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2020 On the construction of measure-valued dual processes
Laurent Miclo
Electron. J. Probab. 25(none): 1-64 (2020). DOI: 10.1214/20-EJP419

Abstract

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.

Citation

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Laurent Miclo. "On the construction of measure-valued dual processes." Electron. J. Probab. 25 1 - 64, 2020. https://doi.org/10.1214/20-EJP419

Information

Received: 9 November 2018; Accepted: 15 January 2020; Published: 2020
First available in Project Euclid: 28 January 2020

zbMATH: 1437.60048
MathSciNet: MR4059184
Digital Object Identifier: 10.1214/20-EJP419

Subjects:
Primary: 60J10
Secondary: 37A25, 60G17, 60J05, 60J25, 60J60, 60J65

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