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January 2004 Local extinction versus local exponential growth for spatial branching processes
János Engländer, Andreas E. Kyprianou
Ann. Probab. 32(1A): 78-99 (January 2004). DOI: 10.1214/aop/1078415829

Abstract

Let X be either the branching diffusion corresponding to the operator $Lu+\beta (u^2-u)$ on $D\subseteq $ $\mathbb{R}^{d}$ [where $\beta (x) \geq 0$ and $\beta\not\equiv 0$ is bounded from above] or the superprocess corresponding to the operator $Lu+\beta u -\alpha u^2$ on $D\subseteq $ $\mathbb{R}^{d}$ (with $\alpha>0$ and $\beta$ is bounded from above but no restriction on its sign). Let $\lambda _{c}$ denote the generalized principal eigenvalue for the operator $L+\beta $ on $D$. We prove the following dichotomy: either $\lambda _{c}\leq 0$ and X exhibits local extinction or $\lambda _{c}> 0$ and there is exponential growth of mass on compacts of D with rate $\lambda _{c}$. For superdiffusions, this completes the local extinction criterion obtained by Pinsky [Ann. Probab. 24 (1996) 237--267] and a recent result on the local growth of mass under a spectral assumption given by Engländer and Turaev [Ann. Probab. 30 (2002) 683--722]. The proofs in the above two papers are based on PDE techniques, however the proofs we offer are probabilistically conceptual. For the most part they are based on "spine'' decompositions or "immortal particle representations'' along with martingale convergence and the law of large numbers. Further they are generic in the sense that they work for both types of processes.

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János Engländer. Andreas E. Kyprianou. "Local extinction versus local exponential growth for spatial branching processes." Ann. Probab. 32 (1A) 78 - 99, January 2004. https://doi.org/10.1214/aop/1078415829

Information

Published: January 2004
First available in Project Euclid: 4 March 2004

zbMATH: 1056.60083
MathSciNet: MR2040776
Digital Object Identifier: 10.1214/aop/1078415829

Subjects:
Primary: 60J80

Rights: Copyright © 2004 Institute of Mathematical Statistics

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Vol.32 • No. 1A • January 2004
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