Intermittency and ageing for the symbiotic branching model
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 \(\alpha>0\), or \(\varrho\leq0\) and \(\alpha>1\). For the four cases \(\varrho=0\), \(\varrho<0\) and \(\alpha=1\), \(\alpha<1\), there is ageing and the dependence of the limit on the sequence \(s(t)\) is computed explicitly.
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