Entropic measure and Wasserstein diffusion (Q2270611)

The authors construct a random probability measure on the circle and the unit interval which has a Gibbs structure and with the relative entropy functional as hamiltonian. It satisfies a quasi-invariance formula with respect to the action of smooth diffeomorphism of the sphere and the interval, respectively. The associated integration by parts formula is used to construct two classes of diffusions on the probability measures on the sphere and the unit interval, respectively, by Dirichlet form methods. The first is closely related to Malliavin's Brownian motion on the homeomorphism group; the second is a probability-valued stochastic perturbation of the heat flow whose intrinsic metric is the quadratic Wasserstein distance. It can be regarded as the canonical diffusion process on the Wasserstein space.