A new approach for the construction of a Wasserstein diffusion

Abstract

We propose in this paper a construction of a diffusion process on the space $\mathcal P_2(\mathbb R)$ of probability measures with a second-order moment. This process was introduced in several papers by Konarovskyi (see e.g. [12]) and consists of the limit as $N$ tends to $+\infty $ 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 $\mathcal P_2(\mathbb R)$ constructed by von Renesse and Sturm in [22]. 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.