Invariant measures of critical spatial branching processes in high dimensions

We consider two critical spatial branching processes on ℝ^d: 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 δ₀, 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 measures.