Abstract
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.
Highlights
Markov assumptions, at the heart of the celebrated Smoluchowski’s article on the equation for a diffusing particle in an external field [1], were so fruitful for the development of the modern theory of statistical physics [2,3,4,5,6], that, as is well-known, the equation has become a workhorse for all sorts of calculations of stochastic phenomena
In appendix A we show that it is possible to find a solution of such a Fokker–Planck equation (FPE) by postulating an ansatz informed by the dynamics of the moments of the Langevin equation, here we construct a closed form FPE
Stochastic delay differential equations have a long tradition. Since their original introduction to study the macrodynamic theory of business cycles [66, 67], they have been employed across a multitude of scientific areas
Summary
At the heart of the celebrated Smoluchowski’s article on the equation for a diffusing particle in an external field [1], were so fruitful for the development of the modern theory of statistical physics [2,3,4,5,6], that, as is well-known, the equation has become a workhorse for all sorts of calculations of stochastic phenomena. An exact solution for such an FPE, so far only obtained at steady state and for the case of one delay [39, 40], cannot be found without supplementing information about the moments of the distribution from the Langevin equation This under-determined nature of the FPE [40] calls into question its practical utility and has recently brought statements about finding the exact time dependent probability distribution equivalent to the DLE as a problem beyond reach [41].
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