Some unsolved and new problems on measure-valued stochastic processes

超过程是一类测度值随机过程，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着广泛的应用。超过程的两个基本例子是Dawson-Watanabe超过程和Fleming-Viot超过程。在本项目中，我们建议对这两个超过程进行以下研究。 在项目的第一部分，我们想要引入一个概率度量值随机过程，它通过在对偶过程中结合同时发生多个碰撞的合并来推广经典的Fleming-Viot超过程。我们计划实施一种新的方法来确定这样一个涉及突变、选择和重组的过程的存在和唯一性。我们将研究这个过程的性质以及它与SPDE的联系。我们还计划进一步利用这一方法来建立和研究相关的广义踏脚石模型。我目前的博士生刘慧丽将致力于这个主题。 研究具有平均场相互作用的超过程，即这种超过程的参数依赖于整个过程的状态，一直是一个非常有趣且经常具有挑战性的问题。在项目的第二部分，我们计划研究这些过程。我们将从依赖空间运动的超过程开始，并根据超过程的当前状态以分支率开始。我将支持一名新的博士生在这个项目的这一部分工作。 本项目的第三部分涉及具有Levy空间运动的一维Dawson-Watanabe超过程的退出问题。对于具有一维谱负Levy空间运动的超布朗运动，我们计划了解它是如何从时空平面上的一组平行直线中出来的。

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.