  This paper proposes a simple mathematical model of non-stationary and
non-linear stochastic dynamics, which approximates a (globally) non-stationary
and non-linear stochastic process by its locally (or \emph{"piecewise"})
stationary version. Profiting from the elegance and simplicity of both, the
exact mathematical model referred to as the Ornstein-Uhlenbeck stochastic
process (which is \emph{globally} stationary, Markov and Gaussian) and of the
Lyapunov criterion associated with the stability of stationarity, we show that
the proposed non-linear non-stationary model provides a natural extension of
the Onsager-Machlup theory of equilibrium thermal fluctuations, to the realm of
non-stationary, non-linear, and non-equilibrium processes. As an illustrative
application, we then apply the extended non-equilibrium Onsager-Machlup theory,
to the description of thermal fluctuations and irreversible relaxation
processes in liquids, leading to the main exact equations employed to construct
the \emph{non-equilibrium self-consistent generalized Langevin equation}
(NE-SCGLE) theory of irreversible processes in liquids. This generic theory has
demonstrated that the most intriguing and long-unsolved questions of the glass
and the gel transitions are understood as a natural consequence of the second
law of thermodynamics, enunciated in terms of the proposed piecewise stationary
stochastic mathematical model.
