Geodesic
Spatial branching populations with long individual lifetimes
Teoriâ veroâtnostej i ee primeneniâ, Tome 43 (1998) no. 4, pp. 655-671

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It is proved that for critical branching particle systems in $\mathbb{R}^{d}$ with symmetric $\alpha$-stable individual motion, $(1+\beta)$-stable branching, and individual lifetime distribution with a tail of exponent $\gamma \le 1$, the system initiated by a Poisson field of particles in $\\mathbb{R}^d$ dies out locally if $d {\alpha \gamma }/\beta$, converges to a Poisson limit of full intensity if $d > {\alpha \gamma }/\beta $, and converges to a nontrivial limit along a subsequence as $d={ \alpha \gamma }/\beta $. Moreover, for a general nonarithmetic lifetime distribution with finite expectation, it is shown that, as $t\rightarrow \infty $, the system converges to a nontrivial limit of full intensity if $ d > \alpha /\beta $ and goes to local extinction otherwise.
Mots-clés : extinction, persistence, stable distributions
Keywords: survival, regularly varying functions, renewal equations. 
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     author = {A. Wakolbinger and V. A. Vatutin},
     title = {Spatial branching populations with long individual lifetimes},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {655--671},
     publisher = {mathdoc},
     volume = {43},
     number = {4},
     year = {1998},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1998_43_4_a1/}
}
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A. Wakolbinger; V. A. Vatutin. Spatial branching populations with long individual lifetimes. Teoriâ veroâtnostej i ee primeneniâ, Tome 43 (1998) no. 4, pp. 655-671. http://geodesic.mathdoc.fr/item/TVP_1998_43_4_a1/