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Using a time-averaging technique we obtain exactly the probabilitydistribution for position and velocity of a Brownian particle under theinfluence of two heat baths at different temperatures. These baths areexpressed by a white noise term, representing the fast dynamics, and a colorednoise term, representing the slow dynamics. Our exact solution scheme accountsfor inertial effects, that are not present in approaches that assume theBrownian particle in the over-damped limit. We are also able to obtain thecontribution associated with the fast noise that are usually neglected by otherapproaches.

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