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