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

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- by Stanley A. Sawyer PDF
- Trans. Amer. Math. Soc.
**152**(1970), 1-38 Request permission

## 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 $(P,\{ {\phi _t}\} ,K)$.## References

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## Additional Information

- © Copyright 1970 American Mathematical Society
- 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