Ergodicity of critical spatial branching processes in low dimensions - Bramson, Maury; Cox, J. Theodore; Greven, Andreas (Annals of Probability, 1993)
We consider two critical spatial branching processes on Rd" role="presentation">ℝd: 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≤2" role="presentation">d≤2, the unique invariant measure with finite intensity is δ0" role="presentation">δ0, the unit point mass on the empty state. In high dimensions, d≥3" role="presentation">d≥3, there is a one-parameter family of nondegenerate invariant measures. We prove here that for d≤2,δ0" role="presentation">d≤2,δ0 is the only invariant measure. In our proof we make use of sub- and super-solutions of the partial differential equation ∂u/∂t=(1/2)Δu−bu2" role="presentation">∂u/∂t=(1/2)Δu−bu