Strong law of large number of a class of super-diffusions

TL;DR

This paper establishes a strong law of large numbers for a class of super-diffusions governed by a specific evolution equation, under certain boundedness conditions on the branching mechanism.

Contribution

It proves a strong law of large numbers for super-diffusions with a particular branching mechanism and diffusion generator, extending previous results to a broader class.

Findings

Strong law of large numbers proved for super-diffusions.

Applicable to super-diffusions with quadratic and integral branching mechanisms.

Results hold under boundedness conditions on the branching measure.

Abstract

In this paper we prove that, under certain conditions, a strong law of large numbers holds for a class of super-diffusions $X$ corresponding to the evolution equation $\partial_t u_t=L u_t+\beta u_t-\psi(u_t)$ on a bounded domain $D$ in $\R^d$, where $L$ is the generator of the underlying diffusion and the branching mechanism $\psi(x,\lambda)=1/2\alpha(x)\lambda^2+\int_0^\infty (e^{-\lambda r}-1+\lambda r)n(x, {\rm d}r)$ satisfies $\sup_{x\in D}\int_0^\infty (r\wedge r^2) n(x,{\rm d}r)<\infty$.