  Whereas semiclassical gravity is based on the semiclassical Einstein equation
with sources given by the expectation value of the stress-energy tensor of
quantum fields, stochastic semiclassical gravity is based on the Einstein-
Langevin equation, which has in addition sources due to the noise kernel. The
noise kernel is the expectation value of the stress-energy bi-tensor which
describes the fluctuations of quantum matter fields. Here, we first give an
axiomatic derivation of the Einstein-Langevin equations and then show how they
can also be derived by the original method based on the influence functional.
As a first application we solve these equations and compute the two- point
correlation functions for the linearized Einstein tensor and for the metric
perturbations in a Minkowski background. We then turn to the important issue of
the validity of semiclassical gravity by examining the criteria based on the
ratio of the variance of fluctuations of the stress-energy tensor of a quantum
field to its mean. We show a calculation of these quantities for a massless
scalar field in the Minkowski and the Casimir vacua as a function of an
intrinsic scale defined by introducing a smeared field or by point separation.
Contrary to prior claims, the ratio of variance to mean-squared being of the
order unity does not necessarily imply the failure of semiclassical gravity.
Next we use the point-separation method to derive a general expression for a
regularized noise-kernel for quantum fields in general curved spacetimes. Using
these expressions one could investigate a host of important problems in early
universe and black hole physics.
