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We consider an infrared truncated massless minimally coupled scalar fieldwith a quartic self-interaction in the locally de Sitter background of aninflating universe. We compute the two-point correlation function of the scalarat one and two-loop order applying quantum field theory. The tree-ordercorrelator at a fixed comoving separation (that is at increasing physicaldistance) freezes in to a nonzero value. At a fixed physical distance, it growslinearly with comoving time. The one-loop correlator, which is the dominantquantum correction, implies a negative temporal growth in the correlationfunction, at this order, at a fixed comoving separation and at a fixed physicaldistance. We also obtain quantitative results for variance in space and time ofone and two-loop correlators and infer that the contrast between the vacuumexpectation value and the variance becomes less pronounced when the loopcorrections are included. Finally, we repeat the analysis of the model applyinga stochastic field theory and reach the same conclusions.

Vacuum Fluctuations of a Scalar Field during Inflation: Quantum versus Stochastic Analysis