Diffusion of Power in Randomly Perturbed Hamiltonian Partial Differential Equations

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.