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 subset of R-d (where beta >= 0 and beta not equivalent to 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 [Ann. Probab. 30 (2002) 683-722, Ann. Inst. H. Poincare Probab. Statist. 42 (2006) 171-185]. We extend significantly the results in [Z. Wahrsch. Verw. Gebiete 36 (1976) 195-212, Math. Scand. 39 (1977) 327-342, J. Funct. Anal. 250 (2007) 374-399] 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."