Studies of the Stochastic Partial Differential Equations

这项工作将研究随机偏微分方程（SPDEs）解的重要性质。这些方程在许多领域都有应用，从物理、化学到生物和金融市场。主要的例子包括随机热方程（SHE）、抛物型Anderson模型（PAM）和kardar - parisii - zhang方程（KPZ）。这些方程及其变体模拟了一些关键现象，如森林火灾的蔓延、烟雾的扩散、肿瘤组织的生长、超导性、宇宙分层结构的形成以及行星或恒星的磁场。本课题将通过建立解的存在唯一性、矩渐近性、解的正性和支持性、概率密度的存在性和平滑性等基本性质，加深我们对这些现象的理解。我们将积极鼓励来自弱势群体的学生参与本研究项目，营造一个包容和多样化的研究环境。研究者还参与了大学预科学生的外展项目。spde依赖于许多技术参数和条件，包括初始条件、噪声结构、扩散系数条件、空间域的几何和解析性质以及相应的边界条件、微分算子等。本项目将重点研究：(1)粗略初始条件下SHE/PAM的研究；(2)研究亚线性生长状态下的SHE，可能存在非lipschitz扩散系数；(3)探讨边界效应对SHE/PAM的影响。这三个目标将通过六个具体项目来实现。该项目由DMS概率计划和促进竞争研究的既定计划（EPSCoR）联合资助。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

The work will study important properties of solutions to stochastic partial differential equations (SPDEs). These equations have applications in many fields, ranging from physics and chemistry to biology and financial markets. Key examples include the stochastic heat equation (SHE), the parabolic Anderson model (PAM), and the Kardar-Parisi-Zhang (KPZ) equation. These equations and their variants model crucial phenomena such as the spread of forest fires, smoke dispersal, growth of tumor tissues, superconductivity, and the formation of the universe's stratified structure and magnetic fields of planets or stars. This project will deepen our understanding of these phenomena by establishing essential properties of solutions, such as existence and uniqueness, moment asymptotics, positivity and support of solutions, existence and smoothness of probability density. Active efforts will be made to encourage participation by students from underrepresented groups in this research program, fostering an inclusive and diverse research environment. The investigator also participates in outreach programs for pre-college students. SPDEs depend on many technical parameters and conditions, including initial conditions, noise structures, conditions on the diffusion coefficients, geometric and analytic properties of the spatial domain and the corresponding boundary conditions, differential operators, among others. This project will focus on: (1) Studying SHE/PAM with rough initial conditions; (2) Investigating SHE in the sublinear growth regime, possibly with non-Lipschitz diffusion coefficients; and (3) Exploring the boundary effects to SHE/PAM. These three objectives will be implemented via six concrete projects. This project is jointly funded by the DMS Probability program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.