Semigroup properties of markov processes with a several dimensional parameter

Abstract

The theory of Markov processes is closely connected with the concept of Itô integrals for Wiener processes. In the two parameter case the fundamental definitions of Itô integrals and an Itô formula for two parameter stochastic processes were given by Ponomarenko (1) and Gichman (2). The Itô theory in the N parameter case was considered e.g. by Surgailis (3). An Itô calculus for N parameter, d dimensional stochastic processes was given by Imkeller (4). Using the basic concepts and notations of the two parameter Itô calculus, we derive Kolmogorov equations for several types of multiparameter Markov processes. We consider these equations for two parameter birth and death processes and two parameter diffusion processes. Two parameter semigroup properties are obtained. Applications are possible e.g. in the theory of reliability of systems with several components.