The Kramers-Moyal Equation of the Cosmological Comoving Curvature
  Perturbation
  Fluctuations of the comoving curvature perturbation with wavelengths larger
than the horizon length are governed by a Langevin equation whose stochastic
noise arise from the quantum fluctuations that are assumed to become classical
at horizon crossing. The infrared part of the curvature perturbation performs a
random walk under the action of the stochastic noise and, at the same time, it
suffers a classical force caused by its self-interaction. By a path-interal
approach and, alternatively, by the standard procedure in random walk analysis
of adiabatic elimination of fast variables, we derive the corresponding
Kramers-Moyal equation which describes how the probability distribution of the
comoving curvature perturbation at a given spatial point evolves in time and is
a generalization of the Fokker-Planck equation. This approach offers an
alternative way to study the late time behaviour of the correlators of the
curvature perturbation from infrared effects.
