Measure dynamics with Probability Vector Fields and sources

TL;DR

This paper introduces Measure Differential Equations with sources, a unified framework combining transport, diffusion, and mass creation, ensuring existence and uniqueness of solutions under certain conditions.

Contribution

It proposes a novel formulation for measure dynamics incorporating sources and proves existence and uniqueness results using a Wasserstein-like functional.

Findings

Existence of solutions under Lipschitz conditions.

Uniqueness when measure dynamics align with Dirac mass measures.

Unified treatment of transport, diffusion, and mass creation.

Abstract

We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term, in which a vector field is replaced by a Probability Vector Field, that is a probability distribution on the tangent bundle; on the other side, a source term. Such new formulation allows to write in a unified way both classical transport and diffusion with finite speed, together with creation of mass. The main result of this article shows that, by introducing a suitable Wasserstein-like functional, one can ensure existence of solutions to Measure Differential Equations with sources under Lipschitz conditions. We also prove a uniqueness result under the following additional hypothesis: the measure dynamics needs to be compatible with dynamics of measures that are sums of Dirac masses.