  In the first part of this paper, we show that the semiclassical
Einstein-Langevin equation, introduced in the framework of a stochastic
generalization of semiclassical gravity to describe the back reaction of matter
stress-energy fluctuations, can be formally derived from a functional method
based on the influence functional of Feynman and Vernon. In the second part, we
derive a number of results for background solutions of semiclassical gravity
consisting of stationary and conformally stationary spacetimes and scalar
fields in thermal equilibrium states. For these cases, fluctuation-dissipation
relations are derived. We also show that particle creation is related to the
vacuum stress-energy fluctuations and that it is enhanced by the presence of
stochastic metric fluctuations.
