Lévy噪声驱动的随机偏微分方程的若干问题

本项目拟以跳扩散过程为主题，研究Lévy过程驱动的随机偏微分方程解的性质，主要包括遍历性、大偏差、热核估计等。具体有以下3个问题：(1)非局部算子的Dirichlet热核估计，主要研究非局部非对称算子的Dirichlet热核双边估计和梯度扰动的Dirichlet热核估计。(2)带跳的奇异系数随机偏微分方程，主要研究可加柱状α-稳定过程驱动的Holder连续系数、有界系数、满足更弱正则性条件系数、或可乘噪声情形等随机偏微分方程强解的轨道唯一性及其性质。(3)退化噪声和Lévy噪声驱动的随机偏微分方程，主要研究退化噪声或Lévy噪声驱动随机偏微分方程 (包括随机Navier-Stokes方程) 解的性质，如解的存在唯一性、遍历性、大偏差原理，以及泛函不等式等，并希望在一般退化乘法噪声下二维随机Navier-Stokes方程遍历性研究有所突破。

Based on the topic of diffusion processes with jumps, this project intends to study the properties of stochastic partial differential equations (SPDEs) driven by Lévy processes, such as ergodicity, large deviation principle, heat kernel estimates and so on. Specifically, following problems will be investigated in this project: (1) Dirichlet heat kernel estimates of nonlocal operators. To study the two-sided Dirichlet heat kernel estimates for nonlocal and non-symmetric operators, and the Dirichlet heat kernel estimates for the operators perturbed by gradient operators. (2) The study of singular SPDEs with jumps. We intend to study the pathwise uniqueness and properties for bounded Holder continuous coefficients or weaker regular coefficients SPDEs driven by additive cylindrical α-stable processes. And SPDEs with multiplicative noise are researched also in this project. (3) The study of SPDEs (including the stochastic Navier-Stokes equation) driven by Lévy processes or degenerate Brownian motion. In this case, we’ll study the properties of the solutions, such as the existence and uniqueness of the solution, ergodicity, large deviation principle, functional inequalities and so on. Meanwhile, we hope some breakthroughs in the research on the ergodicity of stochastic 2D Navier-Stokes equations driven by degenerate multiplicative Brownian motion.