Trichotomous noise-induced transitions

A nonlinear one-dimensional process driven by a multiplicative exponentially correlated three-level Markovian noise (trichotomous noise) is considered. An explicit second-order linear ordinary differential equation for the stationary probability density distribution is obtained for the process. In the case of a linear process with an additive trichotomous noise the exact formula for the steady-state distribution is obtained. The well-known dichotomous noise can be regarded as a special case of the trichotomous noise. As a rule, the system variable has three specific values where the probability density distribution can be singular. For the case of the Hongler model the dependence of the behavior of the stationary probability density on the noise parameters is investigated in detail and illustrated by a phase diagram. Applications to the Gompertz and Verhulst models are also discussed.