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 τ. In particular, the decay of the diffusion coefficients with the form (11+τ)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 (11+τ)2 by dynamical system approaches using the Laplace transform and stochastic differential equations.