Fractional Fokker–Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises

Schertzer, D.; Larchev, M.; Duan, Jinqiao; Yanovsky, Vladimir; Lovejoy, S.

doi:10.1063/1.1318734

Cited by 185 publications

(182 citation statements)

References 41 publications

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“…Alternatively, one can obtain the fractional FokkerPlanck equation using the methods of Refs. [26,45,46]. The fractional derivative in the Fokker-Planck equation appears as a consequence of the increments of Lévy α-stable motion in Eq.…”

Section: A Position-dependent Trapping Timementioning

confidence: 99%

“…Two-dimensional fractional Fokker-Planck equation (25) for the PDF of two stochastic variables x and t can be rigorously derived from the SDEs (2) and (23) driven by Lévy stable noises as in Refs. [26,45,46] (the Gaussian noise in Eq. (2) is a particular case of a Lévy stable noise with index of stability α = 2).…”

Section: A Position-dependent Trapping Timementioning

confidence: 99%

“…More complex situation is the diffusion in nonhomogeneous media, for example diffusion on fractals and multifractals [26]. Nonhomogeneous systems exhibit not only subdiffusion related to traps, but also enhanced diffusion can occur: for example, transport of interacting particles in a weakly disordered media is superdiffusive due to the disorder and subdiffusive without the disorder [27].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Anomalous diffusion in nonhomogeneous media: Power spectral density of signals generated by time-subordinated nonlinear Langevin equations

Kazakevičius

Ruseckas

2015

Physica A: Statistical Mechanics and its Applications

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Subdiffusive behavior of one-dimensional stochastic systems can be described by timesubordinated Langevin equations. The corresponding probability density satisfies the time-fractional Fokker-Planck equations. In the homogeneous systems the power spectral density of the signals generated by such Langevin equations has power-law dependency on the frequency with the exponent smaller than 1. In this paper we consider nonhomogeneous systems and show that in such systems the power spectral density can have power-law behavior with the exponent equal to or larger than 1 in a wide range of intermediate frequencies.

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Section: A Position-dependent Trapping Timementioning

confidence: 99%

Section: A Position-dependent Trapping Timementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Anomalous diffusion in nonhomogeneous media: Power spectral density of signals generated by time-subordinated nonlinear Langevin equations

Kazakevičius

Ruseckas

2015

Physica A: Statistical Mechanics and its Applications

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“…(2)) and get the equation dy(τ ) = ν(x(y)) 1/α η(dτ ). It contains a multiplicative noise in II and corresponds to the Fokker-Planck equation [35] …”

Section: A Multiplicative Noisementioning

confidence: 99%

Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps

Srokowski

2015

Phys. Rev. E

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The stochastic motion in a nonhomogeneous medium with traps is studied and diffusion properties of that system are discussed. The particle is subjected to a stochastic stimulation obeying a general Lévy stable statistics and experiences long rests due to nonhomogeneously distributed traps. The memory is taken into account by subordination of that process to a random time; then the subordination equation is position-dependent. The problem is approximated by a decoupling of the medium structure and memory and exactly solved for a power-law position dependence of the memory. In the case of the Gaussian statistics, the density distribution and moments are derived: depending on geometry and memory parameters, the system may reveal both the subdiffusion and enhanced diffusion. The similar analysis is performed for the Lévy flights where the finiteness of the variance follows from a variable noise intensity near a boundary. Two diffusion regimes are found: in the bulk and near the surface. The anomalous diffusion exponent as a function of the system parameters is derived.

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“…[17]. Another motivation is a study of the large time behavior of solutions of linear (as well as nonlinear) equations v t + Lv = 0 (1.5) with general diffusion operators L. Here, suitable space-time rescaling leads to an equation of type (1.1) with V (x) =…”

Section: Introductionmentioning

confidence: 99%

Generalized Fokker-Planck equations and convergence to their equilibria

Biler

Karch

2003

Evolution Equations Propagation Phenomena - Global Existence - Influence of Non-Linearities

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Abstract. We consider extensions of the classical Fokker-Planck equation u t + Lu = ∇ · (u∇V (x)) on R d with L = −∆ and V (x) = 1 2 |x| 2 , where L is a general operator describing the diffusion and V is a suitable potential. Introduction.We consider the initial value problem for the generalized FokkerPlanck equationwith a sufficiently regular potential V = V (x), u = u(x, t) and u 0 (x) ≥ 0. This equation generalizes the classical Fokker-Planck equationin two ways. First, the second order elliptic differential operator −∆ is replaced by a Markov diffusion operator L so that −L generates a positivity and mass preserving semigroup ewith ∇V (x) = x is replaced by a more general potential V which is large enough as |x| → ∞ so that V is confining.Our assumptions below will guarantee the existence of the unique steady state u ∞ = The paper is in final form and no version of it will be published elsewhere.[307] 308 P. BILER AND G. KARCH u ∞ (x) ≥ 0 of (1.1) with given M > 0,is the steady state for (1.3) for each M > 0, and u(t) tends to u ∞ at an exponential rate in all the L p norms, 1 ≤ p ≤ ∞, cf. [21].The motivations to study extensions (1.1) of (1.3) stem from the probability theory where Fokker-Planck equations are deeply connected with (nonlinear) stochastic differential equations driven by Gaussian and Lévy processes, see e.g. [17]. Another motivation is a study of the large time behavior of solutions of linear (as well as nonlinear) equations v t + Lv = 0 (1.5) with general diffusion operators L. Here, suitable space-time rescaling leads to an equation of type (1.1) with V (x) =

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Fractional Fokker–Planck equation for nonlinear stochastic differential equations driven by non-Gaussian Lévy stable noises

Cited by 185 publications

References 41 publications

Anomalous diffusion in nonhomogeneous media: Power spectral density of signals generated by time-subordinated nonlinear Langevin equations

Anomalous diffusion in nonhomogeneous media: Power spectral density of signals generated by time-subordinated nonlinear Langevin equations

Anomalous diffusion in stochastic systems with nonhomogeneously distributed traps

Generalized Fokker-Planck equations and convergence to their equilibria

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