Exact results are obtained for a large class of non-linear stochastic processes with constant diffusion. Special attention is given to the intrinsic effects induced by the non-linearities. In particular, a boomerang type behaviour of the mean path and time-controlled transitions from unimodal to bimodal probability densities are observed. Finally, the Onsager-Machlup stationary phase approximation is discussed and is recognised to provide a rather poor information for our particular class of models. © 1981.