Strong Law of Large Numbers for branching diffusions

Abstract

Let $X$ be the branching particle diffusion corresponding to the operator $Lu+\beta (u^2-u)$ on $D\subseteq {\mathbb{R}}^d$ (where $\beta \ge 0$ and $\beta \not\equiv 0$). Let ${\lambda }_c$ denote the generalized principal eigenvalue for the operator $L+\beta$ on $D$ and assume that it is finite. When ${\lambda }_c>0$ and $L+\beta -{\lambda }_c$ satisfies certain spectral theoretical conditions, we prove that the random measure $\exp \{-{\lambda }_{c}t\}{X}_t$ converges almost surely in the vague topology as $t$ tends to infinity. This result is motivated by a cluster of articles due to Asmussen and Hering dating from the mid-seventies as well as the more recent work concerning analogous results for superdiffusions of \cite{ET,EW}. We extend significantly the results in \cite{AH76,AH77} and include some key examples of the branching process literature. As far as the proofs are concerned, we appeal to modern techniques concerning martingales and `spine' decompositions or `immortal particle pictures'.

21 pages

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