Adiabatic theorem for a class of quantum stochastic equations

We derive an adiabatic theory for a stochastic differential equation, $ \varepsilon, $\mathrm{d}$ X(s) = L_1(s) X(s), $\mathrm{d}$ s + \sqrt{\varepsilon} L_2(s) X(s) , $\mathrm{d}$ 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 Schr"{o}dinger equation describing a dephasing process.