  The notion of a successful coupling of Markov processes, based on the idea
that both components of the coupled system ``intersect'' in finite time with
probability one, is extended to cover situations when the coupling is
unnecessarily Markovian and its components are only converging (in a certain
sense) to each other with time. Under these assumptions the unique ergodicity
of the original Markov process is proven. A price for this generalization is
the weak convergence to the unique invariant measure instead of the strong one.
Applying these ideas to infinite interacting particle systems we consider even
more involved situations when the unique ergodicity can be proven only for a
restriction of the original system to a certain class of initial distributions
(e.g. translational invariant ones). Questions about the existence of invariant
measures with a given particle density are discussed as well.
