  In this more pedagogical study we want to elucidate on stochastic aspects
inherent to the (non-)equilibrium real time Green's function description (or
`closed time path Green's function' -- CTPGF) of transport equations, the so
called `Kadanoff-Baym equations'. As a toy model we couple a free scalar boson
quantum field to an exemplaric heat bath with some given temperature T. It will
be shown in detail that the emerging transport equations have to be understood
as the ensemble average over stochastic equations of Langevin type. This
corresponds to the equivalence of the influence functional approach by Feynman
and Vernon and the CTP technique. The former, however, gives a more intuitive
physical picture. In particular the physical role of (quantum) noise and the
connection of its correlation kernel to the Kadanoff-Baym equations will be
discussed. The inherent presence of noise and dissipation related by the
fluctuation-dissipation theorem guarantees that the modes or particles become
thermally populated on average in the long-time limit. For long wavelength
modes with momenta much less than the temperature the emerging wave equation do
behave nearly as classical. On the other hand, a kinetic transport description
can be obtained in the semi-classical particle regime. Including fluctuations,
its form resembles that of a phenomenological Boltzmann-Langevin description.
However, we will point out some severe discrepancies in comparison to the
Boltzmann- Langevin scheme. As a further byproduct we also note how the
occurrence of so called pinch singularities is circumvented by a clear physical
necessity of damping within the one-particle propagator.
