Ergodic properties of \(\lambda\)-recurrent Markov processes. (English) Zbl 0691.60065

Summary: We discuss the existence and uniqueness of the \(\lambda\)-invariant function and measure of a \(\lambda\)-recurrent Markov process. In spite of the strong overlapping with the results of S. Niemi and E. Nummelin [Stochastic Processes Appl. 22, 177-202 (1986; Zbl 0606.60080)] and P. Tuominen and R. L. Tweedie [Adv. Appl. Probab. 11, 784-803 (1979; Zbl 0421.60065)] a unified treatment in the framework of right processes seems to be of interest.
Rather than using results from other fields we make extensive use of results from J. Azema, H. Kaplan-Duflo and D. Revuz [Z. Wahrscheinlichkeitstheor. Verw. Geb. 8, 157-181 (1967; Zbl 0178.203)], R. K. Getoor [Séminaire de probabilités XIV, 1978/79, Lect. Notes Math. 784, 397-409 (1980; Zbl 0431.60067)] and M. Sharpe [“General theory of Markov processes” (1988; Zbl 0649.60079)] concerning right processes. It will be implicit in our approach that any \(\lambda\)-recurrent Markov process may be thought of as a recurrent process on another probability space. Asymptotic properties are then derived from the classical ergodic theory of conservative contraction properties.


60J25 Continuous-time Markov processes on general state spaces