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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Absolute continuity results for superprocesses with some applications
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by Steven N. Evans and Edwin Perkins PDF
Trans. Amer. Math. Soc. 325 (1991), 661-681 Request permission

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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Additional Information
  • © Copyright 1991 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 325 (1991), 661-681
  • MSC: Primary 60G30; Secondary 60J80
  • DOI: https://doi.org/10.1090/S0002-9947-1991-1012522-2
  • MathSciNet review: 1012522