## The Annals of Applied Probability

- Ann. Appl. Probab.
- Volume 2, Number 3 (1992), 669-694.

### Fluctuations in a Nonlinear Reaction-Diffusion Model

#### Abstract

A law of large numbers and a central limit theorem are proved for a locally interacting particle system. This system describes a chemical reaction with diffusion with linear creation and quadratic annihilation of particles. The deterministic limit is the solution of a nonlinear reaction-diffusion equation defined on an $n$-dimensional unit cube. The law of large numbers holds for any dimension $n$ and arbitrary times, whereas the central limit theorem holds only for dimension $n \leq 3$ and on a certain bounded time interval (depending on the initial distribution and on the creation rate). A propagation of chaos expansion of the correlation functions is used.

#### Article information

**Source**

Ann. Appl. Probab., Volume 2, Number 3 (1992), 669-694.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoap/1177005654

**Digital Object Identifier**

doi:10.1214/aoap/1177005654

**Mathematical Reviews number (MathSciNet)**

MR1177904

**Zentralblatt MATH identifier**

0758.60108

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60G57: Random measures 60H15: Stochastic partial differential equations [See also 35R60] 35K55: Nonlinear parabolic equations 60F17: Functional limit theorems; invariance principles 60J70: Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.) [See also 92Dxx]

**Keywords**

Nonlinear reaction-diffusion equation locally interacting particle system thermodynamic limit Gaussian limit van Kampen's approximation propagation of chaos BBGKY hierearchy stochastic evolution equations spatially inhomogeneous population growth

#### Citation

Kotelenez, Peter. Fluctuations in a Nonlinear Reaction-Diffusion Model. Ann. Appl. Probab. 2 (1992), no. 3, 669--694. doi:10.1214/aoap/1177005654. https://projecteuclid.org/euclid.aoap/1177005654