The rate of escape for some Gaussian processes and the scattering theory for their small perturbations

A detailed estimate on the growth of the position coordinate in a nonlinearly perturbed Ornstein-Uhlenbeck phase process is given in all dimensions d ≥ 3. The corresponding stochastic wave operators are constructed. A corresponding asymptotics of the phase space process for stochastically perturbed oscillator processes, and systems of semilinear stochastic differential equations with nondiagonal constant diffusion matrix is also studied. Extensions to infinite dimensional cases (stochastically perturbed wave equation and Schrödinger equations) are given.