Coupling and harmonic functions in the case of continuous time Markov processes
Let \((X^\mu_t)\) and \((X^\nu_t)\), \(t \geq 0\), be continuous-time transient Markov processes taking values in Polish state space with initial distributions \(\mu\) and \(\nu\) and common transition kernel. It is shown that there exist (randomized) stopping times \(S\) and \(T\) for respective processes with the following properties: (i) the distributions of the processes at the stopping times are equal: \(L(X_S^\mu \mid S < \infty) = L(X^\nu_T \mid T < \infty)\); (ii) (maximality property) before coupling (i.e. for \(t < S\), resp. \(t < T\)) the processes move in disjoint subsets of the state space. The proof of the result is based essentially on the potential theoretic approach. Some examples and applications of the result are given. The discrete-time case was considered by the second author earlier [Probab. Theory Relat. Fields 75, 195-212 (1987; Zbl 0596.60070)].
scientific article; zbMATH DE number 859317
scientific article