Recurrence, Transience and Bounded Harmonic Functions for Diffusions in the Plane

Abstract

We give conditions for transience and recurrence for certain diffusion processes in the plane which "look" recurrent in certain sectors and transient in others. We also give conditions for the convergence of diffusion paths to paths of the deterministic dynamical system corresponding to the first-order part of $L$, the generator of the process. This in turn is related to the question of existence of bounded harmonic functions for the operator $L$. Conditions are given for the existence and nonexistence of bounded harmonic functions for $L$.