Strong Law of Large Numbers for Branching Diffusions

Let X be the branching particle diffusion corresponding to the operator Lu + β(u2 − u) on D ⊆ Rd (where β ≥ 0 and β 6≡ 0). Let λc denote the generalized principal eigenvalue for the operator L + β on D and assume that it is finite. When λc > 0 and L+β−λc satisfies certain spectral theoretical conditions, we prove that the random measure exp{−λct}Xt 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 [ET, EW]. We extend significantly the results in [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’.