Lévy型过程的遍历性与泛函不等式

Lévy型过程因其广泛应用，正在受到越来越多的关注。本项目主要研究Lévy型过程遍历性，其中对称Lévy型过程的泛函不等式和非对称Lévy型过程遍历性的象征表示是两大研究问题。对称Lévy型过程遍历性研究依据的是Dirichlet型理论，我们试图根据Hunt过程样本轨道关于Dirichlet型刻画，拟采用截断、时间变换等方法给出对称Lévy型过程保守、常返和正常返的充分条件；通过研究非局部对称Dirichlet型泛函不等式，拟采用subordinate变换方法给出对称Lévy型过程各种遍历的判别准则。非对称Lévy型过程遍历性研究则利用拟微分算子中象征的工具，我们试图通过Lévy型算子鞅问题解公式和象征的性质给出Lévy型过程非爆炸和常返的充分条件；依据带跳随机积分理论和马氏过程Lyapunov漂移条件，通过估计过程回返时尾概率事件和各种矩得到Lévy型过程遍历、指数遍历等性质的象征刻画。

Much attention has been paid to Lévy-type processes because of their wide applications.The main goal of this research proposal is to study the ergodicity of Lévy-type processes, which includes functional inequalities for symmetric Lévy-type processes and criteria for the ergodicity of non-symmetric Lévy-type processes via symbol. In the symmetric case, the investigation of the ergodicity is based on the theory of Dirichlet forms. We will apply the characterization for sample path of Hunt processes via Dirichlet forms, and use the cut-off and the time change approaches to obtain sufficient conditions for the conservativeness, the recurrence and the positive recurrence of symmetric Lévy type processes. We will also consider functional inequalites for non-local symmetric Dirichlet forms, and use the subordination technique to derive criteria for the ergodicity of symmetric Lévy-type processes.The study of the ergodicity for non-symmetric Lévy-type processes will be based on the symbol of pseudo-differential operators. Using the formula of the martingale problem for Lévy-type operators and the property of the symbol, we will give conditions in terms of the symbol for the non-explosion and the recurrence of Lévy-type processes. In order to establish the ergodicity and the exponential ergodicity of Lévy-type processes, we will provide explicit estimates for the tail and the moment of the return time. For this we use stochastic integration with jumps and the Lyapunov drift condition for Markov processes.