约束Markov过程的大偏差与拟遍历性及相关问题

Large deviation and ergodicity are two important topics in the theory of Markov process, with close relationship between them. Since the founding of morden theory of large deviations for Markov processes by Dosker-Varadhan,most of successive important results were obtained under certain assumptions on strong ergodicity or reversibility. In this project, large deviations and Quasi-ergodicity for constrained Markov processes, including absorbing Markov processes, will be investigated. Such kinds of processes have wide range of applications, but are not ergodic in the classical sense. The study of large deviation and ergodic behaviors for such processes under constrains is of both theoretical and applied great significance.Since various uniform estimation are usually no longer available due to the lack of strong ergodicity, refined local estimation are needed. The main object of this project is indeed to establish some intrinsic relationship among these two topics and two other important topics- - Quasi-ergodicity and Dirichlet eigenvalues.

大偏差与遍历性是现代Markov过程理论中的重要课题，不但具有丰富的理论，也有广泛的应用。它们之间有着密切的联系。自从Donsker-Varadhan创立了Markov过程大偏差的现代理论以来，大部分重要结果都建立在对过程较强的遍历性或可逆性假设之下。本项目主要研究约束Markov过程的大偏差与拟遍历性，其中包括吸收Markov过程。这类过程也具有广泛的实际背景，但不具备经典意义下的遍历性。研究这类过程的条件大偏差行为和遍历性具有重要的理论和应用价值。由于缺乏强遍历条件下的各种一致估计，需要对过程的演化作更精细的局部估计。本项目的主要目标是建立这种条件大偏差、拟遍历性和另外两个重要课题- - 拟平稳分布和Dirichlet特征值- - 之间的内在联系。

本项目的主要研究一类重要的非遍历Markov过程--吸收Markov过程的条件大偏差, 建立了这种条件大偏差、拟遍历性和另外两个重要课题- - 拟平稳分布和Dirichlet特征值- - 之间的内在联系, 刻划了拟遍历分布于拟平稳分布之间的相互关系与本质区别. 证明了拟遍历分布是某种大偏差速率函数的极小值点, 而这一极小值恰好是过程算子.的Dirichlet特征值, 并给出了这一Dirichlet特征值的边分刻划. 这些结果进一步丰富和完善了Markov过程理论.