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
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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