On Quasi-successful Couplings of Markov Processes

The notion of a successful coupling of Markov processes, based on the idea that both components of a coupled system “intersect” in finite time with probability 1, is extended to cover situations where the coupling is not necessarily Markovian and its components only converge (in a certain sense) to each other with time. Under these assumptions the unique ergodicity of the original Markov process is proved. The price for this generalization is the weak convergence to the unique invariant measure instead of the strong convergence. Applying these ideas to infinite interacting particle systems, we consider even more involved situations where the unique ergodicity can be proved only for a restriction of the original system to a certain class of initial distributions (e.g., translation-invariant). Questions about the existence of invariant measures with a given particle density are also discussed.