A condition for the equivalence of coupling and shift coupling

Abstract

It is proved in this paper that a weak parabolic Harnack inequality for a Markov semigroup implies the existence of a coupling and a shift coupling for the corresponding process with equal chances of success. This implies equality of the tail and invariant $\sigma$-fields for the diffusion as well as equality of the class of bounded parabolic functions and the class of bounded harmonic functions.