Probability distributions for second-order processes driven by Gaussian noise

We derive a Fokker-Planck equation for the joint probability density of the displacement and the velocity of a free particle subjected to an exponentially correlated Gaussian force. This equation is solved analytically in the limits t≪τ, t≫τ and for τ=0 (white noise), where τ is the correlation time. The parameters (moments) which determine the joint density are calculated including terms up to order t²/τ² for t≪τ, and up to order τ/t for t≫τ. For t≪τ the marginal distribution of displacements is exactly Gaussian, to the considered order. A Gaussian distribution derived approximately for t≫τ is suggested to be exact, on the basis of independent, exact calculations of low-order moments. For Gaussian white noise, the joint density is obtained exactly and yields a Gaussian distribution of displacements, with the familiar superdiffusive form for the mean-square deviation. The marginal distribution of velocity obeys an exact diffusion equation with a variable diffusion coefficient, for arbitrary τ.