Fokker-Planck representations of non-Markov Langevin equations: application to delayed systems

Luca Giuggioli*, Zohar Neu

*Corresponding author for this work

Research output: Contribution to journalArticle (Academic Journal)peer-review

2 Citations (Scopus)
259 Downloads (Pure)


Noise and time delays, or history-dependent processes, play an integral part of many natural and man-made systems. The resulting interplay between random fluctuations and time non-locality are essential features of the emerging complex dynamics in non-Markov systems. While stochastic differential equations in the form of Langevin equations with additive noise for such systems exist, the corresponding probabilistic formalism is yet to be developed. Here we introduce such a framework via an infinite hierarchy of coupled Fokker-Planck equations for the n-time probability distribution. When the non-Markov Langevin equation is linear, we show how the hierarchy can be truncated at n= 2 by converting the time non-local Langevin equation to a timelocal one with additive coloured noise. We compare the resulting Fokker-Planck equations to an earlier version, solve them analytically and analyse the temporal features of the probability distributions that would allow to distinguish between Markov and non-Markov features.
Original languageEnglish
Article number2018131
Number of pages23
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Issue number2153
Early online date22 Jul 2019
Publication statusPublished - 1 Sep 2019


  • Non-Markov processes
  • Generalised Langevin equation
  • Fokker-Planck equation
  • Delayed Langevin equation
  • Stochastic systems


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