SPINES, SKELETONS AND THE STRONG LAW OF LARGE NUMBERS FOR SUPERDIFFUSIONS

• 成果类型: Article
• 署名作者: Eckhoff, Maren; Kyprianou, Andreas E.; Winkel, Matthias
• 署名单位: University of Bath; University of Oxford
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/14-AOP944
• 发表日期: 2015
• 页码: 2545-2610
• 关键词: super-brownian motion; branching diffusions; exponential-growth; local extinction; continuous-time; LIMIT-THEOREMS; exit measures; DECOMPOSITION; CONVERGENCE; systems

摘要

Consider a supercritical superdiffusion (X-t)(t >= 0) on a domain D subset of R-d with branching mechanism (x, z) bar right arrow -beta(x)z + alpha(x)z(2) + integral((0, infinity))(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, <(f, X-t)>/P-mu[< f, X-t >] -> W-infinity P-mu-almost surely as t -> infinity, where W-infinity 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.