Nonlinear Stochastic Partial Differential Equations and Applications

非线性随机偏微分方程及其应用

• 批准号: 2307610
• 负责人: Benjamin Seeger
• 金额: $ 16.94万
• 依托单位: University of Texas at Austin
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-09-01 至 2026-08-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2307610&HistoricalAwards=false
• 关键词: Nonlinear; Stochastic; Partial; Differential; Equations

Statistical uncertainty plays a significant role in a diverse range of models for complicated dynamic phenomena, leading to wild, stochastic behavior. Such probabilistic effects are caused, for instance, by unpredictable market shifts in the global economy, or turbulent or chaotic weather patterns at the evolving front of a massive forest fire. The investigator will develop a mathematical understanding for the equations arising in these applications, while also studying the stabilizing and regularizing effects of stochastic noise, for which there is often experimental or numerical evidence. This project will generate opportunities to mentor graduate and undergraduate students by providing both professional advice and mathematical knowledge related to the project. Dynamical random behavior under various complex influences is often described by nonlinear stochastic partial differential equations. Such equations cannot be solved through the superposition of simple formulae and are therefore not yet well-understood mathematically. The project will draw on tools from functional analysis and probability to resolve the well-posedness of nonlinear stochastic partial differential equations arising in competitive large population dynamics and in stochastically forced interface evolutions. The effects of stochasticity will be further analyzed by studying the long-time behavior of solutions, probabilistic averaging and regularizing phenomena, and stochastic selection principles for models with a small level of background noise. The material influence of stochasticity indicates that the statistical fluctuations in experimental data cannot be completely ignored, thereby justifying the technical study of those stochastic partial differential equations involved in this project.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.

统计不确定性在复杂动态现象的各种模型中发挥着重要作用，导致疯狂的随机行为。例如，这种概率效应是由全球经济中不可预测的市场变化，或者大规模森林火灾演变过程中的动荡或混乱的天气模式引起的。研究人员将对这些应用中出现的方程建立数学理解，同时还研究随机噪声的稳定和正则效应，这通常有实验或数值证据。该项目将通过提供与该项目相关的专业建议和数学知识来为研究生和本科生提供指导的机会。各种复杂影响下的动态随机行为通常用非线性随机偏微分方程来描述。此类方程无法通过简单公式的叠加来求解，因此在数学上尚未得到很好的理解。该项目将利用泛函分析和概率分析工具来解决竞争性大群体动态和随机强迫界面演化中出现的非线性随机偏微分方程的适定性。通过研究解的长期行为、概率平均和正则化现象以及小背景噪声模型的随机选择原则，将进一步分析随机性的影响。随机性的实质性影响表明实验数据的统计波动不能完全忽略，从而证明了本项目涉及的随机偏微分方程技术研究的合理性。该奖项体现了 NSF 的法定使命，经评估认为值得支持。基金会的智力价值和更广泛的影响审查标准。