Intermittency and ageing for the symbiotic branching model
type: Item
instance of: scholarly article
title: Intermittency and ageing for the symbiotic branching model
zbMATH Open document ID: 1222.60075
DOI: 10.1214/09-AIHP356
author: Frank Aurzada
author: Leif Döring
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2011-05-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/0905.1003
full work available at URL: https://eudml.org/doc/240811
review text: In [Stochastic Processes Appl. 114, No.~1, 127--160 (2004; Zbl 1072.60086)], \textit{A. M. Etheridge} and \textit{K. Fleischmann} introduced the symbiotic branching model as the solution of the following system of coupled stochastic differential equations: \[ du_{i,t}(k)=\sum_{l\in\mathbb{Z}^d}a(k,l)\big[u_{i,t}(l)-u_{i,t}(k)\big]\,dt\;+\;\sqrt{\kappa u_{1,t}(k)u_{2,t}(k)}\,dW_{i,t}(k),\quad k\in\mathbb{Z}^d,\;t\geq 0,\;i=1,2. \] Here, \(\kappa>0\) is a fixed parameter of the model and \(((W_1(k),W_2(k)),k\in\mathbb{Z}^d)\) is an independent family of pairs of correlated standard Brownian motions; i.e., \(\mathrm{Cov}[W_{1,s}(k),W_{2,t}(k)]=\min(s,t)\varrho\), where \(\varrho\in[-1,1]\) is also a fixed parameter of the model. Furthermore, \(a\) is the transition matrix of a random walk on \(\mathbb{Z}^d\). It is well known that, for suitable initial conditions (Liggett-Spitzer condition), there exists a unique weak solution that takes nonnegative values. In particular, this is true for the constant inital condition \(u_{1,0}=u_{2,0}\equiv1\) that will be considered henceforth. Consider the second Lyapunov exponent \[ \gamma_2=\gamma_2(\kappa,\varrho):=\lim_{t\to\infty}t^{-1}\log{\text E}\big[u_{1,t}(k)^2\big]. \] The model is said to be intermittent if \(\gamma_2>0\). Let \(\bar G\) denote the Green function of the difference of two continuous time random walks with transition rates \(a\). The authors show that the model is intermittent if and only if \[ \kappa\varrho>1/\bar G. \] In particular, it is not intermittent for \(\varrho\leq 0\). The model is said to exhibit ageing if the limit \[ \lim_{s,t\to\infty}\mathrm{cor}\big[u_{1,{s+t}}(k),\,u_{1,t}(k)\big] \] depends on the choice of how \(s=s(t)\to\infty\) as \(t\to\infty\). Consider the special situation of a symmetric interaction kernel \(a\) whose transition probabilities fulfil \(p_t(0,0)\sim ct^{-\alpha}\) as \(t\to\infty\) for some \(\alpha>0\). The authors show that there is no ageing if \(\varrho>0\) and…
reviewed by: Achim Klenke
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60J80
zbMATH DE Number: 5897742
zbMATH Keywords: symbiotic branching model
zbMATH Keywords: interacting diffusions
zbMATH Keywords: intermittency
zbMATH Keywords: ageing
zbMATH Keywords: parabolic Anderson model
zbMATH Keywords: mutually catalytic branching model
MaRDI profile type: Publication
arXiv ID: 0905.1003
cites work: Q3992980
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OpenAlex ID: W4297800305
description: scientific article