Generalized Superprocesses

超过程是测度值随机过程。作为超过程的一个基本例子， Dawson-Watanabe超布朗运动作为空间分布的分支粒子系统的经验测量的时空质量标度极限而出现。另一个基本的例子，也出现了作为一个缩放限制的粒子系统，是概率测度值的弗莱明-维奥特超过程，它描述了在一个大的人口经历遗传漂移（重新采样）的不同基因型的相对频率的演变连同可能的突变，选择和重组。 我们计划对一般Fleming-Viot过程的相互作用超布朗运动和支撑性质进行研究。 理解具有平均场相互作用的超布朗运动，即分支机制依赖于测度值过程状态的超布朗运动，一直是一个有趣的问题。这样一个过程的概率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）最近的工作通过将超布朗运动的"分布函数"所满足的SPDE与一个后向双凸函数相联系，证明了SPDE解的路径唯一性.我们计划采用Xiong（2013）的方法来建立具有平均场分支率的超布朗运动的鞅问题的解的唯一性。我们还将研究这些过程的灭绝行为。 直到最近，对Fleming-Voit过程的支持特性的了解相对较少。Liu and Zhou（2012，2015）和Zhou（2014）利用Donnelly和Kurtz的下视粒子表示研究了具有一般再生机制和布朗空间运动的Lambda-Fleming-Viot过程的支撑性质。我们计划研究具有布朗空间运动的Lambda-Fleming-Voit过程的击中概率的渐近估计。我们还计划进一步研究在低维的支持这样一个过程的不连通性和支持传播现象的Fleming-Viot过程与Levy空间运动。 这些研究将为测度值过程理论的发展做出重要贡献，为具有平均场相互作用的超过程提供新的认识，为具有一般再生机制的Fleming-Viot过程提供新的技术和更好的理解.

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.