Stochastic systems with delay: Perturbation theory for second order statistics

• A perturbation theory for stochastic delay systems is presented.
• The perturbation theory yields second order statistical quantities.
• The theory is developed within the framework of delay Fokker–Planck equations.
• The effective Fokker–Planck operator is a non-local operator in space.
• Negative autocorrelations can be induced by time-delayed feedback.

Within the framework of delay Fokker–Planck equations, a perturbation theoretical method is developed to determine second-order statistical quantities such as autocorrelation functions for stochastic systems with delay. Two variants of the perturbation theoretical approach are presented. The first variant is based on a non-local Fokker–Planck operator. The second variant requires to solve a Fokker–Planck equation with source term. It is shown that the two variants yield consistent results. The perturbation theoretical approaches are applied to study negative autocorrelations that are induced by feedback delays and mediated by the strength of the fluctuating forces that act on the feedback systems.