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.