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

 

A space-time property of a class of measure-valued branching diffusions


Author: Edwin A. Perkins
Journal: Trans. Amer. Math. Soc. 305 (1988), 743-795
MSC: Primary 60G57; Secondary 60J60, 60J80
MathSciNet review: 924777
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Abstract: If $ d > \alpha $, it is shown that the $ d$-dimensional branching diffusion of index $ \alpha $, studied by Dawson and others, distributes its mass over a random support in a uniform manner with respect to the Hausdorff $ {\phi _\alpha }$-measure, where $ {\phi _\alpha }(x) = {x^\alpha }\log \log 1/x$. More surprisingly, it does so for all positive times simultaneously. Slightly less precise results are obtained in the critical case $ d = \alpha $. In particular, the process is singular at all positive times a.s. for $ d \geqslant \alpha $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1988-0924777-0
PII: S 0002-9947(1988)0924777-0
Keywords: Measure-valued diffusion, Hausdorff measure, critical branching process, Loeb space
Article copyright: © Copyright 1988 American Mathematical Society