  The concept of stochastic Lagrangian and its use in statistical dynamics is
illustrated theoretically, and with some examples.
  Dynamical variables undergoing stochastic differential equations are
stochastic processes themselves, and their realization probability functional
within a given time interval arises from the interplay between the
deterministic parts of dynamics and noise statistics. The stochastic Lagrangian
is a tool to formulate realization probabilities via functional integrals, once
the statistics of noises involved in the stochastic dynamical equations is
known. In principle, it allows to highlight the invariance properties of the
statistical dynamics of the system. In this work, after a review of the
stochastic Lagrangian formalism, some applications of it to physically relevant
cases are illustrated.
