ERGODICITY OF CRITICAL SPATIAL BRANCHING-PROCESSES IN LOW DIMENSIONS

We consider two critical spatial branching processes on R(d): critical branching Brownian motion, and the Dawson-Watanabe process. A basic feature of these processes is that their ergodic behavior is highly dimension-dependent. It is known that in low dimensions, d less than or equal to 2, the unique invariant measure with finite intensity is delta(0), the unit point mass on the empty state. In high dimensions, d greater than or equal to 3, there is a one-parameter family of nondegenerate invariant measures. We prove here that for d less than or equal to 2, delta(0) is the only invariant measure. In our proof we make use of sub- and super-solutions of the partial differential equation partial derivative u/partial derivative t = (1/2)Delta u - bu(2).