Markov Systems and Their Additive Functionals

Abstract

For certain classes of Markov systems (that is, stochastic systems which have Markov representations with transition and cotransition probabilities) considered by the author in previous papers, a correspondence was established between additive functionals of any such system and measures on a certain measurable space. We now prove analogous results for arbitrary Markov systems. Measures corresponding to the additive functionals are defined on a certain $\sigma$-algebra in the product space $R \times \Omega$ where $R$ is the real line and $\Omega$ is the sample space (we call it the central $\sigma$-algebra). The theory is applicable not only to traditional processes but also to a number of generalized stochastic processes introduced by Gelfand and Ito. A situation where the observations are performed over a random time interval and the measure $P$ can be infinite is considered in the concluding section. These generalizations are of special importance for the homogeneous case which will be treated in another publication.