Markov chains X and Y with images equal in law to the same Markov chain Z admit a homogeneous Markov coupling where f(X_t) equals g(Y_t) for all t, with X and Y conditionally independent given the image trajectory.
Markovian maximal coupling of Markov processes
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Coupling Markov chains with a common image chain
Markov chains X and Y with images equal in law to the same Markov chain Z admit a homogeneous Markov coupling where f(X_t) equals g(Y_t) for all t, with X and Y conditionally independent given the image trajectory.