Vacuum Fluctuations of a Scalar Field during Inflation: Quantum versus
  Stochastic Analysis
  We consider an infrared truncated massless minimally coupled scalar field
with a quartic self-interaction in the locally de Sitter background of an
inflating universe. We compute the two-point correlation function of the scalar
at one and two-loop order applying quantum field theory. The tree-order
correlator at a fixed comoving separation (that is at increasing physical
distance) freezes in to a nonzero value. At a fixed physical distance, it grows
linearly with comoving time. The one-loop correlator, which is the dominant
quantum correction, implies a negative temporal growth in the correlation
function, at this order, at a fixed comoving separation and at a fixed physical
distance. We also obtain quantitative results for variance in space and time of
one and two-loop correlators and infer that the contrast between the vacuum
expectation value and the variance becomes less pronounced when the loop
corrections are included. Finally, we repeat the analysis of the model applying
a stochastic field theory and reach the same conclusions.
