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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 ofstochastic dynamics, advanced statistical physics and uncertaintyquantification of macroscopic systems. In the present work, such equations arederived for a scalar, nonlinear RDE under additive coloured Gaussian noiseexcitation, through the stochastic Liouville equation. The latter is an exact,yet non-closed equation, involving aver-ages over the time history of thenon-Markovian response. This nonlocality is treated by applying an extension ofthe Novikov-Furutsu theorem and a novel approximation, employing a stochasticVolterra-Taylor functional expansion around instantaneous response moments,leading to efficient, closed, approximate equations for the response pdf. Theseequations retain a tractable amount of nonlocality and nonlinearity, and theyare valid in both the transient and long-time regimes for any correlationfunction of the excitation. Also, they include as special cases variousexisting relevant models, and generalize Hanggi's ansatz in a rational way.Numerical results for a bistable nonlinear RDE confirm the accuracy and theefficiency of the new equations. Extension to the multidimensional case(systems of RDEs) is feasible, yet laborious.

A systematic path to non-Markovian dynamics: New response pdf evolution equations under Gaussian coloured noise excitation