  Using the influence functional formalism we show how to derive a generalized
Einstein equation in the form of a Langevin equation for the description of the
backreaction of quantum fields and their fluctuations on the dynamics of curved
spacetimes. We show how a functional expansion on the influence functional
gives the cumulants of the stochastic source, and how these cumulants enter in
the equations of motion as noise sources. We derive an expression for the
influence functional in terms of the Bogolubov coefficients governing the
creation and annihilation operators of the Fock spaces at different times, thus
relating it to the difference in particle creation in different histories. We
then apply this to the case of a free quantum scalar field in a spatially flat
Friedmann- Robertson-Walker universe and derive the Einstein-Langevin equations
for the scale factor for these semiclassical cosmologies. This approach based
on statistical field theory extends the conventional theory of semiclassical
gravity based on a semiclassical Einstein equation with a source given by the
average value of the energy momentum tensor, thus making it possible to probe
into the statistical properties of quantum fields like noise, fluctuations,
entropy, decoherence and dissipation. Recognition of the stochastic nature of
semiclassical gravity is an essential step towards the investigation of the
behavior of fluctuations, instability and phase transition processes associated
with the crossover to quantum gravity.
