A pair (X; Y) of Markov processes is called a Markov coupling if X and Y have the same transition probabilities and (X;Y) is a Markov process. We say that a coupling is "shy" if there exists a (random) Epsilon > 0 such that dist(X subscript t; Y subscript t) > Epsilon for all t is greater than or equal to 0. We investigate whether shy couplings exist for several classes of Markov processes.
Benjamini et al. (Sat,) studied this question.