Strong law of large number for branching Hunt processes

TL;DR

This paper establishes a strong law of large numbers for a class of branching Hunt processes using spine decomposition, requiring only an $L ext{log}L$ condition, thus advancing understanding of their long-term behavior.

Contribution

It proves a strong law of large numbers for branching Hunt processes under minimal integrability conditions, extending previous results in the field.

Findings

Strong law of large numbers proven for branching Hunt processes

Uses spine decomposition as main analytical tool

Requires only an $L ext{log}L$ integrability condition

Abstract

In this paper we prove that, under certain conditions, a strong law of large number holds for a class of branching particle systems $X$ corresponding to the parameters $(Y,\beta,\psi)$, where $Y$ is a Hunt process and $\psi$ is the generating function for the offspring. The main tool of this paper is the spine decomposition and we only need a $L\log L$ condition.