Strong law of large numbers for branching diffusions

DOI10.1214/09-AIHP203zbMATH Open1196.60139arXiv0709.0272OpenAlexW2009685090MaRDI QIDQ974778

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Publication date: 7 June 2010

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Abstract: Let $X$ be the branching particle diffusion corresponding to the operator $$ on $Dsubseteqmathbb{R}^d$ (where $$ and $$). Let $lambd{a}_c$ denote the generalized principal eigenvalue for the operator $$ on $D$ and assume that it is finite. When $lambd{a}_c>0$ and $$ satisfies certain spectral theoretical conditions, we prove that the random measure $exp-lambd{a}_{c}t{X}_t$ 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'.

Full work available at URL: https://arxiv.org/abs/0709.0272

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