Additive functionals of Markov processes and stochastic systems
Evgeny B. Dynkin · Annales de l’institut Fourier · 1975
Intuitively, an additive functional of a stochastic process ( x t , P ) gives a method to measure time taking into account the development of the process. We associate with any set of states C the mathematical expectation of time x t belongs to C . In this way, we establish to one-to-one correspondence between all the normal additive functionals of a Markov process and all the δ -finite measures on the state space which charge no inaccessible set. This is proved under the condition that transition probabilities are almost all paths. According to the previous results of the author, if two-dimensional probability distributions of a Markov process are absolutely continuous with respect to products of corresponding one-dimensional distributions, then the process can be modified in such a way that the set of additive functionals does not change and transition and cotransition probabilities acquire the above-mentioned properties.