Generalized Superprocesses

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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.