Density-dependent limits for a nonlinear reaction-diffusion model
type: Item
instance of: scholarly article
title: Density-dependent limits for a nonlinear reaction-diffusion model
zbMATH Open document ID: 0843.60057
published in: The Annals of Probability
publication date: 1996-08-13T00:00:00Z
review text: A density-dependent stochastic reaction-diffusion model describing a chemical reaction is studied. The model is constructed by partitioning \([0,1]^p\), \(p \geq 1\), into \(N^p\) congruent cells, distributing approximately \(N^{p_l}\) particles in the system and allowing particles to diffuse between cells by rate \(N^2\) random walks and to react within cells as density dependent birth and death processes. Dividing cell numbers by \(l\) one gets a step-function-valued Markov process \(X_{N,l} (t,r)\), \(t \geq 0\), \(r \in [0,1]^p\). Following Kotelenez in comparing \(X_{N,l}\) with its deterministic counterpart \(\psi\) two laws of large numbers for \(X_{N,l}\), when \(l \to \infty\) as \(N \to \infty\) and for constant \(l\), are proved. Moreover, in the case \(l = N\), \(p = 1\), a central limit theorem for \(X_{N,l}\) is obtained, thus completing the case \(N/l \to 0\) considered earlier by \textit{P. Kotelenez} [Probab. Theory Relat. Fields 78, No. 1, 11-37 (1988; Zbl 0628.60108)] and by the author [Stochastic Processes Appl. 45, No. 2, 193-207 (1993; Zbl 0773.60052)].
reviewed by: T. Inglot
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60F17
Mathematics Subject Classification ID: 60G15
zbMATH DE Number: 828376
zbMATH Keywords: stochastic partial differential equation
zbMATH Keywords: reaction-diffusion model
zbMATH Keywords: central limit theorem
zbMATH Keywords: law of large numbers
author: Douglas Blount
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176988494
OpenAlex ID: W2034994228
DOI: 10.1214/AOP/1176988494
description: scientific article; zbMATH DE number 828376