We construct a new random probability measure on the circle and on the unit interval which in both cases has a Gibbs structure 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 diffusion processes on probability measures (on the sphere or the unit interval) by Dirichlet form methods. The first one is closely related to Malliavin’s Brownian motion on the homeomorphism group. The second one is a probability valued stochastic perturbation of the heat flow, whose intrinsic metric is the quadratic Wasserstein distance. It may be regarded as the canonical diffusion process on the Wasserstein space.
Publié le : 2009-05-15
Classification:
Wasserstein space,
optimal transport,
entropy,
Dirichlet process,
change of variable formula,
measure-valued diffusion,
Brownian motion on the homeomorphism group,
stochastic heat flow,
Wasserstein diffusion,
entropic measure,
60J60,
60G57,
35R60,
47D07,
60H15,
60J45,
58J65
@article{1245434030,
author = {von Renesse, Max-K. and Sturm, Karl-Theodor},
title = {Entropic measure and Wasserstein diffusion},
journal = {Ann. Probab.},
volume = {37},
number = {1},
year = {2009},
pages = { 1114-1191},
language = {en},
url = {http://dml.mathdoc.fr/item/1245434030}
}
von Renesse, Max-K.; Sturm, Karl-Theodor. Entropic measure and Wasserstein diffusion. Ann. Probab., Tome 37 (2009) no. 1, pp. 1114-1191. http://gdmltest.u-ga.fr/item/1245434030/