Smoothing properties of a Wasserstein diffusion

Diffusion processes on the L2-Wasserstein space were introduced by Sturm and von Renesse in 2009 and have been subject of much interest in recent years. In particular, Konarovskyi proposed in 2017 a construction of a diffusion based on a system of massive coalescing particles. The aim of my talk is to present a Girsanov Theorem and a Bismut-Elworthy formula for a diffusion process inspired by Konarovskyi's model.

We will first introduce a diffusion on the Wasserstein space which is a regularized version of Konarovskyi's model. It can be viewed as a continuum of particles evolving on the real line according to a Gaussian interaction kernel weighted by the mass associated to each particle.

Second, we will prove a Girsanov-type Theorem on this process, using an appropriate Fourier inversion of the Gaussian kernel. As a Corollary, we obtain weak existence and weak uniqueness of the solution to a Fokker-Planck equation on the L2-Wasserstein space with very irregular drift.

Third, we will present a regularization result on the semi-group associated to this process with non-smooth drift and prove a Bismut-Elworthy formula which provides an upper bound for the gradient of the semi-group in smooth directions of differentiation.