Stochastic inflation and dimensional reduction

We adopt methods that are well-known in statistical physics to the problem of stochastic inflation. The effective power spectrum for the classical, stochastic long-wavelength fluctuations is calculated for free scalar fields in a de Sitter background. For a smooth separation into long and short wavelengths, we identify an infrared divergence of the effective power spectrum, which has its correspondence in statistical physics in the phenomenon of dimensional reduction. The inflationary dynamics pushes the affected scales exponentially fast to large superhorizon scales, and establishes scale-invariant behavior for smaller scales (for massless fields). In the limit of a sharp separation of wavelengths, the scale of the infrared divergence is pushed to infinity.