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

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 49; no. 38; pp. 384002 - 384030
• Main Authors: Giuggioli, Luca, McKetterick, Thomas John, Kenkre, V M, Chase, Matthew
• Format: Journal Article
• Language: English
• Published: IOP Publishing 23.09.2016
• Subjects: ageing dynamics; delay Langevin equation; Fokker-Planck equation for delay proceeses; generalized Green-Kubo formula; generalized Langevin equation; non-Markov processes; one and two-time probability distribution delayed processes
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/49/38/384002

We construct an equivalent probability description of linear multi-delay Langevin equations subject to additive Gaussian white noise. By exploiting the time-convolutionless transform and a time variable transformation we are able to write a Fokker-Planck equation (FPE) for the 1-time and for the 2-time probability distributions valid irrespective of the regime of stability of the Langevin equations. We solve exactly the derived FPEs and analyze the aging dynamics by studying analytically the conditional probability distribution. We discuss explicitly why the initially conditioned distribution is not sufficient to describe fully out a non-Markov process as both preparation and observation times have bearing on its dynamics. As our analytic procedure can also be applied to linear Langevin equations with memory kernels, we compare the non-Markov dynamics of a one-delay system with that of a generalized Langevin equation with an exponential as well as a power law memory. Application to a generalization of the Green-Kubo formula is also presented.