Spines, skeletons and the strong law of large numbers for superdiffusions
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Publication:888536
DOI10.1214/14-AOP944zbMATH Open1330.60052arXiv1309.6196OpenAlexW2168621054MaRDI QIDQ888536
Author name not available (Why is that?)
Publication date: 30 October 2015
Published in: (Search for Journal in Brave)
Abstract: Consider a supercritical superdiffusion (X_t) on a domain D subset R^d with branching mechanism -�eta(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.
Full work available at URL: https://arxiv.org/abs/1309.6196
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