  By revising the application of the open quantum system approach to the early
universe and extending it to the conditions beyond the Markovian approximation,
we obtain a new non-Markovian quantum Boltzmann equation. Throughout the paper,
we also develop an extension of the quantum Boltzmann equation to describe the
processes that are irreversible at the macroscopic level. This new kinetic
equation is, in principle, applicable to a wide variety of processes in the
early universe. For instance, using this equation one can accurately study the
microscopic influence of a cosmic environment on a system of cosmic background
photons or stochastic gravitational waves. In this paper, we apply the
non-Markovian quantum Boltzmann equation to study the damping of gravitational
waves propagating in a medium consisting of decoupled ultra-relativistic
neutrinos. For such a system, we study the time evolution of the intensity and
the polarization of the gravitational waves. It is shown that, in contrast to
intensity and linear polarization which are damped, the circular polarization
(V-mode) of the gravitational wave (if present) is amplified by propagating
through such a medium.
