Invariant measures of critical spatial branching processes in high dimensions - Bramson, Maury; Cox, J. Theodore; Greven, Andreas (Annals of Probability, 1997)
We consider two critical spatial branching processes on ℝ: critical branching Brownian motion, and the critical 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, the only invariant measure is δ, the unit point mass on the empty state. In high dimensions, d ≥ 3, there is a family vθ, θ ∈ [0, ∞) of extremal invariant measures; the measures vθ are translation invariant and indexed by spatial intensity. We prove here, for d ≥ 3, that all invariant measures are convex combinations of these measure
Keywords: Critical branching Brownian motion; Critical Dawson-Watanabe process; Invariant measures