A phase transition for the coupled branching process. I: The ergodic theory in the range of finite second moments
We consider a certain Markov process \(\eta_ t^{\mu}\) on \({\mathbb{N}}^ S\), \(S={\mathbb{Z}}^ n\). The random variable \(\eta_ t^{\mu}(x)\) is interpreted as the number of particles at x at time t. The initial distribution of this process is a translation invariant measure \(\mu\) with \(\int \eta (x)d\mu <\infty\). The evolution is as follows: At rate \(b\eta\) (x) a particle is born at x but moves instantaneously to y chosen with probability q(x,y). All particles at a site die at rate pd with \(p\in [0,1]\), \(d\in {\mathbb{R}}^+\), and individual particles die independently from each other at rate (1-p)d. Every particle moves independently of everything else according to a continuous time random walk. We are mainly interested in the case \(b=d\) and \(n\geq 3\). The process exhibits a phase transition with respect to the parameter p: For \(p<p^*\) all weak limit points of \({\mathcal L}(\eta_ t^{\mu})\) as \(t\to \infty\) still have particle density \(\int \eta (x)d\mu\). For \(p>p^*\), \({\mathcal L}(\eta_ t^{\mu})\) converges as \(t\to \infty\) to the measure concentrated on the configuration identically 0. We calculate \(p^*\) as well as \(p^{(n)}\), the points with the property that the extremal invariant measures have for \(p>p^{(n)}\) infinite n-th moment of \(\eta\) (x) and for \(p<p^{(n)}\) finite n-th moment. We show the case \(1>p^*>p^{(2)}>p^{(3)}\geq...\geq p^{(n)}>0\), \(p^{(n)}\downarrow 0\) occurs for suitable values of the other parameters. For \(p<p^{(2)}\) we prove the system has a one parameter set \((\nu_{\rho})_{\rho \in {\mathbb{R}}^+}\) of extremal invariant measures and we determine their domain of attraction. This Part I contains statements of all results but only the proofs of the results about the process for values of p with \(p<p^{(2)}\) and the behaviour of moments.
scientific article; zbMATH DE number 4138919
scientific article