The authors study the effect of quantum fluctuations on the roll-down rate of the inflation field in a semiclassical approximation; this is done by treating the inflation field as a classical random field. The quantum fluctuations are simulated by a noise term in the equation of motion. They consider two different inflationary scenarios (new and chaotic inflation) and find that the roll-down rate of the median value of the inflation field is increased by the quantum fluctuations. Non-linear effects may become important in the later stages of the inflationary regime.
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Bardeen et al. (1987) studied this question.
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