An effective stochastic semiclassical theory for the gravitational field

Assuming that the mechanism proposed by Gell-Mann and Hartle works as a mechanism for decoherence and classicalization of the metric field, we formally derive the form of an effective theory for the gravitational field in a semiclassical regime. This effective theory takes the form of the usual semiclassical theory of gravity, based on the semiclassical Einstein equation, plus a stochastic correction which accounts for the back reaction of the lowest order matter stress-energy fluctuations.