The hydrodynamical behavior of the coupled branching process - Greven, Andreas (Annals of Probability, 1984)
The coupled branching process (it') is a Markov process on (M)s (S = ad) with initial distribution t and the following time evolution: At rate btq(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 p', q are transient. We show that the measures +([( +[a'itx])), (a E R+, X E Rd) converge weakly for t - X to VT(y,X). Here v, is the invariant measure of the process with:Ev,(n(x)) = p and which is also extremal in the set of all translationin- variant invariant measures of the process. The density profile T(a, x) is calculated explicitly; it is governed by the diffusion equation. <br