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We raise a question on whether a dynamical system driven by Markov process isMarkovian, for which we are able to propose a criterion and examples ofpositive case. This investigation leads us to develop (i) a generalconstruction of intertwining dual via Liggett duality, and (ii) a realizationof $\Lambda$-linked coupling in a form of dynamical system. We show thisconstruction of intertwining dual and $\Lambda$-linked coupling for an$n$-dimensional drifting Brownian motion when it is a characteristic diffusion.In particular, it includes an extension of Pitman's $2M-W$ theorem by Rogersand Pitman as a special case.

$\Lambda$-linked coupling for drifting Brownian motions