  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 over-damped limit. We are also able to obtain the
contribution associated with the fast noise that are usually neglected by other
approaches.
