Correlation functions and masterequations of generalized (non-Markovian) Langevin equations

Abstract

For the statistical behavior of macrovariables described in terms of Langevin equations with a in general colored random force we deduce useful formulas which simplify the calculation of correlation functions. Utilizing these results and the stochastic properties of the random force we derive an exact time-convolutionless masterequation for the probability hereby showing the mathematical equivalence of the formally different approaches of a Langevin description and a masterequation description. We study in detail the class of time-instantaneous Langevin equations and the important class of retarded (Mori-type) Langevin equations with both, Gaussian and general colored random forces. Using the generalization of the nonlinear Langevin equation for continuous Markov processes with white Gaussian noise and white generalized Poisson noise we show that the resulting masterequation can be recast in the Kramers-Moyal form. Interpreting this Langevin equation in the Stratonovitch sense we deduce the fluctuation induced drift (spurious drift) which can be divided up into two parts, the well known part induced by white Gaussian noise and the one induced by white generalized Poisson noise.