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A decomposition of the (1 + β)-superprocess conditioned on survival

Published online by Cambridge University Press:  12 July 2007

A. M. Etheridge
Affiliation:
Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK
D. R. E. Williams
Affiliation:
Department of Statistics, University of Oxford, 1 South Parks Road, Oxford OX1 3TG, UK

Abstract

There is a large catalogue of decompositions of conditioned superprocesses in terms of an ‘immortal backbone’ or ‘skeleton’ along the branches of which mass is constantly immigrated. We add to this with a study of the (infinite-variance) (1 + β)-superprocess, conditioned on survival until some fixed time T. As one would expect, we see a Poisson number of immortal trees (conditioned on there being at least one), along which mass (conditioned to die before time T) is immigrated. However, here we see a new source of immigration. Not only is mass immigrated along the branches of the immortal trees, but also there is an extra burst of immigration whenever the immortal tree branches. Moreover, the rate of immigration along the branches is no longer deterministic. In the limit as T → ∞, the immortal trees degenerate to the Evans immortal particle and the immigration (of unconditioned mass) along the particle is dictated by a stable subordinator.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2003

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