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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Absolute continuity results for superprocesses with some applications


Authors: Steven N. Evans and Edwin Perkins
Journal: Trans. Amer. Math. Soc. 325 (1991), 661-681
MSC: Primary 60G30; Secondary 60J80
MathSciNet review: 1012522
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Abstract: Let $ {X^1}$ and $ {X^2}$ be instances of a measure-valued Dawson-Watanabe $ \xi $-super process where the underlying spatial motions are given by a Borel right process, $ \xi $, and where the branching mechanism has finite variance. A necessary and sufficient condition on $ X_0^1$ and $ X_0^2$ is found for the law of $ X_s^1$ to be absolutely continuous with respect to the law of $ X_t^2$. The conditions are the natural absolute continuity conditions on $ \xi $, but some care must be taken with the set of times $ s$, $ t$ being considered. The result is used to study the closed support of super-Brownian motion and give sufficient conditions for the existence of a nontrivial "collision measure" for a pair of independent super-Lévy processes or, more generally, for a super-Lévy process and a fixed measure. The collision measure gauges the extent of overlap of the two measures. As a final application, we give an elementary proof of the instantaneous propagation of a super-Lévy process to all points to which the underlying Lévy process can jump. This result is then extended to a much larger class of superprocesses using different techniques.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1012522-2
PII: S 0002-9947(1991)1012522-2
Keywords: Superprocesses, absolute continuity, measure-valued diffusion, measure-valued branching process, random measure
Article copyright: © Copyright 1991 American Mathematical Society