## Local extinction versus local exponential growth for spatial branching processes.(English)Zbl 1056.60083

Let $$X$$ be either the branching diffusion with generator $$Lu+ \beta(u^2-u)$$ on a domain $$D$$ in $$R^d$$, $$\beta$$ nonnegative, bounded, and not identically $$0$$, or the superprocess with generator $$Lu+\beta u-\alpha u^2$$, $$\alpha$$ positive and $$\beta$$ bounded from above. Let the starting measure $$\mu$$ be not identically vanishing, finite, with support compactly embeddable in $$D$$. $$X$$ is said to exhibit weak local extinction under $$P_\mu$$, if $$P_\mu(\lim\sup_{t\uparrow \infty} X_t(B)=0)=1$$ for every Borel set $$B$$ embeddable in $$D$$. The process $$X$$ exhibits weak local extinction if and only if the generalized principal eigenvalue $$\lambda_c$$ of $$L+\beta$$ is nonpositive. In particular, this property does not depend on the starting measure. If $$\lambda_c>0$$, then for any $$\lambda<\lambda_c$$ and any nonempty set $$B$$ compactly embeddable in $$D$$, $P_\mu \Bigl(\lim\sup_{t\uparrow\infty} \exp\{-\lambda t\}X_t(B)=\infty\Bigr)>0, \quad\text{and}\quad P_\mu \Bigl(\lim\sup_{t\uparrow \infty} \exp\{-\lambda_ct\} X_t(B)<\infty\Bigr)=1.$ The method of proof is probabilistic, using “spine” or “immortal particle” decomposition.

### MSC:

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

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