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 an increasing physical distance) freezes into a nonzero value. At a fixed physical distance, it grows linearly with the 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.