Exactly solvable nonlinear model with two multiplicative Gaussian colored noises

An overdamped system with a linear restoring force and two multiplicative colored noises is considered. Noise amplitudes depend on the system state \(x\) as \(x\) and \(|x|^{\alpha}\). An exactly soluble model of a system is constructed due to consideration of a specific relation between noises. Exact expressions for the time-dependent univariate probability distribution function and the fractional moments are derived. Their long-time asymptotic behavior is investigated analytically. It is shown that anomalous diffusion and stochastic localization of particles, not subjected to a restoring force, can occur.