The Annals of Applied Probability

Hydrodynamic limits and propagation of chaos for interacting random walks in domains

Zhen-Qing Chen and Wai-Tong (Louis) Fan

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Abstract

A new non-conservative stochastic reaction–diffusion system in which two families of random walks in two adjacent domains interact near the interface is introduced and studied in this paper. Such a system can be used to model the transport of positive and negative charges in a solar cell or the population dynamics of two segregated species under competition. We show that in the macroscopic limit, the particle densities converge to the solution of a coupled nonlinear heat equations. For this, we first prove that propagation of chaos holds by establishing the uniqueness of a new BBGKY hierarchy. A local central limit theorem for reflected diffusions in bounded Lipschitz domains is also established as a crucial tool.

Article information

Source
Ann. Appl. Probab., Volume 27, Number 3 (2017), 1299-1371.

Dates
Received: September 2014
Revised: February 2016
First available in Project Euclid: 19 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1500451224

Digital Object Identifier
doi:10.1214/16-AAP1208

Mathematical Reviews number (MathSciNet)
MR3678472

Zentralblatt MATH identifier
1372.60133

Subjects
Primary: 60F17: Functional limit theorems; invariance principles 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 92D15: Problems related to evolution

Keywords
Hydrodynamic limit propagation of chaos interacting particle system random walk annihilation reflecting diffusion boundary local time heat kernel coupled nonlinear partial differential equation BBGKY hierarchy Duhamel tree expansion isoperimetric inequality

Citation

Chen, Zhen-Qing; Fan, Wai-Tong (Louis). Hydrodynamic limits and propagation of chaos for interacting random walks in domains. Ann. Appl. Probab. 27 (2017), no. 3, 1299--1371. doi:10.1214/16-AAP1208. https://projecteuclid.org/euclid.aoap/1500451224


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