Article

Fractional Fokker--Planck Equation for Nonlinear Stochastic Differential Equations Driven by Non-Gaussian Levy Stable Noises

09/2004; DOI:doi:10.1063/1.1318734
Source: arXiv

ABSTRACT The Fokker-Planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by Gaussian noises. In this paper, we derive a Fractional Fokker--Planck equation for the probability distribution of particles whose motion is governed by a {\em nonlinear} Langevin-type equation, which is driven by a non-Gaussian Levy-stable noise. We obtain in fact a more general result for Markovian processes generated by stochastic differential equations.}

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    Article: SDEs Driven by a Time-Changed Lévy Process and Their Associated Time-Fractional Order Pseudo-Differential Equations
    Marjorie Hahn, Kei Kobayashi, Sabir Umarov
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    ABSTRACT: It is known that the transition probabilities of a solution to a classical Itô stochastic differential equation (SDE) satisfy in the weak sense the associated Kolmogorov equation. The Kolmogorov equation is a partial differential equation with coefficients determined by the corresponding SDE. Time-fractional Kolmogorov-type equations are used to model complex processes in many fields. However, the class of SDEs that is associated with these equations is unknown except in a few special cases. The present paper shows that in the cases of either time-fractional order or more general time-distributed order differential equations, the associated class of SDEs can be described within the framework of SDEs driven by semimartingales. These semimartingales are time-changed Lévy processes where the independent time-change is given respectively by the inverse of a single or mixture of independent stable subordinators. Examples are provided, including a fractional analogue of the Feynman–Kac formula. KeywordsTime-change–Stochastic differential equation–Semimartingale–Kolmogorov equation–Fractional order differential equation–Pseudo-differential operator–Lévy process–Stable subordinator
    Journal of Theoretical Probability 04/2012; 25(1):262-279. · 0.68 Impact Factor
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27 Apr 2013

Keywords

dynamic behavior
 
Gaussian noises
 
non-Gaussian Levy-stable noise
 
probability distribution
 
{\em nonlinear} Langevin-type equation