Generalized Fokker-Planck equation: Derivation and exact solutions

AbstractWe derive the generalized Fokker-Planck equation associated with the Langevin equation (in the Ito sense) for an overdamped particle in an external potential driven by multiplicative noise with an arbitrary distribution of the increments of the noise generating process. We explicitly consider this equation for various specific types of noises, including Poisson white noise and Lévy stable noise, and show that it reproduces all Fokker-Planck equations that are known for these noises. Exact analytical, time-dependent and stationary solutions of the generalized Fokker-Planck equation are derived and analyzed in detail for the cases of a linear, a quadratic, and a tailored potential.

[1]  N. Kampen,et al.  Stochastic processes in physics and chemistry , 1981 .

[2]  Peter Hänggi,et al.  Correlation functions and masterequations of generalized (non-Markovian) Langevin equations , 1978 .

[3]  Steady-state Lévy flights in a confined domain. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Raisa E. Feldman,et al.  Limit Distributions for Sums of Independent Random Vectors , 2002 .

[6]  S. Sharma,et al.  The Fokker-Planck Equation , 2010 .

[7]  J. Klafter,et al.  Fundamentals of Lévy Flight Processes , 2006 .

[8]  N. Kampen,et al.  Process with delta-correlated cumulants , 1980 .

[9]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[10]  V. Zolotarev.  One-dimensional stable distributions , 1986 .

[11]  Y. Gliklikh.  The Langevin Equation , 1997 .

[12]  A. Erdélyi,et al.  Higher Transcendental Functions , 1954 .

[13]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[14]  T. MacRobert.  Higher Transcendental Functions , 1955, Nature.

[15]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[16]  B. Gnedenko,et al.  Limit Distributions for Sums of Independent Random Variables , 1955 .

[17]  J. Klafter,et al.  Lévy-Driven Langevin Systems: Targeted Stochasticity , 2003 .

[18]  A. Dubkov,et al.  GENERALIZED WIENER PROCESS AND KOLMOGOROV'S EQUATION FOR DIFFUSION INDUCED BY NON-GAUSSIAN NOISE SOURCE , 2005 .

[19]  B. Gnedenko,et al.  Limit distributions for sums of shrunken random variables , 1954 .

[20]  J. Klafter,et al.  Lévy, Ornstein–Uhlenbeck, and Subordination: Spectral vs. Jump Description , 2005 .

[21]  W. Horsthemke,et al.  Nonequilibrium transitions induced by the cross-correlation of white noises. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Yu. A. Brychkov,et al.  Integrals and series , 1992 .

[23]  Igor Sokolov,et al.  Lévy flights in external force fields: from models to equations , 2002 .

[24]  Peter Hänggi,et al.  Langevin description of markovian integro-differential master equations , 1980 .

[25]  Peter Hänggi,et al.  Stochastic processes: Time evolution, symmetries and linear response , 1982 .

[26]  P. Jung,et al.  Colored Noise in Dynamical Systems , 2007 .

[27]  David R. Cox,et al.  The Theory of Stochastic Processes , 1967, The Mathematical Gazette.

[28]  Stationary states in Langevin dynamics under asymmetric Lévy noises. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  J. Keinonen,et al.  Experimental temperature and time dependence of implantations: Al in Zn , 1980 .

[30]  Sune Jespersen,et al.  LEVY FLIGHTS IN EXTERNAL FORCE FIELDS : LANGEVIN AND FRACTIONAL FOKKER-PLANCK EQUATIONS AND THEIR SOLUTIONS , 1999 .

[31]  W. Ebeling.  Stochastic Processes in Physics and Chemistry , 1995 .

[32]  Kiyosi Itô.  Stochastic Differential Equations in a Differentiable Manifold , 1950, Nagoya Mathematical Journal.

[33]  P D Ditlevsen.  Anomalous jumping in a double-well potential. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[34]  Ken-iti Sato.  Lévy Processes and Infinitely Divisible Distributions , 1999 .

[35]  A. Polyanin,et al.  Handbook of First-Order Partial Differential Equations , 2001 .

[36]  Bernardo Spagnolo,et al.  Lévy Flight Superdiffusion: an Introduction , 2008, Int. J. Bifurc. Chaos.

[37]  Werner Horsthemke,et al.  Noise-induced transitions , 1984 .

[38]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[39]  A. V. Skorohod,et al.  The theory of stochastic processes , 1974 .

[40]  Feller William,et al.  An Introduction To Probability Theory And Its Applications , 1950 .

[41]  E. Jakeman,et al.  Non-Gaussian models for the statistics of scattered waves , 1988 .

[42]  A. V. Tour,et al.  Lévy anomalous diffusion and fractional Fokker–Planck equation , 2000, nlin/0001035.

[43]  B. L. Hijmans.  Apuleiana Groningana V , 1974 .