Ergodicity of critical spatial branching processes in low dimensions

Bramson M, Cox JT, Greven A (1993)


Publication Language: English

Publication Type: Journal article, Original article

Publication year: 1993

Journal

Publisher: Institute of Mathematical Statistics (IMS)

Book Volume: 21

Pages Range: 1946-1957

Journal Issue: 4

URI: https://projecteuclid.org/euclid.aop/1176989006

DOI: 10.1214/aop/1176989006

Abstract

We consider two critical spatial branching processes on Rd" role="presentation">ℝd: critical branching Brownian motion, and the 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" role="presentation">d≤2, the unique invariant measure with finite intensity is δ0" role="presentation">δ0, the unit point mass on the empty state. In high dimensions, d≥3" role="presentation">d≥3, there is a one-parameter family of nondegenerate invariant measures. We prove here that for d≤2,δ0" role="presentation">d≤2,δ0 is the only invariant measure. In our proof we make use of sub- and super-solutions of the partial differential equation ∂u/∂t=(1/2)Δu−bu2" role="presentation">∂u/∂t=(1/2)Δu−bu2.

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APA:

Bramson, M., Cox, J.T., & Greven, A. (1993). Ergodicity of critical spatial branching processes in low dimensions. Annals of Probability, 21(4), 1946-1957. https://doi.org/10.1214/aop/1176989006

MLA:

Bramson, Maury, J. Theodore Cox, and Andreas Greven. "Ergodicity of critical spatial branching processes in low dimensions." Annals of Probability 21.4 (1993): 1946-1957.

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