Markov intertwining is an important tool in stochastic processes: it enables to prove equalities in law, to assess convergence to equilibrium in a probabilistic way, to relate apparently distinct random models or to make links with wave equations, see Carmona, Petit and Yor , Aldous and Diaconis , Borodin and Olshanski  and Pal and Shkolnikov  for examples of applications in these domains. Unfortunately the basic construction of Diaconis and Fill  is not easy to manipulate. An alternative approach, where the underlying coupling is first constructed, is proposed here as an attempt to remedy to this drawback, via random mappings for measure-valued dual processes, first in a discrete time and finite state space setting. This construction is related to the evolving sets of Morris and Peres  and to the coupling-from-the-past algorithm of Propp and Wilson . Extensions to continuous frameworks enable to recover, via a coalescing stochastic flow due to Le Jan and Raimond , the explicit coupling underlying the intertwining relation between the Brownian motion and the Bessel-3 process due to Pitman . To generalize such a coupling to more general one-dimensional diffusions, new coalescing stochastic flows would be needed and the paper ends with challenging conjectures in this direction.
"On the construction of measure-valued dual processes." Electron. J. Probab. 25 1 - 64, 2020. https://doi.org/10.1214/20-EJP419