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On the construction of measure-valued dual processes. (English) Zbl 1437.60048
Summary: 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 [P. Carmona et al., Rev. Mat. Iberoam. 14, No. 2, 311–367 (1998; Zbl 0919.60074); D. Aldous and P. Diaconis, Adv. Appl. Math. 8, 69–97 (1987; Zbl 0631.60065); A. Borodin and G. Olshanski, J. Funct. Anal. 263, No. 1, 248–303 (2012; Zbl 1260.60149); S. Pal and M. Shkolnikov, “tertwining diffusions and wave equations”, Preprint, arXiv:1306.0857] for examples of applications in these domains. Unfortunately the basic construction of P. Diaconis and J. A. Fill [Ann. Probab. 18, No. 4, 1483–1522 (1990; Zbl 0723.60083)] 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 B. Morris and Y. Peres [Probab. Theory Relat. Fields 133, No. 2, 245–266 (2005; Zbl 1080.60071)] and to the coupling-from-the-past algorithm of J. G. Propp and D. B. Wilson [Random Struct. Algorithms 9, No. 1–2, 223–252 (1996; Zbl 0859.60067)]. Extensions to continuous frameworks enable to recover, via a coalescing stochastic flow due to Y. Le Jan and O. Raimond [ALEA, Lat. Am. J. Probab. Math. Stat. 1, 21–34 (2006; Zbl 1105.60038)], the explicit coupling underlying the intertwining relation between the Brownian motion and the Bessel-3 process due to J. W. Pitman [Adv. Appl. Probab. 7, 511–526 (1975; Zbl 0332.60055)]. 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.
60J60 Diffusion processes
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60J25 Continuous-time Markov processes on general state spaces
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60J65 Brownian motion
05C81 Random walks on graphs
Full Text: DOI Euclid
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