Some unsolved and new problems on measure-valued stochastic processes

测值随机过程的一些未解决的新问题

• 批准号: 249554-2011
• 负责人: Zhou, Xiaowen
• 金额: $ 0.95万
• 依托单位: Concordia University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=570644
• 关键词: Some; unsolved; new; problems; measure

Superprocesses are measure-valued stochastic processes that have found many applications in the studies of interacting particle systems, stochastic partial differential equations and population genetics. Two fundamental examples for superprocesses are the Dawson-Watanabe superprocess and the Fleming-Viot superprocess. In this project, we propose the following research on these two superprocesses. In the first part of the project, we want to introduce a probability-measure-valued stochastic process that generalizes the classical Fleming-Viot superprocess for population genetics by incorporating the coalescent with simultaneous multiple collisions in its dual process. We plan to implement a novel approach to establish the existence and uniqueness for such a process involving mutation, selection and recombination. We are going to study the properties for this process and its connection to SPDE. We also plan to further exploit this approach to set up and study the related generalized stepping-stone model. My current PhD student, Huili Liu, is going to work on this topic. It has been a very interesting and often challenging problem to study superprocesses with mean field interactions, i.e. the parameters of such a superprocess depend on the whole state of the process. In the second part of the project, we plan to study these processes. We will start with the superprocess with dependent spatial motion and with branching rate depending on the current state of the superprocess. I am going to support a new PhD student to work on this part of the project. The third part of this project concerns the exit problem for one-dimensional Dawson-Watanabe superprocess with Levy spatial motion. For superBrownian motion with one-dimensional spectrally negative Levy spatial motion starting with a point mass at the origin, we plan to understand how it exits from a collection of parallel straight lines on the space-time plane.

超过程是测量值随机过程，在相互作用粒子系统、随机偏微分方程和种群遗传学的研究中有许多应用。超过程的两个基本例子是Dawson-Watanabe超过程和Fleming-Viot超过程。在这个项目中，我们建议对这两个超过程进行以下研究。