  We propose a method to obtain the equilibrium distribution for positions and
velocities of a one-dimensional particle via time-averaging and Laplace
transformations. We apply it to the case of a damped harmonic oscillator in
contact with a thermal bath. The present method allows us to treat, among other
cases, a Gaussian noise function exponentially correlated in time, e.g.,
Gaussian colored noise. We obtain the exact equilibrium solution and study some
of its properties.
