Branching particle systems, stochastic partial differentiable equations and nonlinear rough path analysis

Abstract

In this dissertation, we study some problems related to the stochastic partial differential equations, branching particle systems and rough path analysis. In Chapter 1, we provide a brief introduction and background of the topics considered in this dissertation. In Charter 2, a branching particle system in a random environment has been studied. Under the Mytnik-Sturm branching mechanism, we prove that the scaling limit of this particle system exists. This limit has a Lebesgue density that is a weak solution to a stochastic partial equation. We also investigate the Hölder continuity of this limit, and prove it is 1/2 − ε in time and 1 − ε in space. In Chapter 3, a theory of nonlinear rough paths is developed. Following the idea of Lyons and Gubinelli, we define a nonlinear integral of rough functions. Then we study a rough differential differential equation, and obtain the local and global existence and uniqueness of this solution under suitable sufficient conditions. As an application, we consider the transport equation with rough vector field and observe the classical solution formula does not satisfy the rough equation. Indeed, it is the solution to the transport equation with compensators. In Chapter 4, we study the parabolic Anderson model of Skorokhod type with very rough noise in time. By using the Feynman-Kac formula for moments, we obtain the upper and lower bounds for moments of the solution.