Miclo, Laurent (2020) On the construction of set-valued dual processes. Electronic Journal of Probability, vol. 25. pp. 1-64.

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Identification Number : 10.1214/20-EJP419


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 [8], Aldous and Diaconis [2], Borodin and Olshanski [7] and Pal and Shkolnikov [23] for examples of applications in these domains. Unfortunately the basic construction of Diaconis and Fill [10] 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 [22] and to the coupling-from-the-past algorithm of Propp and Wilson [27].
Extensions to continuous frameworks enable to recover, via a coalescing stochastic flow due to Le Jan and Raimond [16], the explicit coupling underlying the intertwining relation between the Brownian motion and the Bessel-3 process due to Pitman [25]. 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.

Item Type: Article
Language: English
Date: 2020
Refereed: Yes
Divisions: Institut de mathématiques de Toulouse, TSE-R (Toulouse)
Site: UT1
Date Deposited: 29 Sep 2020 14:07
Last Modified: 09 Sep 2021 15:51
OAI Identifier: oai:tse-fr.eu:124239
URI: https://publications.ut-capitole.fr/id/eprint/34916
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