The Annals of Probability

Local extinction versus local exponential growth for spatial branching processes

János Engländer and Andreas E. Kyprianou

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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.

Article information

Ann. Probab., Volume 32, Number 1A (2004), 78-99.

First available in Project Euclid: 4 March 2004

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Zentralblatt MATH identifier

Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Spatial branching processes branching diffusion superdiffusion local extinction spine decomposition immortal particle decomposition generalized principal eigenvalue


Engländer, János; Kyprianou, Andreas E. Local extinction versus local exponential growth for spatial branching processes. Ann. Probab. 32 (2004), no. 1A, 78--99. doi:10.1214/aop/1078415829.

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