Fractional Laplace motion

Kozubowski, Tomasz J.; Meerschaert, Mark M.; Podgórski, Krzysztof

doi:10.1017/s000186780000104x

Cited by 31 publications

(62 citation statements)

References 30 publications

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“…an exponentially weighted derivative, which is similar to the fractional derivative formula (31) but with different weights (Kozubowski et al, 2006). For an exponential or Gamma dispersal kernel, the generator formula is the same as (37) except that the integral is taken over y > 0.…”

Section: Fractional Reproduction-dispersal Equationsmentioning

confidence: 99%

Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

2007

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Reproduction-Dispersal equations, called reaction-diffusion equations in the physics literature, model the growth and spreading of biological species. IntegroDifference equations were introduced to address the shortcomings of this model, since the dispersal of invasive species is often more widespread than what the classical RD model predicts. In this paper, we extend the RD model, replacing the classical second derivative dispersal term by a fractional derivative of order 1 < α ≤ 2. Fractional derivative models are used in physics to model anomalous super-diffusion, where a cloud of particles spreads faster than the classical diffusion model predicts. This paper also establishes a connection between the new RD model and a corresponding ID equation with a heavy tail dispersal kernel. The general theory developed here accommodates a wide variety of infinitely divisible dispersal kernels that adapt to any scale. Each one corresponds to a generalised RD model with a different dispersal operator. The connection established here between RD and ID equations can also be exploited to generate convergent numerical solutions of RD equations along with explicit error bounds.

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Section: Fractional Reproduction-dispersal Equationsmentioning

confidence: 99%

Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

2007

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“…This is because the underlying increment PDFs, while all being members of the generalized Laplace family, change dramatically with scale, something that does not occur with Gaussian or Lévy stable fractals. Also, fLam/fLan does not fall completely within the class of multifractals that have been of much interest lately [Kozubowski et al, 2005].…”

Section: Discussionmentioning

confidence: 97%

Do Heterogeneous Sediment Properties and Turbulent Velocity Fluctuations Have Something in Common? Some History and a New Stochastic Process

Molz

Meerschaert

Kozubowski

et al. 2013

Dynamics of Fluids and Transport in Fractured Rock

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It is increasingly apparent that sediment property distributions on sufficiently small scales are probably irregular. This has led to the development of stochastic theory in subsurface hydrology, including statistically heterogeneous concepts based mainly on the Gaussian and Lévy-stable probability density functions (PDFs), the mathematical basis for stochastic fractals. Gaussian and Levy-stable stochastic fractals have been applied both in the field of turbulence and subsurface hydrology. However, measurements have shown that the increment frequency distributions do not always follow Gaussian or Lévy-stable PDFs. Provided herein is an overview of the origin and development of a new non-stationary stochastic process, called fractional Laplace motion (flam) with stationary, correlated, increments called fractional Laplace noise (fLan). It is based on the Laplace PDF and known generalizations, and does not display self-similarity. Uncorrelated versions are equivalent to a Brownian motion subordinated to the gamma process. In analogy to the development of fractional Brownian motion (fBm) from Brownian motion, fLam is equivalent to fBm subordinated to a gamma process. The new stochastic fractal has increment PDFs that compare better with measurements, the moments of the PDF family remain bounded, and decay of the increment distribution tails vary from being slower than exponential through exponential and on to a Gaussian decay as the lag size increases. This leads to increasingly more intermittent fluctuations as the lag size decreases. It may be that the geometric central limit theorem, and possible generalizations, will play an important role in connecting the abstract mathematics to the physics underlying applications. 1 1. HISTORICAL OVERVIEW Development of TheoryDuring the past 3 decades, it has become increasingly apparent that heterogeneity in natural sediments is pervasive, with the implication that property distributions on sufficiently small scales are probably irregular. (Within this chapter, the term "irregular" will mean continuous functions with discontinuous first derivatives or discontinuous functions.) As this concept developed, various individuals were motivated to characterize heterogeneous property distributions, such as ln(K); K = hydraulic conductivity, using stochastic concepts. In subsurface hydrology, a significant step forward was taken by Freeze [1975], who began to apply time-series

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“…For example, see Gardiner (2009); Kozubowski et al (2006); Lenzi et al (2003); Sabatier et al (2007); Yan (2013) and the references therein. Such equations are related to α-stable processes belonging to the Lévy processes.…”

Section: R\{0}mentioning

confidence: 99%

“…Then, the corresponding Lévy process becomes symmetric. As examples of the Lévy process with such measure, we can give the variance gamma (VG) process (Applebaum, 2009), also known as the symmetric Laplace motion (Kozubowski et al, 2006), with γ = 1 and µ(y) = e −y , and the normal inverse Gaussian (NIG) process (Applebaum, 2009) with γ = 2 and µ(y) = y K 1 (y)/π, where K 1 is the modified Bessel function of the second kind. Then, applying (1.8) to the operator A in (1.4) and taking its adjoint, we have a special form of equation (1.6) as…”

Section: R\{0}mentioning

confidence: 99%

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

Tanaka¹

2015

IMA J Numer Anal

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In this paper, we propose a fast and accurate numerical method based on Fourier transform to solve Kolmogorov forward equations of symmetric scalar Lévy processes. The method is based on the accurate numerical formulas for Fourier transform proposed by Ooura. These formulas are combined with nonuniform fast Fourier transform (FFT) and fractional FFT to speed up the numerical computations. Moreover, we propose a formula for numerical indefinite integration on equispaced grids as a component of the method. The proposed integration formula is based on the sinc-Gauss sampling formula, which is a function approximation formula. This integration formula is also combined with the FFT. Therefore, all steps of the proposed method are executed using the FFT and its variants. The proposed method allows us to be free from some special treatments for a non-smooth initial condition and numerical time integration. The numerical solutions obtained by the proposed method appeared to be exponentially convergent on the interval if the corresponding exact solutions do not have sharp cusps. Furthermore, the real computational times are approximately consistent with the theoretical estimates.

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Fractional Laplace motion

Cited by 31 publications

References 30 publications

Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

Fractional Reproduction-Dispersal Equations and Heavy Tail Dispersal Kernels

Do Heterogeneous Sediment Properties and Turbulent Velocity Fluctuations Have Something in Common? Some History and a New Stochastic Process

A fast and accurate numerical method for symmetric Lévy processes based on the Fourier transform and sinc-Gauss sampling formula

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