In [254] it was shown that
the correlation functions that follow from the Einstein-Langevin
equation which emerges in the framework of stochastic gravity
coincide with that obtained with the usual quantization
procedures [218
], when both the
metric perturbations and the inflaton fluctuations are both
linearized. Stochastic gravity, however, can naturally deal with
the fluctuations of the inflaton field even beyond the linear
approximation.
Here we will illustrate the equivalence with the
usual formalism, based on the quantization of the linear
cosmological and inflaton perturbations, with one of the simplest
chaotic inflationary models in which the background spacetime is a
quasi-de Sitter universe [253, 254
].