Langevin description of Markov master equations II: Noise correlations

Abstract

In this paper we examine the cumulant properties of generally multiplicative noise of stochastically equivalent stochastic differential equations (SDE) for a given (integro) master equation. For an Ito-SDE we obtain as a necessary consequence that the noisef _I (t) possesses a δ-correlated 2-nd order conditioned cumulant 〈f _I(t ₁)f _I(t ₂)|x(t ^*)=x〉 ift ^*≦max{t ₂, t₁}. For time points {t ₁≦t₂...≦t_n−1=t_n} the conditioned cumulants off _I (t) of ordern>2 generally contain memory contributions, but vanish ift _n−1<t _n andt ^*≦t_n. These memory terms are not of relevance for the measure of the macroscopic processx(t). Focussing on an alternative non-Ito SDE description we discuss the resulting facts. The character of multiplicative noise is clearly not removable by choosing a different stochastic calculus. The cumulants of ordern>1 of the noisef _NI (t) generally contain memory contributions which are different from the corresponding possibly non-zero (Ito)-memory terms.