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The topic of this PhD thesis is the derivation of evolution equations forprobability density functions (pdfs) describing the non-Markovian response todynamical systems under Gaussian coloured (smoothly-correlated) noise. Thesepdf evolution equations are derived from the stochastic Liouville equations(SLEs), which are formulated by representing the pdfs as averaged random deltafunctions. SLEs are exact yet non-closed, since they contain averaged termsthat are expressed via higher-order pdfs. These averaged terms are furtherevaluated by employing generalizations of the Novikov-Furutsu (NF) theorem.After the NF theorem, SLE averages are expressed equivalently as nonlocal termsdepending on the whole history of the response (in some cases, on the historyof excitation too). Then, nonlocal terms are approximated by a novel closurescheme, employing the history of appropriate moments of the response (or jointresponse-excitation moments). Application of this scheme results in a family ofnovel pdf evolution equations. These equations are nonlinear and retain atractable amount of the original nonlocality of SLEs, being also in closed formand solvable. Last, the new evolution equations for the one-time response pdfare solved numerically and their results are compared to Monte Carlo (MC)simulations, for the case of a scalar bistable random differential equationunder Ornstein-Uhlenbeck excitation. The results show that the novel evolutionequations are in very good agreement with the MC simulations, even for highnoise intensities and large correlation times of the excitation, that is, awayfrom the white noise limit, where the existing pdf evolution equations found inliterature fail. It should be noted that the computational effort for solvingthe new pdf evolution equations is comparable to the effort required forsolving the respective classical Fokker-Planck-Kolmogorov equation.

Probabilistic responses of dynamical systems subjected to Gaussian coloured noise excitation. Foundations of a non-Markovian theory