A Formula for Semigroups, with an Application to Branching Diffusion Processes

A Markov process <!-- MATH $P = \{ {x_t}\}$ --> proceeds until a random time , where the distribution of given is <!-- MATH $\exp ( - {\phi _t})$ --> for finite additive functional <!-- MATH $\{ {\phi _t}\}$ --> , at which time it jumps to a new position given by a substochastic kernel <!-- MATH $K({x_\tau },A)$ --> . A new time is defined, the process again jumps at a time <!-- MATH $\tau + \tau '$ --> and so forth, producing a new Markov process . 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 <!-- MATH $\{ {\phi _t}\}$ --> , 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 for some triple <!-- MATH $(P,\{ {\phi _t}\} ,K)$ --> .