  We raise a question on whether a dynamical system driven by Markov process is
Markovian, for which we are able to propose a criterion and examples of
positive case. This investigation leads us to develop (i) a general
construction of intertwining dual via Liggett duality, and (ii) a realization
of $\Lambda$-linked coupling in a form of dynamical system. We show this
construction 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 Rogers
and Pitman as a special case.
