Nonlinear Stochastic Partial Differential Equations and Applications

统计不确定性在各种复杂动力学现象的模型中起着重要作用，导致狂野的随机行为。这种概率效应是由全球经济中不可预测的市场变化或大规模森林火灾前沿的动荡或混乱的天气模式造成的。研究人员将对这些应用中出现的方程进行数学理解，同时还将研究随机噪声的稳定和正则化效应，这些效应通常有实验或数值证据。该项目将通过提供与项目相关的专业建议和数学知识，为研究生和本科生提供指导机会。在各种复杂影响下的动力学随机行为通常用非线性随机偏微分方程来描述。这些方程不能通过简单公式的叠加来求解，因此在数学上还没有得到很好的理解。该项目将利用泛函分析和概率的工具来解决竞争性大种群动力学和随机强迫界面演化中产生的非线性随机偏微分方程的适定性。随机性的影响将通过研究解决方案的长期行为，概率平均和正则化现象，以及具有小背景噪声水平的模型的随机选择原则来进一步分析。随机性的实质性影响表明，实验数据中的统计波动不能完全忽略，从而证明了对该项目中涉及的随机偏微分方程进行技术研究的合理性。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.