S. Albeverio et Vn. Kolokoltsov, THE RATE OF ESCAPE FOR SOME GAUSSIAN-PROCESSES AND THE SCATTERING-THEORY FOR THEIR SMALL PERTURBATIONS, Stochastic processes and their applications, 67(2), 1997, pp. 139-159
A detailed estimate on the growth of the position coordinate in a nonl
inearly perturbed Ornstein-Uhlenbeck phase process is given in all dim
ensions d greater than or equal to 3. The corresponding stochastic wav
e operators are constructed. A corresponding asymptotics of the phase
space process for stochastically perturbed oscillator processes, and s
ystems of semilinear stochastic differential equations with nondiagona
l constant diffusion matrix is also studied, Extensions to infinite di
mensional cases (stochastically perturbed wave equation and Schrodinge
r equations) are given.