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Transactions of the American Mathematical Society

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

 

 

A formula for semigroups, with an application to branching diffusion processes


Author: Stanley A. Sawyer
Journal: Trans. Amer. Math. Soc. 152 (1970), 1-38
MSC: Primary 60.67
DOI: https://doi.org/10.1090/S0002-9947-1970-0266319-8
MathSciNet review: 0266319
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Abstract: A Markov process $ P = \{ {x_t}\} $ proceeds until a random time $ \tau $, where the distribution of $ \tau $ given $ P$ is $ \exp ( - {\phi _t})$ for finite additive functional $ \{ {\phi _t}\} $, at which time it jumps to a new position given by a substochastic kernel $ K({x_\tau },A)$. A new time $ \tau '$ is defined, the process again jumps at a time $ \tau + \tau '$ and so forth, producing a new Markov process $ P'$. A formula for the infinitesimal generator of the new process (in terms of the i.g. of the old) is then derived. Using branching processes and local times $ \{ {\phi _t}\} $, classical solutions of some linear partial differential equations with nonlinear boundary conditions are constructed. Also, conditions are given guaranteeing that a given Markov process is of type $ P'$ for some triple $ (P,\{ {\phi _t}\} ,K)$.


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DOI: https://doi.org/10.1090/S0002-9947-1970-0266319-8
Keywords: Markov processes, diffusion processes, semigroups, branching processes, infinitesimal generator
Article copyright: © Copyright 1970 American Mathematical Society