## The Annals of Probability

- Ann. Probab.
- Volume 37, Number 3 (2009), 1114-1191.

### Entropic measure and Wasserstein diffusion

Max-K. von Renesse and Karl-Theodor Sturm

#### Abstract

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.

#### Article information

**Source**

Ann. Probab., Volume 37, Number 3 (2009), 1114-1191.

**Dates**

First available in Project Euclid: 19 June 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1245434030

**Digital Object Identifier**

doi:10.1214/08-AOP430

**Mathematical Reviews number (MathSciNet)**

MR2537551

**Zentralblatt MATH identifier**

1177.60066

**Subjects**

Primary: 60J60: Diffusion processes [See also 58J65] 60G57: Random measures 35R60: Partial differential equations with randomness, stochastic partial differential equations [See also 60H15] 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60H15: Stochastic partial differential equations [See also 35R60] 60J45: Probabilistic potential theory [See also 31Cxx, 31D05] 58J65: Diffusion processes and stochastic analysis on manifolds [See also 35R60, 60H10, 60J60]

**Keywords**

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

#### Citation

von Renesse, Max-K.; Sturm, Karl-Theodor. Entropic measure and Wasserstein diffusion. Ann. Probab. 37 (2009), no. 3, 1114--1191. doi:10.1214/08-AOP430. https://projecteuclid.org/euclid.aop/1245434030