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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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