Electronic Journal of Probability

On the construction of measure-valued dual processes

Laurent Miclo

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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.

Article information

Electron. J. Probab., Volume 25 (2020), paper no. 6, 64 pp.

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

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Digital Object Identifier

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 37A25: Ergodicity, mixing, rates of mixing 60J05: Discrete-time Markov processes on general state spaces 60J25: Continuous-time Markov processes on general state spaces 60J60: Diffusion processes [See also 58J65] 60G17: Sample path properties 60J65: Brownian motion [See also 58J65]

Markov intertwining relations measure-valued dual processes set-valued dual processes Diaconis-Fill couplings random mappings coalescing stochastic flows Pitman’s theorem one-dimensional diffusions

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Miclo, Laurent. On the construction of measure-valued dual processes. Electron. J. Probab. 25 (2020), paper no. 6, 64 pp. doi:10.1214/20-EJP419. https://projecteuclid.org/euclid.ejp/1580202285

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