Fokker-Planck description for a linear delayed Langevin equation with additive Gaussian noise (Q2826710)

In non-Markovian systems, there is no general procedure for constructing a transport (Fokker-Planck) equation, while starting from the Langevin equation. The authors address this particular point for linear delayed systems (multi-delayed systems in one dimension), which are known to display remarkably different behavior compared to their Markov counterparts. The focus is on so-called generalized Langevin equations, where time-nonlocality is given by a time convolution of the stochastic variable with a memory kernel. The related FPE is known to be under-determined and supplementary data must be obtained about the moments of the distribution, directly from the Langevin equation. The proposed solution of this problem amounts to the construction and solution of the closed FPE for the 1- and 2-time joint probability distributions associated with a DLE. The resultant FPE representation of the DLE is given in terms of the conditional probability distribution of the delay process and it is shown how aging effects appear. The formalism allows to derive analytically the Green-Kubo formula.