A new approach for the construction of a Wasserstein diffusion

TL;DR

This paper introduces a novel construction of a Wasserstein diffusion process on the space of probability measures, using particle systems with short-range interactions that approximate a singular process, and compares it to existing models.

Contribution

It presents a new particle-based method to construct Wasserstein diffusion, providing a smoother approximation to a previously singular process and establishing connections with existing Wasserstein diffusions.

Findings

Constructed a Wasserstein diffusion via particle systems with short-range interactions.

Showed the approximation of the singular process by smoother processes.

Compared the new process with the Wasserstein diffusion by von Renesse and Sturm.

Abstract

We propose in this paper a construction of a diffusion process on the Wasserstein space P\_2(R) of probability measures with a second-order moment. This process was introduced in several papers by Konarovskyi (see e.g. "A system of coalescing heavy diffusion particles on the real line", 2017) and consists of the limit when N tends to infinity of a system of N coalescing and mass-carrying particles. It has properties analogous to those of a standard Euclidean Brownian motion, in a sense that we will precise in this paper. We also compare it to the Wasserstein diffusion on P\_2(R) constructed by von Renesse and Sturm (see Entropic measure and Wasserstein diffusion). We obtain that process by the construction of a system of particles having short-range interactions and by letting the range of interactions tend to zero. This construction can be seen as an approximation of the singular process of Konarovskyi by a sequence of smoother processes.