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

Clemson University, CEU, South Carolina, United States
Journal of Mathematical Physics (Impact Factor: 1.24). 09/2004; 42(1). DOI: 10.1063/1.1318734
Source: arXiv


The Fokker-Planck equation has been very useful for studying dynamic behavior of stochastic differential equations driven by Gaussian noises. However, there are both theoretical and empirical reasons to consider similar equations driven by strongly non-Gaussian noises. In particular, they yield strongly non-Gaussian anomalous diffusion which seems to be relevant in different domains of Physics. In this paper, we therefore derive a fractional Fokker-Planck equation for the probability distribution of particles whose motion is governed by a nonlinear Langevin-type equation, which is driven by a Levy stable noise rather than a Gaussian. We obtain in fact a general result for a Markovian forcing. We also discuss the existence and uniqueness of the solution of the fractional Fokker-Planck equation. (C) 2001 American Institute of Physics.

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Available from: Shaun Lovejoy, Apr 27, 2013
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    • "Moreover, nonlocal reaction-diffusion equations have been also considered in the monograph [4] but the integral operators there are generally smooth or only mildly singular (i.e., the kernel is at least integrable over R N ). On the other hand, the linear parabolic equation ∂ t u + (−∆) s u = 0, s ∈ (0, 1), instead of the usual parabolic equation ∂ t u − ∆u = 0, is a much studied topic of anomalous diffusion in physics, probability and finance (see, e.g., [1] [33] [40] [42]). We also refer the reader to an interesting tutorial in [47] which introduces the main concepts behind normal and anomalous diffusion. "
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    • "However, Fokker-Planck equations for SDEs driven by general Lévy processes are not readily available due to the difficulty in obtaining the expressions for the adjoint operators of the infinitesimal generators associated with these SDEs [1]. For Ito SDEs driven by Lévy processes, the Fokker-Planck equations have been discussed by many authors, see [10] [9] among others. For Marcus SDEs [5] [6] [4] [1], the research is relatively few. "
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