Abstract
We consider two critical spatial branching processes on ℝd: 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.
Original language | English (US) |
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Pages (from-to) | 56-70 |
Number of pages | 15 |
Journal | Annals of Probability |
Volume | 25 |
Issue number | 1 |
DOIs | |
State | Published - Jan 1997 |
Keywords
- Critical Dawson-Watanabe process
- Critical branching Brownian motion
- Invariant measures
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty