Spines, skeletons and the strong law of large numbers for superdiffusions

Abstract

Consider a supercritical superdiffusion $(X_{t})_{t\ge0}$ on a domain $D\subseteq\mathbb{R}^{d}$ with branching mechanism

\[(x,z)\mapsto-\beta(x)z+\alpha(x)z^{2}+\int_{(0,\infty)}(e^{-zy}-1+zy)\Pi(x,dy).\] The skeleton decomposition provides a pathwise description of the process in terms of immigration along a branching particle diffusion. We use this decomposition to derive the strong law of large numbers (SLLN) for a wide class of superdiffusions from the corresponding result for branching particle diffusions. That is, we show that for suitable test functions $f$ and starting measures $\mu$,

\[\frac{\langle f,X_{t}\rangle}{P_{\mu}[\langle f,X_{t}\rangle]}\to W_{\infty}\qquad P_{\mu}\mbox{-}\mathrm{almost}\ \mathrm{surely}\ \mathrm{as}\ t\to\infty,\] where $W_{\infty}$ is a finite, non-deterministic random variable characterized as a martingale limit. Our method is based on skeleton and spine techniques and offers structural insights into the driving force behind the SLLN for superdiffusions. The result covers many of the key examples of interest and, in particular, proves a conjecture by Fleischmann and Swart [Stochastic Process. Appl. 106 (2003) 141–165] for the super-Wright–Fisher diffusion.