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 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 O(epsilon(-2)). Here epsilon 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 O(epsilon(-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.