Langevin Description of Markov Master Equations II: Noise Correlations

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 I to-SDE we obtain as a necessary consequence that the noise f i ( t ) possesses a ~-correlated 2-nd order conditioned cumulant ( f l ( t l ) f i ( t 2 ) l x ( t * ) = x ) if t* < max {t 2, t~ }. For time points {t I < t2... __< t,_ 1 = t,} the conditioned cumulants of f i ( t ) of order n > 2 generally contain memory contributions, but vanish if t,_ 1 < t, and t* < t,. These memory terms are not of relevance for the measure of the macroscopic process x(t). Focussing on an alternative non-l to 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 order n > l of the noise f~l( t) generally contain memory contributions which are different from the corresponding possibly non-zero (Ito)-memory terms.