  We discuss the stability of semiclassical gravity solutions with respect to
small quantum corrections by considering the quantum fluctuations of the metric
perturbations around the semiclassical solution. We call the attention to the
role played by the symmetrized two-point quantum correlation function for the
metric perturbations, which can be naturally decomposed into two separate
contributions: intrinsic and induced fluctuations. We show that traditional
criteria on the stability of semiclassical gravity are incomplete because these
criteria based on the linearized semiclassical Einstein equation can only
provide information on the expectation value and the intrinsic fluctuations of
the metric perturbations. By contrast, the framework of stochastic
semiclassical gravity provides a more complete and accurate criterion because
it contains information on the induced fluctuations as well. The
Einstein-Langevin equation therein contains a stochastic source characterized
by the noise kernel (the symmetrized two-point quantum correlation function of
the stress tensor operator) and yields stochastic correlation functions for the
metric perturbations which agree, to leading order in the large N limit, with
the quantum correlation functions of the theory of gravity interacting with N
matter fields. These points are illustrated with the example of Minkowski
spacetime as a solution to the semiclassical Einstein equation, which is found
to be stable under both intrinsic and induced fluctuations.
