Energy Growth in Schrödinger's Equation with Markovian Forcing

Schrödinger's equation is considered on a one-dimensional torus with time dependent potential v(θ, t) = λ V(θ) X (t), where V (θ) is an even trigonometric polynomial and X(t) is a stationary Markov process. It is shown that when the coupling constant λ is sufficiently small, the average kinetic energy grows as the square-root of time. More generally, the H^s norm of the wave function is shown to behave as t^s/4.