Physica l12A (1982) 557-587 North-Holland Publishing Co. A CUMULANT EXPANSION FOR THE TIME CORRELATION FUNCTIONS OF SOLUTIONS TO LINEAR STOCHASTIC DIFFERENTIAL EQUATIONS J.B.T.M. ROERDINK Institute for Theoretical Physics, Princetonplein 5, Utrecht, The Netherlands Received 23 October 1981 It is shown that the cumulant expansion for linear stochastic differential equations, hitherto used to compute one-time averages of the solution process, is also capable of yielding the two-time correlation and probability density functions. The general case with a coefficient matrix, an inhomogeneous part and an initial condition which are all random and mutually correlated, is discussed. Two examples are given, the latter of which treats the harmonic oscillator with stochastic frequency and driving term studied before. Finally we investigate the relation of our method with the so-called smoothing method. 1. Introduction This article is concerned with linear stochastic differential equations of the form d u(t) = A(t, to)u(t) + f(t, to), (1.1) dt where u(t) is a vector, A(t, to) a random coefficient matrix or linear operator and/(t, to) a random vector*. The random nature of these quantities is indicated by the parameter to which will often be omitted in the following. The initial condition U(to) may be taken as fixed or in general as a random quantity u0(to). In a previous article1), hereafter referred to as I, we considered the case in which A(t, to), f(t, to) and u0(to) are mutually correlated. It was shown that the average of u(t) obeys itself a differential equation of the form d (u(t)) = K(t/to)(U(t)) + F(t/to) + I(t/to), (1.2) provided that a~'c is small, where a is a measure for the strength of the * Although the variable t in (1.1) in this article is interpreted as denoting a physical time, it could be any one-dimensional physical variable. 0378-4371/82/0000-0000l$02.75 © 1982 North-Holland