  Determining evolution equations governing the probability density function
(pdf) of non-Markovian responses to random differential equations (RDEs)
excited by coloured noise, is an important issue arising in various problems of
stochastic dynamics, advanced statistical physics and uncertainty
quantification of macroscopic systems. In the present work, such equations are
derived for a scalar, nonlinear RDE under additive coloured Gaussian noise
excitation, through the stochastic Liouville equation. The latter is an exact,
yet non-closed equation, involving aver-ages over the time history of the
non-Markovian response. This nonlocality is treated by applying an extension of
the Novikov-Furutsu theorem and a novel approximation, employing a stochastic
Volterra-Taylor functional expansion around instantaneous response moments,
leading to efficient, closed, approximate equations for the response pdf. These
equations retain a tractable amount of nonlocality and nonlinearity, and they
are valid in both the transient and long-time regimes for any correlation
function of the excitation. Also, they include as special cases various
existing relevant models, and generalize Hanggi's ansatz in a rational way.
Numerical results for a bistable nonlinear RDE confirm the accuracy and the
efficiency of the new equations. Extension to the multidimensional case
(systems of RDEs) is feasible, yet laborious.
