Multiple Points of Operator Semistable Lévy Processes

Luks, Tomasz; Xiao, Yimin

doi:10.1007/s10959-018-0859-4

Cited by 3 publications

(1 citation statement)

References 25 publications

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“…In recent years the fractal path behavior of semistable Lévy processes has been investigated, complementing previous classical results for their stable counterparts. It turned out that in terms of fractal dimension (mainly Hausdorff dimension) the range, the graph and multiple points of the sample paths almost surely are not affected by the log-periodic perturbations [9,10,19,30], even in terms of exact Hausdorff measure [11]. Nevertheless, semistable Lévy processes show a different behavior when turning from fractality to fractionality.…”

Section: Introductionmentioning

confidence: 99%

Space-time duality for semi-fractional diffusions

Kern¹,

Lage²

2019

Preprint

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Almost sixty years ago Zolotarev proved a duality result which relates an α-stable density for α ∈ (1, 2) to the density of a 1 α -stable distribution on the positive real line. In recent years Zolotarev duality was the key to show space-time duality for fractional diffusions stating that certain heat-type fractional equations with a fractional derivative of order α in space are equivalent to corresponding timefractional differential equations of order 1 α . We review on this space-time duality and take it as a recipe for a generalization from the stable to the semistable situation.

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Section: Introductionmentioning

confidence: 99%

Space-time duality for semi-fractional diffusions

Kern¹,

Lage²

2019

Preprint

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Space-Time Duality for Semi-Fractional Diffusions

Kern

Lage

2021

Fractal Geometry and Stochastics VI

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PSD and Cross-PSD of Responses of Seven Classes of Fractional Vibrations Driven by fGn, fBm, Fractional OU Process, and von Kármán Process

2024

Symmetry

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This paper gives its contributions in four stages. First, we propose the analytical expressions of power spectrum density (PSD) responses and cross-PSD responses to seven classes of fractional vibrators driven by fractional Gaussian noise (fGn). Second, we put forward the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators excited by fractional Brownian motion (fBm). Third, we present the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators driven by the fractional Ornstein–Uhlenbeck (OU) process. Fourth, we bring forward the analytical expressions of PSD and cross-PSD responses to seven classes of fractional vibrators excited by the von Kármán process. We show that the statistical dependences of the responses to seven classes of fractional vibrators follow those of the excitation of fGn, fBm, the OU process, or the von Kármán process. We also demonstrate the obvious effects of fractional orders on the responses to seven classes of fractional vibrations. In addition, we newly introduce class VII fractional vibrators, their frequency transfer function, and their impulse response in this research.

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Multiple Points of Operator Semistable Lévy Processes

Cited by 3 publications

References 25 publications

Space-time duality for semi-fractional diffusions

Space-time duality for semi-fractional diffusions

Space-Time Duality for Semi-Fractional Diffusions

PSD and Cross-PSD of Responses of Seven Classes of Fractional Vibrations Driven by fGn, fBm, Fractional OU Process, and von Kármán Process

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