Wasserstein Diffusion on Multidimensional Spaces

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 $$d{\mathbb{P}}^{\beta }(\mu )=\frac{1}{Z}{e}^{-\beta \text{Ent}(\mu \mathrm{\mid }m)}d{\mathbb{P}}^{\ast }(\mu );$$ (ii) associated with a regular Dirichlet form with carré du champ derived from the Wasserstein gradient in the sense of Otto calculus $${\mathcal{E}}_W(f)=\underset{g\to f}{\mathrm{lim\hspace{0.17em}inf}}\frac{1}{2}{\int }_{\mathcal{P}(M)}\parallel {\mathrm{\nabla }}_{W}g{\parallel }^2(\mu )d{\mathbb{P}}^{\beta }(\mu );$$ (iii) non-degenerate, at least in the case of the $n$-sphere and the $n$-torus.