An ergodic theorem for Schlögl models with small migration

We consider a class of reaction-diffusion processes with state space N^Zd. The reaction part is described by a birth and death process where the rates are given by certain polynomials. The diffusion part is an irreducible symmetric random walk. We prove ergodicity in the case of a sufficiently small migration rate. For the proof we couple two processes and show that the density of the discrepancies goes to zero.