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, nondeterministic 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 superWright–Fisher diffusion.
(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, nondeterministic 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 superWright–Fisher diffusion.
Original language  English 

Pages (fromto)  25452610 
Number of pages  66 
Journal  Annals of Probability 
Volume  43 
Issue number  5 
Early online date  9 Sep 2015 
DOIs  
Publication status  Published  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.
Profiles

Andreas Kyprianou
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Probability Laboratory at Bath
 Institute for Mathematical Innovation (IMI)  Director of the Bath Institute for Mathematical Innovation
Person: Research & Teaching, Teaching & Other