Fractional Pearson diffusions

Leonenko, Nikolai; Meerschaert, Mark M.; Sikorskii, Alla

doi:10.1016/j.jmaa.2013.02.046

Cited by 95 publications

(76 citation statements)

References 37 publications

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“…This paper extends the results from Leonenko et al (2013 b ) to two of the heavy-tailed Pearson diffusions. Namely, we define fractional reciprocal gamma and Fisher-Snedecor diffusions by time changing the corresponding non-fractional heavy-tailed diffusions by the inverse of the standard stable subordinator.…”

Section: Introductionsupporting

confidence: 69%

“…The first three types of Pearson diffusions have stationary distributions with all moments, and are studied in detail in Leonenko et al (2013 b ). In this paper we focus on analytical and probabilistic properties of the other two subfamilies, the FS and RG diffusions that have heavy-tailed stationary distributions.…”

Section: Pearson Diffusionsmentioning

confidence: 99%

“…Alternative approaches are required to define fractional diffusions as the stochastic processes. For the three non-heavy-tailed Pearson diffusions (OU, CIR, Jacobi), their fractional analogs were defined in Leonenko et al (2013 b ), where it was also shown that these processes are governed by the time-fractional Fokker-Planck and backward Kolmogorov equations with space-varying polynomial coefficients. The spectral theory for the generators of the non-heavy-tailed Pearson diffusions was used to obtain the explicit strong solutions to the corresponding fractional Cauchy problems.…”

Section: Introductionmentioning

confidence: 99%

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Heavy-tailed fractional Pearson diffusions

Leonenko

Papić

Sikorskii

et al. 2017

Stochastic Processes and their Applications

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We define heavy-tailed fractional reciprocal gamma and Fisher-Snedecor diffusions by a non-Markovian time change in the corresponding Pearson diffusions. Pearson diffusions are governed by the backward Kolmogorov equations with space-varying polynomial coefficients and are widely used in applications. The corresponding fractional reciprocal gamma and Fisher-Snedecor diffusions are governed by the fractional backward Kolmogorov equations and have heavy-tailed marginal distributions in the steady state. We derive the explicit expressions for the transition densities of the fractional reciprocal gamma and Fisher-Snedecor diffusions and strong solutions of the associated Cauchy problems for the fractional backward Kolmogorov equation.

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Section: Introductionsupporting

confidence: 69%

Section: Pearson Diffusionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Heavy-tailed fractional Pearson diffusions

Leonenko

Papić

Sikorskii

et al. 2017

Stochastic Processes and their Applications

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“…Anh and Leonenko [1] presented a spectral representation of the mean-square solution of the fractional kinetic equation with random initial condition. The explicit strong solutions for fractional Pearson diffusions are developed by using spectral methods involving the Mittag-Leffler function in [10]. In most situations, analytical methods do not work well on most FDEs, so the reasonable option is to resort to numerical methods.…”

mentioning

confidence: 99%

A Crank--Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation

Zeng¹,

Liu²,

Li³

et al. 2014

SIAM J. Numer. Anal.

330

143

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In this paper, a new alternating direction implicit Galerkin-Legendre spectral method for the two-dimensional Riesz space fractional nonlinear reaction-diffusion equation is developed. The temporal component is discretized by the Crank-Nicolson method. The detailed implementation of the method is presented. The stability and convergence analysis is strictly proven, which shows that the derived method is stable and convergent of order 2 in time. An optimal error estimate in space is also obtained by introducing a new orthogonal projector. The present method is extended to solve the fractional FitzHugh-Nagumo model. Numerical results are provided to verify the theoretical analysis. Introduction.In the past decades, fractional calculus has been used to model particle transport in porous media. Recently, there has been increasing interest in the study of fractional calculus for its wide application in many fields of science and engineering, such as the physical and chemical processes, materials, control theory, biology, finance, and so on (see [3,9,25,23,33,35]). In physics, fractional derivatives are used to model anomalous diffusion, where particles spread differently than the classical Brownian motion model [23]. Kinetic equations of the diffusion, diffusionadvection, and Fokker-Planck equations with partial fractional derivatives were recognized as a useful approach for the description of transport dynamics in complex systems. Reaction-diffusion models have been used for numerous applications in patten formation in biology, chemistry, physics, and engineering. These systems show that diffusion can produce the spontaneous formation of spatial-temporal patterns. The idea is to use a fractional-order density gradient to recover, at least at a phenomenological level, the nonhomogeneities of the porous media. Given the structural

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“…It is known (see, e.g., Meerschaert and Sikorskii (2012), Leonenko, Meerschaert and Sikorskii (2013a)) that…”

Section: The Fractional Bessel-riesz Motionmentioning

confidence: 99%

Stochastic representation of fractional Bessel-Riesz motion

Anh

Leonenko

Sikorskii

2017

Chaos, Solitons & Fractals

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Abstract. This paper derives the stochastic solution of a Cauchy problem for the distribution of a fractional diffusion process. The governing equation involves the Bessel-Riesz derivative (in space) to model heavy tails of the distribution, and the Caputo-Djrbashian derivative (in time) to depicts the memory of the diffusion process. The solution is obtained as Brownian motion with time change in terms of the Bessel-Riesz subordinator on the inverse stable subordinator. This stochastic solution, named fractional Bessel-Riesz motion, provides a method to simulate a large class of stochastic motions with memory and heavy tails.

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Fractional Pearson diffusions

Cited by 95 publications

References 37 publications

Heavy-tailed fractional Pearson diffusions

Heavy-tailed fractional Pearson diffusions

A Crank--Nicolson ADI Spectral Method for a Two-Dimensional Riesz Space Fractional Nonlinear Reaction-Diffusion Equation

Stochastic representation of fractional Bessel-Riesz motion

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