Trichotomous noise-induced transitions

A nonlinear one-dimensional process driven by a multiplicative exponentiall y correlated three-level Markovian noise (trichotomous noise) is considered . An explicit second-order linear ordinary differential equation for the st ationary probability density distribution is obtained for the process. In t he case of a linear process with an additive trichotomous noise the exact f ormula for the steady-state distribution is obtained. The well-known dichot omous noise can be regarded as a special case of the trichotomous noise. As a rule, the system variable has three specific values where the probabilit y density distribution can be singular. Far the case of the Hongler model t he dependence of the behavior of the stationary probability density on the noise parameters is investigated in detail and illustrated by a phase diagr am. Applications to the Gompertz and Verhulst models are also discussed. [S 1063-651X(99)03708-3].