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May 2009 Entropic measure and Wasserstein diffusion
Max-K. von Renesse, Karl-Theodor Sturm
Ann. Probab. 37(3): 1114-1191 (May 2009). DOI: 10.1214/08-AOP430


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.


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Max-K. von Renesse. Karl-Theodor Sturm. "Entropic measure and Wasserstein diffusion." Ann. Probab. 37 (3) 1114 - 1191, May 2009.


Published: May 2009
First available in Project Euclid: 19 June 2009

zbMATH: 1177.60066
MathSciNet: MR2537551
Digital Object Identifier: 10.1214/08-AOP430

Primary: 35R60 , 47D07 , 58J65 , 60G57 , 60H15 , 60J45 , 60J60

Keywords: Brownian motion on the homeomorphism group , change of variable formula , Dirichlet process , entropic measure , Entropy , measure-valued diffusion , Optimal transport , stochastic heat flow , Wasserstein diffusion , Wasserstein space

Rights: Copyright © 2009 Institute of Mathematical Statistics


Vol.37 • No. 3 • May 2009
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