带切换的随机偏微分方程的强解与强Feller性之研究

本项目主要研究带依状态切换的随机偏微分方程强解的存在唯一性、非爆炸性、强Feller性。我们将分别在Da Prato和 Zabczyk的半群方法框架下，以及Krylov和Rozovskii、Pardoux的变分方法框架下讨论。特别的，我们将证明一些系数非正则(如Dini-连续)的带依状态切换的随机半线性偏微分方程的强解存在唯一性，给出解非爆炸的条件，进一步考虑用可积性刻画的带切换的随机偏微分方程解非爆炸的条件。最后，在解存在唯一且非爆炸的基础上考虑转移半群的强Feller性质。

This project aims to study the existence, uniqueness, non-explosion and strong Feller property of the solutions of stochastic partial differential equations(SPDEs) with state-dependent switching. Our discussion will carry out within the frameworks introduced by Da Prato & Zabczyk, and by Pardoux, Krylov & Rozovskii respectively. Especially, we shall prove the existence and uniqueness, non-explosion of strong solutions of some state-dependent switching semilinear SPDEs with irregular(such as Dini-continuous) coefficients, and furthermore, concern the integrability conditions on the coefficients to ensure the non-explosion. Lastly, we shall study the strong Feller property when the SPDE has a unique non-explosive strong solution.