this context, we prove a novel existence result \`a la Peano for this class of dynamics under mere Carath\'eodory regularity assumptions. The latter is based on a natural, yet previously unexplored set-valued adaptation of the semi-discrete Euler scheme proposed by Filippov to study ordinary differential equations whose right-hand sides are measurable in the time variable. By leveraging some of the underlying methods along with new estimates for solutions of continuity equations, we also bring substantial improvements to the earlier versions of the Filippov estimates, compactness and relaxation properties of the solution sets of continuity inclusions, which are derived in the Cauchy-Lipschitz framework.