Criteria for ergodicity of Lévy type operators in dimension one

Some sufficient conditions for the recurrence, the positive recurrence and the exponential ergodicity of one-dimensional Lévy type operators are presented. The conditions are classified according to different conditions on the ranges and integrability of the Lévy measure, based on the drift inequalities for the extended generator, and on a comparison with diffusion operators. A number of examples are illustrated, including the fractional Laplacian operator and the Ornstein-Uhlenbeck type operator.