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Transactions of the American Mathematical Society

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

 

 

Superprocesses and their linear additive functionals


Author: E. B. Dynkin
Journal: Trans. Amer. Math. Soc. 314 (1989), 255-282
MSC: Primary 60J80; Secondary 60G57, 60H05, 60J55
DOI: https://doi.org/10.1090/S0002-9947-1989-0930086-7
MathSciNet review: 930086
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Abstract: Let $ X = ({X_t},P)$ be a measure-valued stochastic process. Linear functionals of $ X$ are the elements of the minimal closed subspace $ L$ of $ {L^2}(P)$ which contains all $ {X_t}(B)$ with $ \smallint {{X_t}{{(B)}^2}\;dP\; < \infty } $. Various classes of $ L$-valued additive functionals are investigated for measure-valued Markov processes introduced by Watanabe and Dawson. We represent such functionals in terms of stochastic integrals and we derive integral and differential equations for their Laplace transforms. For an important particular case--"weighted occupation times"--such equations have been established earlier by Iscoe.

We consider Markov processes with nonstationary transition functions to reveal better the principal role of the backward equations. This is especially helpful when we derive the formula for the Laplace transforms.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1989-0930086-7
Keywords: Measure-valued Markov processes, additive funtionals, stochastic integrals
Article copyright: © Copyright 1989 American Mathematical Society