A partial Laplacian as an infinitesimal generator on the Wasserstein space

TL;DR

This paper introduces a partial Laplacian as an infinitesimal generator on the Wasserstein space, enabling the definition of heat flow and analysis of smoothing effects for probability measures, with implications for PDEs and Mean Field Games.

Contribution

It develops a new partial Laplacian operator on Wasserstein space and demonstrates its use in defining heat flow and analyzing smoothing effects, advancing the mathematical framework for stochastic processes in this space.

Findings

Defined a partial Laplacian as an infinitesimal generator on Wasserstein space

Established a heat flow with a smoothing effect for certain initial conditions

Developed Fourier analysis and conic surface theory in metric spaces

Abstract

We study stochastic processes on the Wasserstein space, together with their infinitesimal generators. One of these processes is modeled after Brownian motion and plays a central role in our work. Its infinitesimal generator defines a partial Laplacian on the space of Borel probability measures, and we use it to define heat flow on the Wasserstein space. We verify a distinctive smoothing effect of this flow for a particular class of initial conditions. To this end, we will develop a theory of Fourier analysis and conic surfaces in metric spaces. We note that the use of the infinitesimal generators has been instrumental in proving various theorems for Mean Field Games, and we anticipate they will play a key role in future studies of viscosity solutions of PDEs in the Wasserstein space.