Abstract
We 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.
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References
P. Langevin, C. R. Acad. Sci. 146, 530 (1908)
W.T. Coffey, Yu.P. Kalmykov, J.T. Waldron, The Langevin Equation, 2nd edn. (World Scientific, Singapore, 2004)
N.G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1992)
W. Horsthemke, R. Lefever, Noise-Induced Transitions (Springer-Verlag, Berlin, 1984)
P. Hänggi, H. Thomas, Phys. Rep. 88, 207 (1982)
H. Risken, The Fokker-Planck Equation, 2nd edn. (Springer-Verlag, Berlin, 1989)
K. Sato, Lévy processes and infinitely divisible distributions, (Cambridge University Press, Cambridge, 1999)
C.W. Gardiner, Handbook of Stochastic Methods, 2nd edn. (Springer-Verlag, Berlin, 1990)
S. Jespersen, R. Metzler, H.C. Fogedby, Phys. Rev. E 59, 2736 (1999)
P.D. Ditlevsen, Phys. Rev. E 60, 172 (1999)
R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)
V.V. Yanovsky, A.V. Chechkin, D. Schertzer, A.V. Tur, Physica A 282, 13 (2000)
D. Brockmann, I.M. Sokolov, Chem. Phys. 284, 409 (2002)
A.V. Chechkin, V.Y. Gonchar, J. Klafter, R. Metzler, Adv. Chem. Phys. 133, 439 (2006)
S.I. Denisov, W. Horsthemke, P. Hänggi, Phys. Rev. E 77, 061112 (2008)
P. Hänggi, P. Jung, Adv. Chem. Phys. 89, 239 (1995)
I.I. Gikhman, A.V. Skorokhod, The Theory of Stochastic Processes (Springer, Berlin, 2004), Vol. 1
K. Ito, Nagoya Math. J. 1, 35 (1950)
S.I. Denisov, A.N. Vitrenko, W. Horsthemke, Phys. Rev. E 68, 046132 (2003)
I. Eliazar, J. Klafter, J. Stat. Phys. 111, 739 (2003)
I. Eliazar, J. Klafter, J. Stat. Phys. 119, 165 (2005)
A. Dubkov, B. Spagnolo, Fluct. Noise Lett. 5, L267 (2005)
A. Dubkov, B. Spagnolo, V.V. Uchaikin, Int. J. Bifurcat. Chaos 18, 2649 (2008)
W. Feller, An Introduction to Probability Theory and its Applications, 2nd edn. (Wiley, New York, 1971), Vol. 2
P. Hänggi, Z. Phys. B 30, 85 (1978)
P. Hänggi, Z. Phys. B 36, 271 (1980)
N.G. van Kampen, Physica A 102, 489 (1980)
P. Hänggi, Z. Phys. B 31, 407 (1978)
B.V. Gnedenko, A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Cambridge, MA, 1954)
V.M. Zolotarev, One-Dimensional Stable Distributions (American Mathematical Society, Providence, 1986)
S.G. Samko, A.A. Kilbas, O.I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon & Breach, New York, 1993)
S.I. Denisov, W. Horsthemke, Phys. Rev. E 62, 7729 (2000)
A.D. Polyanin, V.F. Zaitsev, A. Moussiaux, Handbook of First-Order Partial Differential Equations (Taylor & Francis, London, 2002)
A.P. Prudnikov, Yu.A. Brychkov, O.I. Marichev, Integrals and Series (Gordon & Breach, New York, 1986), Vol. 1, Eq. (2.5.6.4)
H. Bateman, A. Erdélyi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2
E. Jakeman, R.J.A. Tough, Adv. Phys. 37, 471 (1988)
B. Dybiec, E. Gudowska-Nowak, I.M. Sokolov, Phys. Rev. E 76, 041122 (2007)
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Denisov, S., Horsthemke, W. & Hänggi, P. Generalized Fokker-Planck equation: Derivation and exact solutions. Eur. Phys. J. B 68, 567–575 (2009). https://doi.org/10.1140/epjb/e2009-00126-3
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DOI: https://doi.org/10.1140/epjb/e2009-00126-3