Sweetest taboo processes

Mazzolo, Alain

doi:10.1088/1742-5468/aad19c

Cited by 25 publications

(37 citation statements)

References 66 publications

(161 reference statements)

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“…Verbally the censorship idea resembles random processes conditioned to stay in a bounded domain forever, [16,17]. However, we point out that the "censoring" concept is not the same [38] as that of the (Doob-type) conditioning.…”

Section: Regional Fractional Laplacianmentioning

confidence: 83%

“…The Authors of Ref. [38] exclude from considerations so-called taboo processes which are related to the concept of the Doob h-transform, [16][17][18] and are known not to leave the open set D. It is demonstrated that what is named in Ref. [38] a censored processes (actually, a recurrent censored symmetric α-stable process) is different (in law) from the conditioned not to leave D symmetric stable one, c.f.…”

Section: Regional Fractional Laplacian: Signatures Of Reflecting Bmentioning

confidence: 99%

“…Compare e.g. also [16,17] (Brownian case) and [19][20][21] (Lévy-stable case), where the role of lowest eigenstates and eigenvalues (thence eigenvalue gaps) of the motion generator has appeared to be vital for the description of decay rates of killed stochastic processes. The spectral data of motion generators are relevant for quantifying long-living processes in a spatial trap (e.g.…”

mentioning

confidence: 99%

“…We point out that other analogs of nonlocal Neumann data, imposed directly on the fractional Laplacian, were proposed as well, [43].The existence problem for jump-type processes with an infinite life-time in a bounded domain, seems to have been left aside in the physics literature (see however Refs. [16,17] in connection with diffusion processes and [15,46] for a preliminary discussion of the Cauchy process that is trapped in the interval). To the contrary, permanently trapped arXiv:1810.07028v2 [cond-mat.stat-mech]…”

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confidence: 99%

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Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights

Garbaczewski

Stephanovich

2019

Phys. Rev. E

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The fractional Laplacian (−∆) α/2 , α ∈ (0, 2) has many equivalent (albeit formally different) realizations as a nonlocal generator of a family of α-stable stochastic processes in R n . On the other hand, if the process is to be restricted to a bounded domain, there are many inequivalent proposals for what a boundary-data respecting fractional Laplacian should actually be. This ambiguity holds true not only for each specific choice of the process behavior at the boundary (like e.g. absorbtion, reflection, conditioning or boundary taboos), but extends as well to its particular technical implementation (Dirchlet, Neumann, etc. problems). The inferred jump-type processes are inequivalent as well, differing in their spectral and statistical characteristics, which may strongly influence the ability of the formalism (if uncritically adopted) to provide an unambigous description of real geometrically confined physical systems with disorder. Specifically that refers to their relaxation properties and the near-equilibrium asymptotic behavior. In the present paper we focus on Lévy flight-induced jump-type processes which are constrained to stay forever inside a finite domain. That refers to a concept of taboo processes (imported from Brownian to Lévy -stable contexts), to so-called censored processes and to reflected Lévy flights whose status still remains to be unequivocally settled. As a byproduct of our fractional spectral analysis, with reference to Neumann boundary conditions, we discuss disordered semiconducting heterojunctions as the bounded domain problem. I. MOTIVATIONBrownian motion in a bounded domain is a classic problem with an ample coverage in the literature, specifically concerning the absorbing (Dirichlet) and reflecting (Neumann) boundary data (for the present purpose we disregard other boundary data choices). A coverage concerning their physical relevance is enormous as well [1,2].Anticipating further discussion, we quite inentionally point out source papers dealing with reflected Brownian motion, [3] and exposing at some length the method of eigenfuction expansions for the reflected and other boundarydata problems, [4]- [8], c.f. also [9]. The latter method is as well an indispensable tool in the analysis of spectral properties of fractional Laplacians and related jump-type processes in bounded domains. Its direct link with well developed theory of heat semigroups for jump-type processes (mostly these with absorpion/killing) allows to address the statistics of exits from the domain, like e.g. the first and mean first exit times, large time behavior, stationarity issues, probability of survival and its asymptotic decay, c.f.[10]- [15]. Compare e.g. also [16,17] (Brownian case) and [19][20][21] (Lévy-stable case), where the role of lowest eigenstates and eigenvalues (thence eigenvalue gaps) of the motion generator has appeared to be vital for the description of decay rates of killed stochastic processes. The spectral data of motion generators are relevant for quantifying long-living processes in a spatial...

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Section: Regional Fractional Laplacianmentioning

confidence: 83%

Section: Regional Fractional Laplacian: Signatures Of Reflecting Bmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights

Garbaczewski

Stephanovich

2019

Phys. Rev. E

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“…Verbally, the censorship idea resembles that of random processes conditioned to stay in a bounded domain forever, [37,38]. However, the "censoring" concept is not the same [12] as that of the (Doob-type) conditioning employed in [37,38].…”

Section: Respectively By Introducing (−∆)mentioning

confidence: 99%

Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement

Garbaczewski,

Żaba

2022

Preprint

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We discuss an impact of detailed (path-wise) reflection-from-the barrier scenarios upon confining properties of a paradigmatic family of symmetric α-stable Lévy processes, whose permanent residence in a finite interval on a line is enforced by a two-sided reflection. The analysis is carried out in conjunction with attempts to give meaning to the notion of a reflecting Lévy process, in terms of the domain of its generator. To this end, the fractional Laplacian needs to be constrained by a suitable version of the Neumann boundary condition, and no consesus has been reached as yet. I. MOTIVATIONII. RESTRICTED VERSUS REGIONAL FRACTIONAL LAPLACIANS: WHITHER THE REFLECTIONS ARE GONE ?A. Conundrum: What are stochastic processes connected with singular α-harmonic functions ?

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Exact solutions for the probability density of various conditioned processes with an entrance boundary

Mazzolo

2024

Journal of Mathematical Physics

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The probability density is a fundamental quantity for characterizing diffusion processes. However, it is seldom known except in a few renowned cases, including Brownian motion and the Ornstein–Uhlenbeck process and their bridges, geometric Brownian motion, Brownian excursion, or Bessel processes. In this paper, we utilize Girsanov’s theorem, along with a variation of the method of images, to derive the exact expression of the probability density for diffusions that have one entrance boundary. Our analysis encompasses numerous families of conditioned diffusions, including the Taboo process and Brownian motion conditioned on its growth behavior, as well as the drifted Brownian meander and generalized Brownian excursion.

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Sweetest taboo processes

Cited by 25 publications

References 66 publications

Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights

Fractional Laplacians in bounded domains: Killed, reflected, censored, and taboo Lévy flights

Levy processes in bounded domains: Path-wise reflection scenarios and signatures of confinement

Exact solutions for the probability density of various conditioned processes with an entrance boundary

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