Measure differential equations

TL;DR

This paper introduces Measure Differential Equations (MDEs), a novel class of equations for probability measures that generalize ordinary differential equations and connect to PDEs and multi-particle systems.

Contribution

The paper defines MDEs, establishes existence and uniqueness results, and demonstrates their applications to diffusion, concentration, PDEs, and mean-field limits.

Findings

MDEs generalize ODEs for probability measures.

Existence and uniqueness of solutions are proved.

MDEs relate to PDEs and multi-particle systems.

Abstract

A new type of differential equations for probability measures on Euclidean spaces, called Measure Differential Equations (briefly MDEs), is introduced. MDEs correspond to Probability Vector Fields, which map measures on an Euclidean space to measures on its tangent bundle. Solutions are intended in weak sense and existence, uniqueness and continuous dependence results are proved under suitable conditions. The latter are expressed in terms of the Wasserstein metric on the base and fiber of the tangent bundle. MDEs represent a natural measure-theoretic generalization of Ordinary Differential Equations via a monoid morphism mapping sums of vector fields to fiber convolution of the corresponding Probability Vector Fields. Various examples, including finite-speed diffusion and concentration, are shown, together with relationships to Partial Differential Equations. Finally, MDEs are also natural mean-field limits of multi-particle systems, with convergence results extending the classical Dubroshin approach.