Generalized Fokker-Planck equation: Derivation and exact solutions

Details Generalized Fokker-Planck equation: Derivation and exact solutions Generalized Fokker-Planck equation: Derivation and exact solutions

2008-08-02

arxiv.org/abs/0808.0274v2

Eur. Phys. J. B 68, 567 (2009)

Physics

Condensed Matter

Statistical Mechanics

9 pages

Scientific paper

10.1140/epjb/e2009-00126-3

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 L\'{e}vy 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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