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Two representations of a conditioned superprocess

Published online by Cambridge University Press:  14 November 2011

Steven N. Evans
Steven N. Evans
Affiliation:
Department of Statistics, University of California at Berkeley, 367 Evans Hall, Berkeley, CA 94720, U.S.A.
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Synopsis

We consider a class of measure-valued Markov processes constructed by taking a superprocess over some underlying Markov process and conditioning it to stay alive forever. We obtain two representations of such a process. The first representation is in terms of an “immortal particle” that moves around according to the underlying Markov process and throws off pieces of mass, which then proceed to evolve in the same way that mass evolves for the unconditioned superprocess. As a consequence of this representation, we show that the tail σ-field of the conditioned superprocess is trivial if the tail σ-field of the underlying process is trivial. The second representation is analogous to one obtained by LeGall in the unconditioned case. It represents the conditioned superprocess in terms of a certain process taking values in the path space of the underlying process. This representation is useful for studying the “transience” and “recurrence” properties of the closed support process.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 5 , 1993 , pp. 959 - 971
Copyright
Copyright © Royal Society of Edinburgh 1993

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