Strong Law of Large Numbers for branching diffusions

Let $X$ be the branching particle diffusion corresponding to the operator $Lu+\beta (u^{2}-u)$ on $D\subseteq \mathbb{R}^{d}$ (where $\beta \geq 0$ 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'.