Coupling and harmonic functions in the case of continuous time Markov processes

Consider two transient Markov processes (Xvt)t[epsilon]R·, (X[mu]t)t[epsilon]R· with the same transition semigroup and initial distributions v and [mu]. The probability spaces supporting the processes each are also assumed to support an exponentially distributed random variable independent of the process. We show that there exist (randomized) stopping times S for (Xvt), T for (X[mu]t) with common final distribution, L(XvSS < [infinity]) = L(X[mu]TT < [infinity]), and the property that for t < S, resp. t < T, the processes move in disjoint portions of the state space. For such a coupling (S, T) it is shown where denotes the bounded harmonic functions of the Markov transition semigroup. Extensions, consequences and applications of this result are discussed.