Law of large numbers for branching symmetric Hunt processes with measure-valued branching rates

TL;DR

This paper proves weak and strong laws of large numbers for a broad class of symmetric Hunt processes with measure-valued, state-dependent branching rates, extending previous results to new examples.

Contribution

It introduces new laws of large numbers for symmetric Hunt processes with general, measure-valued branching mechanisms, broadening the scope of prior work.

Findings

Established weak and strong laws of large numbers for the processes.

Extended results to processes with measure-valued, state-dependent branching rates.

Applicable to new examples beyond previous models.

Abstract

We establish weak and strong law of large numbers for a class of branching symmetric Hunt processes with the branching rate being a smooth measure with respect to the underlying Hunt process, and the branching mechanism being general and state-dependent. Our work is motivated by recent work on strong law of large numbers for branching symmetric Markov processes by Chen-Shiozawa [J. Funct. Anal., 250, 374--399, 2007] and for branching diffusions by Engl\"ander-Harris-Kyprianou [Ann. Inst. Henri Poincar\'e Probab. Stat., 46, 279--298, 2010]. Our results can be applied to some interesting examples that are covered by neither of these papers.