Dissipation and fluctuation of quantum fields in expanding universes

A stochastic dynamics of a long-wavelength part of a scalar field in an expanding universe is derived by using the influence functional method. Dissipation as well as fluctuation are derived for general parameters: a mass, a coupling to the scalar curvature, and a cutoff scale parameter. A dissipation-fluctuation relation is found with a temperature which is proportional to the Hawking temperature, but system dependent. The method is further applied to an expanding universe with a power law and yields the dispersion which agrees with that obtained by the regularization method. The back reaction to the background de Sitter space itself is also obtained.