Fokker–Planck Equations for Time-Delayed Systems via Markovian Embedding

• Published in: Journal of statistical physics Vol. 177; no. 1; pp. 95 - 118
• Main Authors: Loos, Sarah A. M., Klapp, Sabine H. L.
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2019; Springer; Springer Nature B.V
• Subjects: Approximation; Discrete time systems; Distribution (Probability theory); Embedded systems; Embedding; Fokker-Planck equation; Markov processes; Mathematical analysis; Mathematical and Computational Physics; Physical Chemistry; Physics; Physics and Astronomy; Probability density functions; Quantum Physics; Statistical analysis; Statistical Physics and Dynamical Systems; Stochastic systems; Theoretical; Time delay systems; Fokker–Planck equations; Non-Markovian processes; Time delay; Stochastic delay differential equations; Novikov’s theorem
• ISSN: 0022-4715; 1572-9613
• DOI: 10.1007/s10955-019-02359-4

For stochastic systems with discrete time delay, the Fokker–Planck equation (FPE) of the one-time probability density function (PDF) does not provide a complete, self-contained probabilistic description. It explicitly involves the two-time PDF, and represents, in fact, only the first member of an infinite hierarchy. We here introduce a new approach to obtain a Fokker–Planck description by using a Markovian embedding technique and a subsequent limiting procedure. On this way, we find a closed, complete FPE in an infinite-dimensional space, from which one can derive a hierarchy of FPEs. While the first member is the well-known FPE for the one-time PDF, we obtain, as second member, a new representation of the equation for the two-time PDF. From a conceptual point of view, our approach is simpler than earlier derivations and it yields interesting insight into both, the physical meaning, and the mathematical structure of delayed processes. We further propose an approximation for the two-time PDF, which is a central quantity in the description of these non-Markovian systems as it directly gives the correlation between the present and the delayed state. Application to a prototypical bistable system reveals that this approximation captures the non-trivial effects induced by the delay remarkably well, despite its surprisingly simple form. Moreover, it outperforms earlier approaches for the one-time PDF in the regime of large delays.