Exactly solvable model with an absorbing state and multiplicative colored Gaussian noise

We study the temporal evolution of a system that has an absorbing state and that is driven by colored Gaussian noise, whose amplitude depends on the system state x as $|x{|}^{\alpha }.$ Exact, analytical expressions for the probability density functions of the system and of the absorption time are derived. We also calculate numerical characteristics of the probability density functions, namely, the fractional moments of the system and the mean absorption time, and analyze the role of the functional form of the noise correlation function.