Ergodic theorems for the multitype contact process

This paper studies a process involving competition of two types of particles (1 and 2) for the empty space (0). Each site of the lattice Z^d is therefore in one of three possible states: 0, 1, or 2. Particles of each type die with rate 1, while an empty site becomes occupied by a particle of type i with rate λ_i (proportion of neighbors of type i). The set of neighbors of a site x is of the form {y:{norm of matrix}x-y{norm of matrix}≦J}, for a positive integer J and a norm {norm of matrix}·{norm of matrix}. Assuming there are only 0's and 1's present at the beginning, the process reduces to the contact process, with the critical rate of survival of 1's being λ_c. The basic problem we address is the existence of equilibria in which both types of particles coexist. Without loss of generality, one can restrict to the case λ₂ ≧ λ₁ > λ_cand in this case we show: (1) If λ₂ > λ₁, and the initial state is translation invariant and contains infinitely many 2's, then the 1's go away and the process approaches the invariant measure of the contact process with only 2's and 0's present, (2) If λ₂ = λ₁, and d≦2, then clustering occurs: starting from a translation invariant initial measure with no mass on all 0's, the process converges weakly to a convex combination of the two invariant measures obtained with only one type of particles present, and (3) If λ₂ = λ₁, and d≧3, then there is a one-parameter family of invariant measures including both types.