  Stochastic dynamics in the energy representation is employed as a method to
study non-equilibrium Brownian-like systems. It is shown that the equation of
motion for the energy of such systems can be taken in the form of the Langevin
equation with multiplicative noise. Properties of the steady states are
examined by solving the Fokker-Planck equation for the energy distribution
functions. The generalized integral fluctuation theorem is deduced for the
systems characterized by the shifted probability flux operator. There are a
number of entropy and fluctuation relations such as the Hatano-Sasa identity
and the Jarzynski's equality that follow from this theorem.
