Fokker-Planck description for a linear delayed Langevin equation with additive Gaussian noise

Luca Giuggioli, Thomas John McKetterick, V. M. Kenkre, Matthew Chase

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

14 Citations (Scopus)
304 Downloads (Pure)


We construct an equivalent probability description of linear multi-delay Langevin equations subject to additive Gaussian white noise. By exploiting the time-convolutionless transform and a time variable transformation we are able to write a Fokker-Planck equation for the 1-time and for the 2-time probability distributions valid irrespective of the regime of stability of the Langevin equations. We solve exactly the derived Fokker-Planck equations and analyze the aging dynamics by studying analytically the conditional probability distribution. We discuss explicitly why the initially conditioned distribution is not sufficient to describe fully out a non-Markov process as both preparation and observation times have bearing on its dynamics. As our analytic procedure can also be applied to linear Langevin equations with memory kernels, we compare the non-Markov dynamics of a one-delay system with that of a
Generalized Langevin equation with an exponential as well as a power law memory. Application to a generalization of the Green-Kubo formula is also presented.
Original languageEnglish
Article number384002
Number of pages29
JournalJournal of Physics A: Mathematical and Theoretical
Issue number38
Publication statusPublished - 30 Aug 2016

Structured keywords

  • Engineering Mathematics Research Group


  • Non-Markov
  • Fokker-Planck
  • Langevin
  • aging
  • Green-Kubo


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