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

Maren Eckhoff, Andreas E. Kyprianou, Matthias Winkel

Research output: Contribution to journalArticlepeer-review

13 Citations (Scopus)
137 Downloads (Pure)

Abstract

Consider a supercritical superdiffusion (X t ) t≥0 on a domain D⊆R d with branching mechanism
(x,z)↦−β(x)z+α(x)z 2 +∫ (0,∞) (e −zy −1+zy)Π(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 μ ,


⟨f,X t ⟩P μ [⟨f,X t ⟩] →W ∞ P μ -almost surely as t→∞,
where W ∞ 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.
Original languageEnglish
Pages (from-to)2545-2610
Number of pages66
JournalAnnals of Probability
Volume43
Issue number 5
Early online date9 Sep 2015
DOIs
Publication statusPublished - 30 Sep 2015

Fingerprint Dive into the research topics of 'Spines, skeletons and the strong law of large numbers for superdiffusions'. Together they form a unique fingerprint.

Cite this