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September 2015 Spines, skeletons and the strong law of large numbers for superdiffusions
Maren Eckhoff, Andreas E. Kyprianou, Matthias Winkel
Ann. Probab. 43(5): 2545-2610 (September 2015). DOI: 10.1214/14-AOP944

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.

Citation

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Maren Eckhoff. Andreas E. Kyprianou. Matthias Winkel. "Spines, skeletons and the strong law of large numbers for superdiffusions." Ann. Probab. 43 (5) 2545 - 2610, September 2015. https://doi.org/10.1214/14-AOP944

Information

Received: 1 September 2013; Revised: 1 May 2014; Published: September 2015
First available in Project Euclid: 9 September 2015

zbMATH: 1330.60052
MathSciNet: MR3395469
Digital Object Identifier: 10.1214/14-AOP944

Subjects:
Primary: 60J68
Secondary: 60F15, 60J80

Rights: Copyright © 2015 Institute of Mathematical Statistics

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Vol.43 • No. 5 • September 2015
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