On the construction of measure-valued dual processes

Miclo, Laurent

doi:10.1214/20-ejp419

Cited by 9 publications

(10 citation statements)

References 24 publications

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“…Finally, we should mention that all of these formulae have in some sense their origin in the study of coalescing stochastic flows of diffusions and Markov chains, see [37]. Recently a general abstract theory has been developed for constructing couplings be-tween intertwined Markov processes based on random maps and coalescing flows, see [36]. It would be interesting to understand to what extend this is related to the present constructions and more generally to analogous couplings in integrable probability.…”

Section: Intermediate Results and Strategy Of Proofmentioning

confidence: 97%

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

Assiotis

2020

Ann. Henri Poincaré

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We introduce a space inhomogeneous generalization of the dynamics on interlacing arrays considered by Borodin and Ferrari in [7]. We show that for a certain class of initial conditions the point process associated to the dynamics has determinantal correlation functions and we calculate explicitly, in the form of a double contour integral, the correlation kernel for one of the most classical initial conditions, the densely packed. En route to proving this we obtain some results of independent interest on non-intersecting general pure-birth chains, that generalize the Charlier process, the discrete analogue of Dyson's Brownian motion. Finally, these dynamics provide a coupling between the inhomogeneous versions of the TAZRP and PushTASEP particle systems which appear as projections on the left and right edges of the array respectively⋆ ⋆ + 1 ⋆ + 2 ⋆ + 3 ⋆ + 4 ⋆ + 5 ⋆ + 6 Figure 1: A configuration of particles in GT 4 . If the clock of the particle labelled X 3,1 4 rings, which happens at rate λ(⋆ + 1), then the move is blocked since interlacing with X 2,1 4 would be violated. On the other hand, if the clock of the particle X 2,2 4 rings, which happens at rate λ(⋆ + 3), then it jumps to the right by one and instantaneously pushes both X 3,3 4 and X 4,4 4 to the right by one as well, for otherwise the interlacing would break.The right edge particle system is called inhomogeneous PushTASEP, see [7], [6] and also [16] for a q-deformation of the homogeneous case. The fact that this particle system has an underlying determinantal structure is new as well, but is also obtained in the independent work of Petrov [46] that uses different methods.

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Section: Intermediate Results and Strategy Of Proofmentioning

confidence: 97%

Determinantal Structures in Space-Inhomogeneous Dynamics on Interlacing Arrays

Assiotis

2020

Ann. Henri Poincaré

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“…the book [3] of Bakry, Gentil and Ledoux, and is based on other geometric considerations, as we will see in the sequel. Other motivations for the couplings of primal and dual processes in the context of diffusions can be found in Machida [11] and [14].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…In particular, we see that the intertwining coupling we have constructed is different from the one proposed by Pitman [16], which is a.s. touching (the upper) boundary repeatedly. Instead we end up with the intertwining dual constructed in [14] via stochastic flows. It is mentioned there how to deduce the classical Pitman's dual, via Lévy's theorem.…”

Section: Intertwined Dual Processes: a Generalized Pitman Theoremmentioning

confidence: 99%

Construction of set-valued dual processes on manifolds

Arnaudon,

Coulibaly-Pasquier,

Miclo

2020

Preprint

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The purpose of this paper is to construct a Brownian motion X :" pXtq tě0 taking values in a Riemannian manifold M , together with a compact valued process D :" pDtq tě0 such that, at least for small enough F D -stopping time τ ě 0 and conditioned to F D τ , the law of Xτ is the normalized Lebesgue measure on Dτ . This intertwining result is a generalization of Pitman theorem. We first construct regular intertwined processes related to Stokes' theorem. Then using several limiting procedures we construct synchronous intertwined, free intertwined, mirror intertwined processes. The local times of the Brownian motion on the (morphological) skeleton or the boundary of D plays an important role. Several example with moving intervals, discs, annulus, symmetric convex sets are investigated. KEYWORDS. Brownian motions on Riemannian manifolds, intertwining relations, setvalued dual processes, couplings of primal and dual processes, stochastic mean curvature evolutions, boundary and skeleton local times, generalized Pitman theorem.

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“…where ( 8) was taken into account in the second equality. Using again (8), we deduce (10). To get the sought intertwining relation, it is sufficient to check it on the pK N `1,n q nP 0,N `1 , which is a base of L 2 pπ N `1q.…”

Section: Proofmentioning

confidence: 99%

“…The latter question is very important for applications, since given X, it amounts to knowing how to extract the important information Y from X. In [10] we proposed a way to do it via the introduction of some random mappings in some particular situations where Y is subset-valued (but then the state space of Y can end up being much larger than the state space of X).…”

Section: Introductionmentioning

confidence: 99%