An effective stochastic semiclassical theory for the gravitational field

Assuming that the mechanism proposed by Gell-Mann and Hartle works as a mec hanism for decoherence and classicalization of the metric field, we formall y derive the form of an effective theory for the gravitational field in a s emiclassical regime. This effective theory takes the form of the usual semi classical theory of gravity, based on the semiclassical Einstein equation, plus a stochastic correction which accounts for the backreaction of the low est order matter stress-energy fluctuations.