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.
"A condition for the equivalence of coupling and shift coupling." Ann. Probab. 28 (4) 1666 - 1679, October 2000. https://doi.org/10.1214/aop/1019160502