随机偏微分方程及其障碍问题的研究

This project contains two parts: the first part concerns the quasi-linear stochastic partial differential equations with two obstacles under classical framework; and the second part will be about the stochastic partial differential equations driven by G-Brownian motion under nonlinear expectation framework. .In the first part, we will use both probabilistic and analytical approaches to prove the existence and uniqueness of the two obstacles problem for quasi-linear SPDE with Lipschitz coefficients and establish a comparison theorem for the solutions. .In the second part, basing on the well-posedness of quasi-linear SPDE driven by G-Brownian motion, we will study the obstacle problem for quasi-linear SPDE under G-framework by analytical approach. Moreover, we will establish a maximum principle for its local solutions.

本项目的目标是在前期工作的基础上，研究经典框架下的随机偏微分方程的双边反射问题以及G-框架下的随机偏微分方程的相关问题。.本项目将研究系数满足Lipschitz条件和适当的可积性条件的拟线性抛物型随机偏微分方程的双边障碍问题的弱解的存在唯一性。为了实现上述研究目标，我们将使用概率和分析两种方法。.本项目还将研究G-框架下的随机偏微分方程及其障碍问题。在G-布朗运动驱动的拟线性抛物型随机偏微分方程的弱解的存在唯一性基础上，用分析的方法研究G-框架下随机偏微分方程的障碍问题。还将进一步建立其局部解所满足的最大值原理。