## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 1 (2017), 100-146.

### Systems of interacting diffusions with partial annihilation through membranes

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

#### Abstract

We introduce an interacting particle system in which two families of reflected diffusions interact in a singular manner near a deterministic interface $I$. This 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. A related interacting random walk model with discrete state spaces has recently been introduced and studied in Chen and Fan (2014). In this paper, we establish the functional law of large numbers for this new system, thereby extending the hydrodynamic limit in Chen and Fan (2014) to reflected diffusions in domains with mixed-type boundary conditions, which include absorption (harvest of electric charges). We employ a new and direct approach that avoids going through the delicate BBGKY hierarchy.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 1 (2017), 100-146.

**Dates**

Received: June 2014

Revised: July 2015

First available in Project Euclid: 26 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1485421330

**Digital Object Identifier**

doi:10.1214/15-AOP1047

**Mathematical Reviews number (MathSciNet)**

MR3601647

**Zentralblatt MATH identifier**

1361.60088

**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 interacting diffusion reflected diffusion Dirichlet form annihilation nonlinear boundary condition coupled partial differential equation martingales

#### Citation

Chen, Zhen-Qing; Fan, Wai-Tong (Louis). Systems of interacting diffusions with partial annihilation through membranes. Ann. Probab. 45 (2017), no. 1, 100--146. doi:10.1214/15-AOP1047. https://projecteuclid.org/euclid.aop/1485421330