The Hydrodynamical Behavior of the Coupled Branching Process

Abstract

The coupled branching process $(\eta^\mu_t)$ is a Markov process on $(\mathbb{N})^S (S = \mathbb{Z}^d)$ with initial distribution $\mu$ and the following time evolution: At rate $b\eta(x)$ a particle is born at site $x$, which moves instantaneously to a site $y$ chosen with probability $q(x, y)$. All particles at a site die at rate $pd$, individual particles die independent from each other at rate $(1 - p)d$. Furthermore, all particles perform independent continuous time random walks with kernel $p(x, y)$. We consider here the case $b = d$ and the symmetrized kernels $\hat p, \hat q$ are transient. We show that the measures $\mathscr{L}(\eta^\mu_t(\cdot + \lbrack\alpha \sqrt{tx}\rbrack)), (\alpha \in \mathbb{R}^+, x \in \mathbb{R}^d)$ converge weakly for $t \rightarrow \infty$ to $\nu_{\tau(a,x)}$. Here $\nu_\rho$ is the invariant measure of the process with: $E^{\nu_\rho}(\eta(x)) = \rho$ and which is also extremal in the set of all translationinvariant invariant measures of the process. The density profile $\tau(\alpha, x)$ is calculated explicitly; it is governed by the diffusion equation.