Generalized Superprocesses

广义超级过程

• 批准号: RGPIN-2016-06704
• 负责人: Zhou, Xiaowen
• 金额: $ 1.6万
• 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: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=710364
• 关键词: Generalized; Superprocesses

Superprocesses are measure-valued stochastic processes. As one of the fundamental examples of superprocesses, Dawson-Watanabe superBrownian motion arises as the space-time-mass scaling limit of empirical measures of spatially distributed branching particle systems. The other fundamental example, also arising as a scaling limit of particle systems, is the probability-measure-valued Fleming-Viot superprocess which describes the evolution of the relative frequencies of different genotypes in a large population undergoing genetic drifting (re-sampling) together with possible mutation, selection and recombination. We plan to carry out researches on interacting superBrownian motions and support properties of general Fleming-Viot processes. It is always an interesting problem to understand superBrownian motions with mean field interactions, i.e. those superBrownian motions whose branching mechanisms depend on the states of the measure-valued processes. The probability law of such a process is often specified as the unique solution to the corresponding martingale problem. It has been a challenging problem to show the uniqueness of solution to the martingale problem. For the one-dimensional superBrownian motion, in recent work of Xiong (2013) the pathwise uniqueness of solution to an SPDE satisfied by the "distribution function'' of the superBrownian motion is proved via associating the SPDE with a backward doubly SDE. We plan to adapt the approach of Xiong (2013) to establish the uniqueness of solution to the associated martingale problem for the superBrownian motion with mean field branching rate. We will also investigate the extinction behaviors of such processes. The support properties of Fleming-Voit processes are relatively less understood until recently. Using the lookdown particle representation of Donnelly and Kurtz, previous progresses have been made in Liu and Zhou (2012, 2015) and in Zhou (2014) on studying the support properties of Lambda-Fleming-Viot processes with general reproduction mechanisms and with Brownian spatial motion. We plan to explore asymptotic estimates on hitting probabilities of the Lambda-Fleming-Voit processes with Brownian spatial motion. We also plan to further study the disconnectedness of support for such a process in lower dimensions and the support propagation phenomena for Fleming-Viot processes with Levy spatial motions. The proposed researches are expected to make remarkable contributions to the theory of measure-valued processes by bringing new insight to superprocesses with mean field interactions and providing new techniques and better understanding to Fleming-Viot processes with general reproduction mechanisms.

超级过程是测值随机过程。作为超级过程的基本例子之一，道森-渡边超布朗运动是作为空间分布分支粒子系统的经验测量的时空质量尺度极限而出现的。另一个基本的例子，也是作为粒子系统的尺度限制而出现的，是概率测量值的 Fleming-Viot 超级过程，它描述了一起经历遗传漂变（重新采样）的大量群体中不同基因型的相对频率的演变可能发生突变、选择和重组。 我们计划开展相互作用的超布朗运动和一般弗莱明-维奥过程的支持性质的研究。 理解具有平均场相互作用的超布朗运动始终是一个有趣的问题，即那些分支机制取决于测值过程的状态的超布朗运动。这种过程的概率定律通常被指定为相应鞅问题的唯一解。证明鞅问题解的唯一性一直是一个具有挑战性的问题。对于一维超布朗运动，Xiong（2013）最近的工作中，通过将 SPDE 与后向双 SDE 相关联，证明了由超布朗运动的“分布函数”满足的 SPDE 解的路径唯一性。采用 Xiong (2013) 的方法来确定具有平均场分支率的超布朗运动的相关鞅问题的解的唯一性。这些过程的灭绝行为。 直到最近，人们对 Fleming-Voit 工艺的支撑特性的了解相对较少。使用 Donnelly 和 Kurtz 的下视粒子表示，Liu 和 Zhou (2012, 2015) 以及 Zhou (2014) 在研究具有一般再生机制和布朗空间的 Lambda-Fleming-Viot 过程的支持特性方面取得了进展。运动。我们计划探索布朗空间运动的 Lambda-Fleming-Voit 过程的命中概率的渐近估计。我们还计划进一步研究低维中此类过程的支持不连续性以及带有 Levy 空间运动的 Fleming-Viot 过程的支持传播现象。 所提出的研究预计将为测值过程理论做出显着贡献，为具有平均场相互作用的超级过程带来新的见解，并为具有一般复制机​​制的弗莱明-维奥过程提供新技术和更好的理解。