Spines, skeletons and the Strong Law of Large Numbers for superdiffusions

TL;DR

This paper establishes a Strong Law of Large Numbers for superdiffusions using skeleton decomposition and spine techniques, providing new insights into their long-term behavior and confirming a conjecture for the super-Wright-Fisher diffusion.

Contribution

It introduces a novel method based on skeleton and spine techniques to prove the SLLN for a broad class of superdiffusions, including key models like the super-Wright-Fisher.

Findings

Proves SLLN for superdiffusions with a wide class of branching mechanisms.

Confirms a conjecture by Fleischmann and Swart for the super-Wright-Fisher diffusion.

Provides structural insights into the driving forces behind the SLLN.

Abstract

Consider a supercritical superdiffusion (X_t) on a domain D subset R^d with branching mechanism -\beta(x) z+\alpha(x) z^2 + int_{(0,infty)} (e^{-yz}-1+yz) 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, < f,X_t>/P_{mu}[< f,X_t>] -> W_{infty}, P_{mu}-almost surely as t->infty, where W_{infty} is a finite, non-deterministic random variable characterised 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 for the super-Wright-Fisher diffusion.