  We discuss dissipative stochastic wave and diffusion equations resulting from
an interaction of the inflaton with an environment in an external expanding
metric. We show that a diffusion equation well approximates the wave equation
in a strong friction limit. We calculate the long wave power spectrum of the
wave equation under the assumption that the perturbations are slowly varying in
time and the expansion is almost exponential. Under the assumption that the
noise has a form invariant under coordinate transformations we obtain the power
spectrum close to the scale invariant one. In the diffusion approximation we go
beyond the slow variation assumption. We calculate the power spectrum exactly
in models with exponential inflation and polynomial potentials and with
power-law inflation and exponential potentials.
