  We study primordial perturbations generated from quantum fluctuations of an
inflaton based on the formalism of stochastic gravity. Integrating out the
degree of freedom of the inflaton field, we analyze the time evolution of the
correlation function of the curvature perturbation at tree level and compare it
with the prediction made by the gauge-invariant linear perturbation theory. We
find that our result coincides with that of the gauge-invariant perturbation
theory if the e-folding from the horizon crossing time is smaller than some
critical value ($\sim |$slow-roll parameter $|^{-1}$), which is the case for
the scales of the observed cosmological structures. However, in the limit of
the superhorizon scale, we find a discrepancy in the curvature perturbation,
which suggests that we should include the longitudinal part of the
gravitational field in the quantization of a scalar field even in stochastic
gravity.
