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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.
MSC:
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
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References:
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