Equilibrium behavior of the sexual reproduction process with rapid diffusion
type: Item
instance of: scholarly article
title: Equilibrium behavior of the sexual reproduction process with rapid diffusion
zbMATH Open document ID: 0757.60104
author: Chris Noble
published in: The Annals of Probability
publication date: 1993-01-16T00:00:00Z
review text: Consider the sexual reproduction process characterized by a birth rate \(\lambda\) and a scale parameter \(\varepsilon\). The state of the process at time \(t\) is denoted by \(_ \varepsilon\xi_ t\subset\varepsilon Z\), where points in \(_ \varepsilon\xi_ t\) are considered as being occupied by a single particle and others are vacant. The case of \(\varepsilon\downarrow 0\) of this model is usually referred to as hydrodynamic limit [cf. \textit{A. DeMasi, P. A. Ferrari} and \textit{J. L. Lebowitz}, J. Stat. Phys. 44, 589-644 (1986; Zbl 0629.60107); \textit{J. L. Lebowitz, E. Presutti} and \textit{H. Spohn}, ibid. 51, 841-862 (1988)]. The main results are that the existence of nonzero equilibrium \((\lim_{t\to\infty} P\{x\in{_ \varepsilon\xi_ t}\}\neq 0)\) depends on the value of \(\lambda\), an upper bound for the critical value of \(\lambda\) is given by \(\limsup_{\varepsilon\downarrow 0}\lambda_ c(\varepsilon)\leq 4.5\) and it is also conjectured that \(\liminf_{\varepsilon\downarrow 0}\lambda_ c(\varepsilon)\geq 4.5\). For all noncritical values of \(\lambda>4.5\) or \(\lambda<4.5\) the equilibrium density of the process is also well studied and estimated.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 35K57
Mathematics Subject Classification ID: 60F99
zbMATH DE Number: 89786
zbMATH Keywords: interacting particle system
zbMATH Keywords: reaction-diffusion equation
zbMATH Keywords: sexual reproduction process
zbMATH Keywords: hydrodynamic limit
zbMATH Keywords: equilibrium density
reviewed by: Chengxun Wu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176989802
OpenAlex ID: W2015774213
DOI: 10.1214/AOP/1176989802
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description: scientific article; zbMATH DE number 89786