  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.
