Invariant measures of critical spatial branching processes in high dimensions

Maury Bramson, J. T. Cox, Andreas Greven

Research output: Contribution to journalArticle

7 Scopus citations

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 languageEnglish (US)
Pages (from-to)56-70
Number of pages15
JournalAnnals of Probability
Volume25
Issue number1
DOIs
StatePublished - Jan 1997

Keywords

  • Critical Dawson-Watanabe process
  • Critical branching Brownian motion
  • Invariant measures

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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