Wasserstein Diffusion on Multidimensional Spaces

Abstract

Given any closed Riemannian manifold $M$, we construct a reversible diffusion process on the space ${\mathcal P}(M)$ of probability measures on $M$ that is (i) reversible w.r.t.~the entropic measure ${\mathbb P}^\beta$ on ${\mathcal P}(M)$, heuristically given as $$d\mathbb{P}^\beta(\mu)=\frac{1}{Z} e^{-\beta \, \text{Ent}(\mu| m)}\ d\mathbb{P}^*(\mu);$$ (ii) associated with a regular Dirichlet form with carr\'e du champ derived from the Wasserstein gradient in the sense of Otto calculus $${\mathcal E}_W(f)=\liminf_{g\to f}\ \frac12\int_{{\mathcal P}(M)} \big\|\nabla_W g\big\|^2(\mu)\ d{\mathbb P}^\beta(\mu);$$ (iii) non-degenerate, at least in the case of the $n$-sphere and the $n$-torus.