Problems in self-similar Markov processes

自相似马尔可夫过程中的问题

• 批准号: 2436441
• 金额: --
• 依托单位: University of Bath
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2436441
• 关键词: Problems; self; similar; Markov; processes

The interplay between stable Lévy processes and self-similar Markov processes has attracted much interest in recent decades, and is what continues to motivate research in the field of Lévy processes. In particular, the problem of characterising the laws of Markov processes via Doob h-transforms which arise from limiting procedures of conditioned processes has been considered. As part of this project, we aim to describe the Markov Additive Process conditioned to continuously hit the unit sphere in d dimensions, building upon existing work which involved conditioning the stable process to continuously approach the unit sphere. With this in hand, we will proceed to develop a Williams type path decomposition for transient self-similar Markov processes which breaks the path of the process into conditioned segments and for which we will give a full description. En route, it will be necessary to develop several new fluctuation identities for Markov Additive Processes, which, themselves will form the basis of e.g. a chapter in the thesis.

稳定Lévy过程和自相似Markov过程之间的相互作用在近几十年来引起了人们的极大兴趣，并继续推动Lévy过程领域的研究。特别是，通过Doob h-变换所产生的限制程序的条件过程的马尔可夫过程的法律特征的问题已被考虑。作为这个项目的一部分，我们的目标是描述马尔可夫加性过程的条件，不断击中单位球在d维，现有的工作，其中涉及条件的稳定过程，不断接近单位球的基础上。有了这个在手，我们将着手开发一个威廉姆斯型路径分解的瞬态自相似马尔可夫过程，打破了该过程的路径成条件段，我们将给出一个完整的描述。在途中，将有必要开发几个新的马尔可夫加性过程的波动身份，这本身将形成的基础，例如在论文中的一章。