Probabilistic treatment of steady states of classical overdamped noisy systems with time-delay
We explore possibilities to describe the dynamical behaviour of
classical overdamped, noisy systems with a time-delayed feedback force
(that depends on the system state at one earlier instant in time
t-τ) via probability densities. Due to the non-Markovian character
of such systems, there is no standard Fokker-Planck (FP) equation
which corresponds to the (delayed) stochastic Langevin equation [1,2].
In my talk, I will first review earlier theoretical work on how the FP
approach for delayed systems yields an infinite hierarchy of coupled
differential equations that involves n-time (joint) probability
densities depending on an increasing number of instances in time n
-> ∞ . Although these equations are not self-sufficient,
they are a valuable starting point for approximation schemes. In
particular, I will discuss a first order perturbation-theoretical
approach  and its application to two exemplary systems involving a
Brownian particle in a one-dimensional potential with delayed
feedback. We compare the perturbation-theoretical results with those
from Brownian dynamics simulations of the underlying delayed Langevin
equation. Further, we discuss properties of the two-time probability
density, an essential ingredient for the first member of the delayed
 S. Guillouzic et al., Phys. Rev. E 59, 3970 (1999).
 M. L. Rosinberg et al., Phys. Rev. E 91, 042114 (2015).
 T. D. Frank, Phys. rev. E 71, 031106 (2005).