PROBABILITY-DISTRIBUTIONS FOR 2ND-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 << tau, t >> tau and for tau = 0 (white noise), where tau is the correlation time. The parameters (moments) which determine the joint density are calculated including terms up to order t2/tau2 for t << tau, and up to order tau/t for t >> tau. For t << tau the marginal distribution of displacements is exactly Gaussian, to the considered order. A Gaussian distribution derived approximately for t >> tau 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 tau.