Generalized Superprocesses

超过程是测值随机过程。作为超过程的基本例子之一，Dawson-Watanabe超布朗运动作为空间分布的分枝粒子系统经验测度的时空质量标度极限而产生。另一个基本的例子也是粒子系统的尺度极限，它描述了在经历遗传漂移(重采样)的大种群中不同基因类型的相对频率的演变以及可能的突变、选择和重组。 我们计划对相互作用的超布朗运动和一般Fleming-Viot过程的支撑性进行研究。 理解具有平均场相互作用的超布朗运动，即那些分支机制依赖于测值过程的状态的超布朗运动一直是一个有趣的问题。这种过程的概率律通常被指定为相应的鞅问题的唯一解。如何证明此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，在熊(2013)最近的工作中，通过将超布朗运动的“分布函数”与倒向二重随机微分方程联系起来，证明了超布朗运动的“分布函数”所满足的一类随机微分方程解的路径唯一性。我们计划采用熊晓松(2013)的方法来建立具有平均场分支率的超布朗运动的相关鞅问题解的唯一性。我们还将研究这类过程的灭绝行为。 直到最近，人们对Fleming-Voit过程的支持性质还知之甚少。利用Donnelly和Kurtz的向下粒子表示，刘和周(2012,2015)和周(2014)在研究具有一般再生机制和布朗空间运动的Lambda-Fleming-Viot过程的支撑性方面取得了进展。我们计划研究具有布朗空间运动的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这类过程在低维空间中支撑点的不连通性，以及带有Levy空间运动的Fleming-Viot过程的支撑点传播现象。 这些研究为平均场相互作用的超过程提供了新的见解，为具有一般复制机制的Fleming-Viot过程提供了新的技术和更好的理解，有望为测值过程理论做出显着贡献。

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.