The rate of escape for some Gaussian processes and the scattering theory for their small perturbations (Q1382530)

The authors investigate the long-time behaviour, more specifically, the growth rates, of some degenerate diffusion processes or of their components. First, they consider the Ornstein-Uhlenbeck process solving the system (1) \(dx=v dt\), \(dv=-\beta v dt +dw\) in \(\mathbb R^{2d}\), where \(w\) is a standard \(d\)-dimensional Wiener process, and they prove that \(\liminf _{t\to \infty } |x(t) |/f(t)=+\infty \) whenever \(d\geq 3\) and \(f\) is an increasing positive function such that \(\int ^{\infty }_{1}(t^{-1/2}f(t))^{d} dt<\infty \). Further, they study a nonlinear perturbation (2) \(dx=v dt\), \(dv = K(x) dt -\beta v dt +dw\) of the equation (1) and under some assumptions on the growth of the continuous function \(K\) they show the existence of a stochastic wave operator: for each solution of (1) there exists a solution to (2) with the same asymptotic behaviour at infinity. Various modifications and generalizations, including infinite-dimensional ones, of these results are then discussed.