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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    ABSTRACT: We investigate the long term behavior in terms of finite dimensional global attractors and (global) asymptotic stabilization to steady states, as time goes to infinity, of solutions to a non-local semilinear reaction-diffusion equation associated with the fractional Laplace operator on non-smooth domains subject to Dirichlet, fractional Neumann and Robin boundary conditions.
    Discrete and Continuous Dynamical Systems 03/2016; 36(3). DOI:10.3934/dcds.2016.36.1279 · 0.83 Impact Factor
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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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    ABSTRACT: Marcus Stochastic differential equations are appropriate models for many engineering and scientific applications. Fokker-Planck equation describes time evolution of probability densities of stochastic dynamical systems and plays an important role in quantifying propagation and evolution of uncertainty. Fokker-Planck equation is developed in this paper for nonlinear systems driven by non-Gaussian Levy processes and modeled by Marcus stochastic differential equations with some mild assupmtion on coefficients of the noise terms.
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    • "The FP equation for the distribution of the conditional probability density p(x, t) = p(x, t|x 0 , 0), i.e., the probability of the process X t has value x at time t given it had value x 0 at time 0, is given by [3] [22] [1] [2] p t = −(f (x) p) x + 1 2 dp xx + ε[k α (−∆) "
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    ABSTRACT: The Fokker-Planck equations for stochastic dynamical systems, with non-Gaussian $\alpha-$stable symmetric L\'evy motions, have a nonlocal or fractional Laplacian term. This nonlocality is the manifestation of the effect of non-Gaussian fluctuations. Taking advantage of the Toeplitz matrix structure of the time-space discretization, a fast and accurate numerical algorithm is proposed to simulate the nonlocal Fokker-Planck equations, under either absorbing or natural conditions. The scheme is shown to satisfy a discrete maximum principle and to be convergent. It is validated against a known exact solution and the numerical solutions obtained by using other methods. The numerical results for two prototypical stochastic systems, the Ornstein-Uhlenbeck system and the double-well system are shown.
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