Couplings of markov chains by randomized stopping times - Part I: Couplings, harmonic functions and the poisson equation - Greven, Andreas (Probability Theory and Related Fields, 1987)
We consider a M a r k o v chain on (E,N) generated by a M a r k o v kernel P. We study the question, when we can find for two initial distri- butions v and p two r a n d o m i z e d stopping times T of (,Xn), N and S of (uX,), N, such that the distribution of X r equals the one of , X s and T, S are b o t h finite. T h e answer is given in terms of ( v - p , h ) with h b o u n d e d harmonic, or in terms of lim " __#) pk . 1 .3"(v F o r stopping times T, S for two chains ( X,)n N, (,X,)n N we consider measures t/, on (E,N) defined as follows: r/(A)=expected n u m b e r of visits of ( X,) to A before T, ( A ) = e x p e c t e d n u m b e r of visits of (,X,) to A before S. We show that we can construct T, S such that t/ and are mutually singular and Lf( Xr)= ( Xs). We relate and to the positive and negative part of certain solutions of the Poisson equation ( I - P ) ( . ) = v-/