Probability distributions for second-order processes driven by Gaussian noise.

Abstract

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\ensuremath{\ll}\ensuremath{\tau}, t\ensuremath{\gg}\ensuremath{\tau} and for \ensuremath{\tau}=0 (white noise), where \ensuremath{\tau} is the correlation time. The parameters (moments) which determine the joint density are calculated including terms up to order ${\mathit{t}}^{2}$/${\mathrm{\ensuremath{\tau}}}^{2}$ for t\ensuremath{\ll}\ensuremath{\tau}, and up to order \ensuremath{\tau}/t for t\ensuremath{\gg}\ensuremath{\tau}. For t\ensuremath{\ll}\ensuremath{\tau} the marginal distribution of displacements is exactly Gaussian, to the considered order. A Gaussian distribution derived approximately for t\ensuremath{\gg}\ensuremath{\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 \ensuremath{\tau}.