Exact time-average distribution for a stationary non-Markovian massive Brownian particle coupled to two heat baths

Using a time-averaging technique we obtain exactly the probability distribution for position and velocity of a Brownian particle under the influence of two heat baths at different temperatures. These baths are expressed by a white noise term, representing the fast dynamics, and a colored noise term, representing the slow dynamics. Our exact solution scheme accounts for inertial effects that are not present in approaches that assume the Brownian particle in the overdamped limit. We are also able to obtain the contributions associated with the fast noise that are usually neglected by other approaches.