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Local extinction in continuous-state branching processes with immigration. (English) Zbl 1314.60146

Let \(\{Y_t\}\) be a one-dimensional continuous-state branching process with immigration, started at zero. Denote by \(\Psi\) the Laplace exponent of the spectrally positive Lévy process describing the reproduction and by \(\Phi\) the Laplace exponent of the subordinator describing the immigration. The zero set is defined as the closure of \(\{t\leq 0: Y_t= 0\}\). Exploiting a connection between the zero set and the random cut-out sets defined by B. B. Mandelbrot [Z. Wahrscheinlichkeitstheor. Verw. Geb. 22, 145–157 (1972; Zbl 0234.60102)], the authors construct the zero set as infinitely regenerative set in terms of \(\Psi\) and \(\Phi\). They derive necessary and sufficient conditions for the zero set to be polar, transient, or recurrent, respectively. More detailed results are obtained under the assumption that \(\Phi/\Psi\) is regularly varying, with special attention being paid to the case of stable and gamma mechanisms.
Finally, the connection between random covering of the real line and the zero set is extended to generalized Ornstein-Uhlenbeck (OU) processes, characterizing, in particular, the zero set of OU processes driven by stable Lévy processes.

MSC:

60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
60G51 Processes with independent increments; Lévy processes
60G52 Stable stochastic processes

Citations:

Zbl 0234.60102
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References:

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