  We derive the stochastic description of a massless, interacting scalar field
in de Sitter space directly from the quantum theory. This is done by showing
that the density matrix for the effective theory of the long wavelength
fluctuations of the field obeys a quantum version of the Fokker-Planck
equation. This equation has a simple connection with the standard Fokker-Planck
equation of the classical stochastic theory, which can be generalised to any
order in perturbation theory. We illustrate this formalism in detail for the
theory of a massless scalar field with a quartic interaction.
