  We study the evolution of the energy (mode-power) distribution for a class of
randomly perturbed Hamiltonian partial differential equations and derive {\it
master equations} for the dynamics of the expected power in the discrete modes.
In the case where the unperturbed dynamics has only discrete frequencies
(finitely or infinitely many) the mode-power distribution is governed by an
equation of discrete diffusion type for times of order $\cO(\ve^{-2})$. Here
$\ve$ denotes the size of the random perturbation. If the unperturbed system
has discrete and continuous spectrum the mode-power distribution is governed by
an equation of discrete diffusion-damping type for times of order
$\cO(\ve^{-2})$. The methods involve an extension of the authors' work on
deterministic periodic and almost periodic perturbations, and yield new results
which complement results of others, derived by probabilistic methods.
