## Electronic Journal of Probability

### On the construction of measure-valued dual processes

Laurent Miclo

#### 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.

#### Article information

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

Dates
Accepted: 15 January 2020
First available in Project Euclid: 28 January 2020

https://projecteuclid.org/euclid.ejp/1580202285

Digital Object Identifier
doi:10.1214/20-EJP419

#### Citation

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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