Statistical properties of a class of nonlinear systems driven by colored multiplicative Gaussian noise

We derive the time-dependent univariate and bivariate probability distribution function for an overdamped system with a quadratic potential driven by colored Gaussian noise, whose amplitude depends on the system state x as $|x{|}^{\alpha }.$ Particular attention is paid to the effect of the correlation function of the noise on the statistical properties of the system. We obtain exact expressions for the fractional moments as well as the correlation function of the system and calculate the fractal dimension. We also consider the special case of a constant potential and determine the criteria for anomalous diffusion and stochastic localization of free particles.