  Semiclassical Einstein-Langevin equations for arbitrary small metric
perturbations conformally coupled to a massless quantum scalar field in a
spatially flat cosmological background are derived. Use is made of the fact
that for this problem the in-in or closed time path effective action is simply
related to the Feynman and Vernon influence functional which describes the
effect of the ``environment'', the quantum field which is coarse grained here,
on the ``system'', the gravitational field which is the field of interest. This
leads to identify the dissipation and noise kernels in the in-in effective
action, and to derive a fluctuation-dissipation relation. A tensorial Gaussian
stochastic source which couples to the Weyl tensor of the spacetime metric is
seen to modify the usual semiclassical equations which can be viewed now as
mean field equations. As a simple application we derive the correlation
functions of the stochastic metric fluctuations produced in a flat spacetime
with small metric perturbations due to the quantum fluctuations of the matter
field coupled to these perturbations.
