## The Annals of Applied Probability

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

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

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

#### 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