Adiabatic theorem for a class of stochastic differential equations on a Hilbert space

Abstract

We derive an adiabatic theory for a stochastic differential equation,

\[ \varepsilon\, \d X(s) = L_1(s) X(s)\, \d s + \sqrt{\varepsilon} L_2(s) X(s) \, \d B_s, \]

under a condition that instantaneous stationary states of $L_1(s)$ are also stationary states of $L_2(s)$. We use our results to derive the full statistics of tunneling for a driven stochastic Schrödinger equation describing a dephasing process.