  This paper presents a new, non-Gaussian formulation of stochastic gravity by
incorporating the higher moments of the fluctuations of the quantum stress
energy tensor for a free quantum scalar field in a consistent way. A scheme is
developed for obtaining realizations of these fluctuations in terms of the
Wightman function, and the behavior of the fluctuations is investigated. The
resulting probability distribution for fluctuations of the energy density in
Minkowski spacetime is found to be similar to a shifted Gamma distribution.
This distribution features a minimum energy density cutoff at a small negative
value, but a sharp peak in the vicinity of this cutoff such that the total
probability of observing a negative value is approximately 62%, balanced by
correspondingly larger but rarer positive values.
