On the construction of set-valued dual processes

Laurent Miclo (1)

1 IMT - Institut de Mathématiques de Toulouse UMR5219

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 [5], Aldous and Diaconis [2], Borodin and Olshanski [4] and Pal and Shkolnikov [18] for examples of applications in these domains. Unfortunately the basic construction the underlying coupling by Diaconis and Fill [7] is not easy to manipulate. A direct approach is proposed here to remedy to this drawback, via random mappings for set-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 [17] and to the coupling-from-the-past algorithm of Propp and Wilson [22]. Extensions to continuous frameworks enable to recover, via a coalescing stochastic flow due to Le Jan and Raimond [13], the explicit coupling underlying the intertwining relation between the Brownian motion and the Bessel-3 process due to Pitman [20]. 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.

Mots clés: Markov intertwining relations; set-valued dual processes; Diaconis-Fill couplings; random mappings; coalescing stochastic flows; Pitman's theorem; one-dimensional diffusions; measure-valued dual processes

Domaines: Probabilités [math.PR]

Laurent Miclo. On the construction of set-valued dual processes. 2020. ⟨hal-01911989v2⟩