Entropic measure and Wasserstein diffusion

We construct a canonical reversible process $({\mu }_t{)}_{t\ge 0}$ on the $L^2$-Wasserstein space of probability measures P(R), regarded as an infinite dimensional Riemannian manifold. This process has an invariant measure $P^{\beta }$ which may be characterized as the 'uniform distribution' on P(R) with weight function $exp(-\beta Ent(.|m))/Z$ where m denotes a given finite measure on R.

One of the key results is the quasi-invariance of this measure $P^{\beta }$ under push forwards $\mu \mapsto h_{\ast }\mu$ by means of smooth diffeomorphisms h of R.