Existence of probability measure valued jump-diffusions in generalized Wasserstein spaces

TL;DR

This paper proves the existence of probability measure valued jump-diffusions in generalized Wasserstein spaces by embedding these spaces into locally compact spaces, enabling classical martingale problem techniques to be applied.

Contribution

It introduces a novel embedding method for Wasserstein spaces into locally compact spaces, facilitating the analysis of complex jump-diffusion processes.

Findings

Established existence of measure-valued jump-diffusions with general dynamics.

Developed verification tools for generator conditions, including maximum principle.

Applied results to large particle systems with mean-field interaction.

Abstract

We study existence of probability measure valued jump-diffusions described by martingale problems. We develop a simple device that allows us to embed Wasserstein spaces and other similar spaces of probability measures into locally compact spaces where classical existence theory for martingale problems can be applied. The method allows for general dynamics including drift, diffusion, and possibly infinite-activity jumps. We also develop tools for verifying the required conditions on the generator, including the positive maximum principle and certain continuity and growth conditions. To illustrate the abstract results, we consider large particle systems with mean-field interaction and common noise.