A PHASE-TRANSITION FOR THE COUPLED BRANCHING-PROCESS .1. THE ERGODIC-THEORY IN THE RANGE OF FINITE 2ND MOMENTS

We consider a particular Markov process eta-t-mu on NS, S = Zn. 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 integral-eta-(x)d-mu < infinity. 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-epsilon [0, 1], d-epsilon-R+, and individual particles die independently from each other at rate (1 - p)d. Every particle moves independently of everthing else according to a continuous time random walk. We are mainly interested in the case b = d and n greater-than-or-equal-to 3. The process exhibits a phase transition with respect to the parameter p: For p < p* all weak limit points of L (eta-t-mu) as t --> infinity still have particle density integral-eta-(x)d-mu. For p > p*, L (eta-t-mu) converges as t --> infinity 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) greater-than-or-equal-to . . . greater-than-or-equal-to p(n) > 0, p(n) down 0 occurs for suitable values of the other parameters. For p < p(2) we prove the system has a one parameter set (v-rho)-rho-epsilon-R+ of extermal invariant measures and we determine their domain of attraction. 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 the n-th moments and p(n).