Random Evolutions with Underlying Semi-Markov Processes

of the random evolution. Extensions of the initial definition and its uses have been completed by various authors and are reported in the surveys of Hersh [13] and Pinsky [27]. In this paper we extend the definition of random evolution to the case in which the underlying process is a semi-Markov process. We prove new representation theorems for solutions of various abstract integral equations in terms of this generalized random evolution. In particular, we use a special type of random evolution with underlying semi-Markov process to give a new representation for the solution of abstract Cauchy problems of the type treated by Griego-Hersh and, motivated through this representation, we generate new limit theorems of 'generalized- central-limit-theorem type' for the abstract Cauchy solutions. In Section one we present background results on Markov renewal processes and semi-Markov processes which we need in this development. In Section two we define the random evolution with underlying semi-Markov process and related notions; we then prove conditioning results for the random evolutions and representation theorems for solutions of abstract integral equations in terms of