Spatial branching populations with long individual lifetimes

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.