Couplings of markov chains by randomized stopping times

Summary We consider a Markov chain on (E, ) generated by a Markov kernel P. We study the question, when we can find for two initial distributions and two randomized stopping times T of (X_n)_nNand S of (X_n)_nN, such that the distribution of X_Tequals the one of X_Sand T, S are both finite.The answer is given in terms of -, h with h bounded harmonic, or in terms of .For stopping times T, S for two chains (X_n)_nN,(X_n)_nNwe consider measures , on (E, ) defined as follows: (A)=expected number of visits of (X_n) toA before T, (A)=expected number of visits of (X_n) toA before S.We show that we can construct T, S such that and are mutually singular and (_vX_T)=(X_S. We relate and to the positive and negative part of certain solutions of the Poisson equation (I-P)(·)=-.