# Superposition principle and schemes for Measure Differential Equations

## TL;DR

This paper investigates Measure Differential Equations (MDE), establishing their connection with nonlocal continuity equations, proving a superposition representation, and proposing schemes for solutions that highlight uniqueness issues.

## Contribution

It introduces a superposition principle for MDEs and develops alternative convergent schemes, advancing understanding of solution properties and uniqueness.

## Key findings

- Established a superposition representation for MDEs.
- Developed alternative schemes converging to solutions.
- Highlighted conditions for uniqueness and non-uniqueness.

## Abstract

Measure Differential Equations (MDE) describe the evolution of probability measures driven by probability velocity fields, i.e. probability measures on the tangent bundle. They are, on one side, a measure-theoretic generalization of ordinary differential equations; on the other side, they allow to describe concentration and diffusion phenomena typical of kinetic equations. In this paper, we analyze some properties of this class of differential equations, especially highlighting their link with nonlocal continuity equations. We prove a representation result in the spirit of the Superposition Principle by Ambrosio-Gigli-Savar\'e, and we provide alternative schemes converging to a solution of the MDE, with a particular view to uniqueness/non-uniqueness phenomena.