Electronic Journal of Probability

A new approach for the construction of a Wasserstein diffusion

Victor Marx

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Abstract

We propose in this paper a construction of a diffusion process on the space $\mathcal P_2(\mathbb R)$ of probability measures with a second-order moment. This process was introduced in several papers by Konarovskyi (see e.g. [12]) and consists of the limit as $N$ tends to $+\infty $ of a system of $N$ coalescing and mass-carrying particles. It has properties analogous to those of a standard Euclidean Brownian motion, in a sense that we will precise in this paper. We also compare it to the Wasserstein diffusion on $\mathcal P_2(\mathbb R)$ constructed by von Renesse and Sturm in [22]. We obtain that process by the construction of a system of particles having short-range interactions and by letting the range of interactions tend to zero. This construction can be seen as an approximation of the singular process of Konarovskyi by a sequence of smoother processes.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 124, 54 pp.

Dates
Received: 17 October 2017
Accepted: 3 December 2018
First available in Project Euclid: 19 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1545188691

Digital Object Identifier
doi:10.1214/18-EJP254

Mathematical Reviews number (MathSciNet)
MR3896861

Zentralblatt MATH identifier
07021680

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60J60: Diffusion processes [See also 58J65] 60B12: Limit theorems for vector-valued random variables (infinite- dimensional case)
Secondary: 60G44: Martingales with continuous parameter 82B21: Continuum models (systems of particles, etc.)

Keywords
Wasserstein diffusion interacting particle system coalescing particles modified Arratia flow Brownian sheet differential calculus on Wasserstein space Itô formula for measure-valued processes

Rights
Creative Commons Attribution 4.0 International License.

Citation

Marx, Victor. A new approach for the construction of a Wasserstein diffusion. Electron. J. Probab. 23 (2018), paper no. 124, 54 pp. doi:10.1214/18-EJP254. https://projecteuclid.org/euclid.ejp/1545188691


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