Invariant measures of critical spatial branching processes in high dimensions

Bramson M, Cox JT, Greven A (1997)

Publication Language: English

Publication Status: Published

Publication Type: Journal article, Original article

Publication year: 1997

Journal

Annals of Probability Institute of Mathematical Statistics (IMS)

Publisher: Institute of Mathematical Statistics (IMS)

Book Volume: 25

Pages Range: 56-70

Journal Issue: 1

URI: https://projecteuclid.org/euclid.aop/1024404278

DOI: 10.1214/aop/1024404278

Abstract

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 measures.

Authors with CRIS profile

Andreas Greven Lehrstuhl für Mathematische Stochastik

Involved external institutions

University of Minnesota (UMN) US United States (USA) (US)

How to cite

APA:

Bramson, M., Cox, J.T., & Greven, A. (1997). Invariant measures of critical spatial branching processes in high dimensions. Annals of Probability, 25(1), 56-70. https://dx.doi.org/10.1214/aop/1024404278

MLA:

Bramson, Maury, J. Theodore Cox, and Andreas Greven. "Invariant measures of critical spatial branching processes in high dimensions." Annals of Probability 25.1 (1997): 56-70.

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