## The Annals of Probability

- Ann. Probab.
- Volume 5, Number 5 (1977), 653-677.

### 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.

#### Article information

**Source**

Ann. Probab., Volume 5, Number 5 (1977), 653-677.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176995711

**Digital Object Identifier**

doi:10.1214/aop/1176995711

**Mathematical Reviews number (MathSciNet)**

MR451415

**Zentralblatt MATH identifier**

0379.60076

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60J55: Local time and additive functionals

Secondary: 60G05: Foundations of stochastic processes

**Keywords**

31-00 Markov system additive functional central projection

#### Citation

Dynkin, E. B. Markov Systems and Their Additive Functionals. Ann. Probab. 5 (1977), no. 5, 653--677. doi:10.1214/aop/1176995711. https://projecteuclid.org/euclid.aop/1176995711