https://doi.org/10.1140/epjb/e2009-00126-3
Generalized Fokker-Planck equation: Derivation and exact solutions
1
Max-Planck-Institut für Physik komplexer Systeme, Nöthnitzer
Straße 38, 01187 Dresden, Germany
2
Sumy State University, 2
Rimsky-Korsakov Street, 40007 Sumy, Ukraine
3
Department of Chemistry, Southern Methodist University, Dallas, Texas, 75275, USA
4
Institut
für Physik, Universität Augsburg, Universitätsstraße 1, 86135 Augsburg, Germany
Corresponding author: a stdenis@pks.mpg.de
Received:
30
December
2008
Published online:
8
April
2009
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.
PACS: 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motion / 05.10.Gg – Stochastic analysis methods / 02.50.-r – Probability theory, stochastic processes, and statistics
© EDP Sciences, Società Italiana di Fisica, Springer-Verlag, 2009
