Strong Law of Large Numbers for branching diffusions

TL;DR

This paper establishes a strong law of large numbers for branching diffusions, showing almost sure convergence of rescaled measures under spectral conditions, extending classical results with modern martingale and spine techniques.

Contribution

It significantly extends previous results on branching diffusions by proving almost sure convergence under spectral conditions using modern probabilistic methods.

Findings

Almost sure convergence of rescaled branching diffusion measures

Extension of classical results to broader spectral conditions

Application of martingale and spine decomposition techniques

Abstract

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