## The Annals of Probability

- Ann. Probab.
- Volume 19, Number 4 (1991), 1440-1462.

### Comparison of Stochastic and Deterministic Models of a Linear Chemical Reaction with Diffusion

#### Abstract

Particles placed in $N$ cells on the unit interval give birth or die according to linear rates. Adjacent cells are coupled by diffusion with a rate proportional to $N^2$. Cell numbers are divided by a density parameter to represent concentrations, and the resulting space-time Markov process is compared to a corresponding deterministic model, the solution to a partial differential equation. The models are viewed as Hilbert space valued processes and compared by means of a law of large numbers and central limit theorem. New and nearly optimal results are obtained by exploiting the Ornstein-Uhlenbeck type structure of the stochastic model.

#### Article information

**Source**

Ann. Probab., Volume 19, Number 4 (1991), 1440-1462.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176990219

**Digital Object Identifier**

doi:10.1214/aop/1176990219

**Mathematical Reviews number (MathSciNet)**

MR1127711

**Zentralblatt MATH identifier**

0741.92022

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F17: Functional limit theorems; invariance principles

Secondary: 60H15: Stochastic partial differential equations [See also 35R60] 60G15: Gaussian processes

**Keywords**

Central limit theorem Ornstein-Uhlenbeck process stochastic partial differential equation reaction diffusion equation

#### Citation

Blount, Douglas. Comparison of Stochastic and Deterministic Models of a Linear Chemical Reaction with Diffusion. Ann. Probab. 19 (1991), no. 4, 1440--1462. doi:10.1214/aop/1176990219. https://projecteuclid.org/euclid.aop/1176990219