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

Consider two transient Markov processes (Xvt)tϵR·, (Xμt)tϵR· with the same transition semigroup and initial distributions v and μ. The probability spaces supporting the processes each are also assumed to support an exponentially distributed random variable independent of the process. w that there exist (randomized) stopping times S for (Xvt), T for (Xμt) with common final distribution, L(XvS|S < ∞) = L(XμT|T < ∞), 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 Prob(S=∝)+Prob(T=∝)=maxhϵHh′⩽1〈y-μh〉 H denotes the bounded harmonic functions of the Markov transition semigroup. Extensions, consequences and applications of this result are discussed.