On a mean reward from a common Markov replacement process
We study the integral representation of potentials by exit laws in the framework of sub-Markovian semigroups of bounded operators acting on . We mainly investigate subordinated semigroups in the Bochner sense by means of -subordinators. By considering the one-sided stable subordinators, we deduce an integral representation for the original semigroup.
Let be a nonnegative function with its only singularity at , e.g. , . We study the behavior of the Wiener process in left and right hand neighborhoods of level crossings by finding necessary and sufficient conditions on for the integrals of to be finite or infinite.