Some unsolved and new problems on measure-valued stochastic processes

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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.