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

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