  We investigate the problem of metric fluctuations in the presence of the
vacuum fluctuations of matter fields and critically assess the usual assertion
that vacuum energy implies a Planckian cosmological constant. A new stochastic
classical approach to the quantum fluctuations of spacetime is developed. The
work extends conceptually Boyer's random electrodynamics to a theory of random
gravity but has a considerably richer structure for inheriting nonlinearity
from general relativity. Attention is drawn to subtleties in choosing boundary
conditions for metric fluctuations in relation to their dynamical consequences.
Those compatible with the observed Lorentz invariance must allow for
spontaneous conformal fluctuations, in addition to stochastic gravitational
waves due to zero point gravitons. This is implemented through an effective
metric defined in terms of the random spacetime metric modulo a fluctuating
conformal factor. It satisfies an effective Einstein equation coupled to an
effective stress-energy tensor incorporating gravitational self energy of
metric fluctuations as well as matter fields. The effective Einstein equation
is expanded up to second order nonlinearity. A UV-cutoff is introduced whose
specific value, however, does not enter into the resulting description of
random gravity. The averaged effective metric satisfies the empty space
Einstein equation with an effective cosmological constant. This vanishes when
only the massless matter fields are included. More generally, a finite
effective cosmological constant compatible with the observational constraints
can be obtained as long as the bare masses of the massive matter fields are
nearly zero, or the conformal invariance of matter is restored at some high
energy scale.
