Fokker-Planck approach to stochastic systems with time delay - challenges and advances

Finding a consistent probabilistic description of stochastic systems with discrete time-
delay, whose Langevin dynamics naturally leads to an infinite hierarchy of Fokker-Planck
(FP) equations for the n-time joint probability distribution functions [1-3], is a theoretical challenge
which still raises many open questions. This is especially true for systems subject to nonlinear
forces, which is at the same time a particularly important case from the viewpoint of applications.
One major issue is that the higher members of the hierarchy contain unknown functional derivatives
between noise and the stochastic state variable.
In this talk, I will review different ways to derive a FP equation [1-3] and introduce a new
derivation that yields an alternative representation of the hierarchy. I will discuss three approaches
[1-3] to find an approximate solution for the one-time probability density, for example a closure
scheme based on a linearization of the deterministic forces in all members of the hierarchy starting
from the second one [3]. Furthermore, I will address the question whether these approaches can be
generalized towards higher members of the hierarchy, aiming to find a suitable approximation for
the two-time probability density. The latter is indeed a very important quantity for the calculation of
dynamical properties and in the context of stochastic thermodynamics [4].

[1] T. D. Frank, Phys. Rev. E 71, 031106 (2005).
[2] S. Guillouzic et al., Phys. Rev. E 59, 3970 (1999).
[3] S. A. M. Loos and S. H. L. Klapp, Phys. Rev. E 96, 012106 (2017).
[4] S. A. M. Loos and S. H. L. Klapp, arXiv:1806.04995 (2017).