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Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck equations

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  • Frank, T.D.

Abstract

We discuss two fundamental aspects of Fokker–Planck equations that are nonlinear with respect to probability densities. First, we show that evolution equations of this kind describe processes involving stochastic feedback and interpret stochastic feedback processes in terms of hitchhiker processes and path integral solutions. Second, we demonstrate that nonlinear Fokker–Planck equations can be interpreted as linear Fokker–Planck equations describing nonlinear families of Markov diffusion processes. We exploit this finding in order to derive complete hierarchies of probability densities from nonlinear Fokker–Planck equations.

Suggested Citation

  • Frank, T.D., 2004. "Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 331(3), pages 391-408.
    • Handle: RePEc:eee:phsmap:v:331:y:2004:i:3:p:391-408
    DOI: 10.1016/j.physa.2003.09.056
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    Cited by:

    1. McCauley, Joseph L., 2007. "A comment on the paper “Stochastic feedback, nonlinear families of Markov processes, and nonlinear Fokker–Planck equations” by T.D. Frank," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 382(2), pages 445-452.
    2. Hashemi, M.S., 2015. "Group analysis and exact solutions of the time fractional Fokker–Planck equation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 417(C), pages 141-149.

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