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.

超级过程是测值随机过程，在相互作用粒子系统、随机偏微分方程和群体遗传学的研究中得到了许多应用。超级进程的两个基本示例是 Dawson-Watanabe 超级进程和 Fleming-Viot 超级进程。在这个项目中，我们提出对这两个超级过程进行以下研究。 在该项目的第一部分中，我们希望引入一种概率测度值随机过程，该过程通过在其对偶过程中结合同时发生多个碰撞的聚结来推广群体遗传学的经典 Fleming-Viot 超级过程。 我们计划实施一种新颖的方法来确定涉及突变、选择和重组的过程的存在性和唯一性。我们将研究该过程的特性及其与 SPDE 的联系。我们还计划进一步利用这种方法来建立和研究相关的广义垫脚石模型。我现在的博士生刘慧丽将研究这个课题。 研究具有平均场相互作用的超级过程是一个非常有趣且经常具有挑战性的问题，即这种超级过程的参数取决于过程的整个状态。在该项目的第二部分，我们计划研究这些过程。我们将从具有相关空间运动的超级过程开始，并且分支率取决于超级过程的当前状态。我将支持一名新的博士生从事该项目的这一部分。 该项目的第三部分涉及具有 Levy 空间运动的一维 Dawson-Watanabe 超级过程的退出问题。对于从原点处的点质量开始的具有一维谱负利维空间运动的超布朗运动，我们计划了解它如何从时空平面上的一组平行直线中退出。