Mathematical analysis of the Wiener processes with time-delayed feedback

It is known that time delays generally make a system unstable. However, it is numerically observed that the diffusion coefficients of the Wiener processes with time-delayed feedback decrease while increasing the time delay tau. In particular, the decay of the diffusion coefficients with the form (1 / 1+tau)(2) has been confirmed by numerical simulations [Ando et al., Phys. Rev. E 96, 012148 (2017)]. In this paper, we present two analytical derivations for the relation (1 / 1+tau)(2) by dynamical system approaches using the Laplace transform and stochastic differential equations. (c) 2024 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International (CC BY-NC-ND) license (https://creativecommons.org/licenses/by-nc-nd/4.0/).