  Recent rapid advances in single particle tracking and supercomputing
techniques resulted in an unprecedented abundance of diffusion data exhibiting
complex behaviours, such the presence of power law tails of the msd and memory
functions, commonly referred to as "fractional dynamics". Motivated by these
developments, we study the stationary solutions of the classical and
generalized Langevin equation as models of the contemporarily observed
phenomena.
  In the first chapter we sketch the historical background of the generalized
Langevin equation. The second chapter is devoted to the brief overview of the
theory of the Gaussian variables and processes. In the third chapter we derive
the generalized Langevin equation from Hamilton's equations of motion. In the
fourth chapter the series of propositions and theorems shows that a large class
of Langevin equations has a solutions that, sampled in discrete time, are the
moving average autoregressive processes, which can be analysed using large
number of available statistical methods. The fifth chapter starts with a short
introduction into the basic notions of ergodic theory. Then we prove a
generalized version of the classical Maruyama's theorem, which describes the
time averages of all stationary Gaussian processes with a finite number of
spectral atoms. In the last, sixth chapter we continue the study of the
non-ergodic solutions of the Langevin equation, this time introducing the
ergodicity breaking through the random parametrisation of the equation itself.
This leads to a specific form of non-Gaussianity which affects only
many-dimensional distributions of the process. The main result of this chapter
is the series of propositions which describes the second order structure of
solutions of the superstatistical Langevin equation.
