A new approach for the construction of a Wasserstein diffusion

Victor Marx (1)

1 LJAD - Laboratoire Jean Alexandre Dieudonné

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.

Mots clés: differential calculus on Wasserstein space; Itô formula for measure-valued processes; modified Arratia flow; Brownian sheet; Wasserstein diffusion; interacting particle system; coalescing particles; measure-valued processes

Domaines: Probabilités [math.PR]

Victor Marx. A new approach for the construction of a Wasserstein diffusion. Electronic Journal of Probability, 2018, 23 (124). ⟨hal-01618228v3⟩