Generalized Superprocesses

超过程是测量值随机过程。作为超过程的基本例子之一，道森-渡边超布朗运动作为空间分布分支粒子系统经验测度的时空-质量尺度极限而出现。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，是概率测量值弗莱明-维奥超过程，它描述了在经历基因漂移（重新采样）以及可能的突变、选择和重组的大群体中不同基因型的相对频率的进化。****我们计划研究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。****理解具有平均场相互作用的超布朗运动一直是一个有趣的问题，即那些分支机制依赖于测量值过程状态的超布朗运动。这种过程的概率律通常被指定为对应鞅问题的唯一解。如何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，在Xiong（2013）的最新工作中，通过将超布朗运动的SPDE与向后双SDE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。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相关鞅问题解的唯一性。我们还将研究这些过程的消光行为。****弗莱明-福伊特过程的支持性质直到最近才得到相对较少的了解。Liu和Zhou（2012, 2015）和Zhou（2014）利用Donnelly和Kurtz的向下粒子表示，在研究具有一般繁殖机制和布朗空间运动的Lambda-Fleming-Viot过程的支持性质方面取得了进展。我们计划探索具有布朗空间运动的Lambda-Fleming-Voit过程命中概率的渐近估计。我们还计划进一步研究这种过程的低维支撑的不连通性以及具有Levy空间运动的Fleming-Viot过程的支撑传播现象。****所提出的研究有望通过为具有平均场相互作用的超过程带来新的见解，并为具有一般再生机制的弗莱明-维奥过程提供新的技术和更好的理解，从而对测量值过程理论做出重大贡献。***************************

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.***************************