Ergodicity For Lévy Type Operators And Related Topics

This paper is to study the ergodicity for general Lévy type operators and related problems. It consists of five parts.In the first part, we present sufficient conditions for the recurrence, the positive recurrence and the exponential ergodicity of one-dimensional Lévy type operators. These conditions are based on drift inequalities for the extended generator and a comparison with diffusion operators. We classify them according to different conditions on the ranges and integrability of the Lévy measure. A number of examples are illustrated, including the fractional Laplacian operator and the Ornstein-Uhlenbeck type operator.The second part firstly describes two typical expressions of general Lévy type operators. They are the integral-differential representation and the pseudo-differential representation. Using both of them, we obtain sufficient conditions for the non-explosion and the recurrence of Lévy type operators. In particular, our result extends Chung-Fuchs's recurrent criteria about Lévy processes. Furthermore, sufficient conditions for the Feller continuity of Lévy type operators are also presented.The study of symmetric property in the L~2-sense for the non-positive definite operator is motivated by the theory of probability and analysis. In part three, we provide sufficient conditions for the existence of symmetric measure for Lévy type operators. Some new examples are constructed.The aim of part four is to discuss sufficient conditions and necessary conditions for the Poincare inequality of one-dimensional symmetric Lévy type operators with Dirichlet boundary. We use the variational formula of the principal eigenvalue for general symmetric Markov processes and the classic Hardy inequality to obtain some sufficient conditions. As for necessary conditions, the weighted Hardy inequality for decreasing functions is employed. At the last, some interesting examples and problems are presented.One of our concerned problems about the symmetric Lévy type operator is when the corresponding Poincare inequality holds. However, as pointed out in previous chapters, this topic is hard in general. We devote our time in the last part to establishing the relationship between functional inequalities and Lyapunov conditions. More precisely, we can derive functional inequalities directly from the classic Lyapunov conditions. This approach is successfully applied to general symmetric Markov processes, and also shows clearly the connection between part one and part four.