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
 

• reversible w.r.t. the entropic measure $\mathbb{p}^\beta$ on $\mathcal{p}(m)$, heuristically given as 

$$d\mathbb{p}^\beta(\mu) =\frac{1}{z} e^{-\beta \, \text{ent}(\mu | m)}\ d\mathbb{p}^0(\mu);$$

• associated with a regular dirichlet form with carré du champ derived from the wasserstein gradient in the sense of otto calculus

$$\mathcal{e}_w(f)=\liminf_{\tilde f\to f}\ \frac12\int_{\mathcal{p}(m)} \big\|\nabla_w \tilde f\big\|^2(\mu)\ d\mathbb{p}^\beta(\mu);$$

• non-degenerate, at least in the case of the $n$-sphere and the $n$-torus.