Stochastic Analysis and Asymptotic Problems

这个研究项目研究随机分析中的问题，随机分析是概率论的一部分，研究随机脉冲作用下的动力系统。该项目的中心目标是研究受粗糙噪声干扰的偏微分方程；这些方程为广泛的现象提供了数学模型，包括界面生长、流体动力学中的湍流和聚合物结构。该研究将调查，除其他主题外，解决方案的间歇性和混沌性质，这与物理系统的重要特征有关。该项目的第二个目标是扩大随机变分法的应用范围，也称为马利文微积分。Malliavin微积分是将经典变分学从函数扩展到随机过程的一种数学理论。它已被证明是一个强大的工具，在推导收敛速度的中心极限定理，这是在统计推断有很大的相关性。重点将放在具有长记忆的随机过程的分析上，这对于分析来自金融、电信和其他领域的数据很有用。该项目涉及三个主题领域的问题。第一个主题是研究在时间上为白色且具有齐次空间协方差的高斯噪声驱动下的随机热方程。一个具有挑战性的目标是推导出解的泛函的变量变换公式，并研究该公式在一系列问题中的应用，包括大时间渐近性和间歇性性质。并计划进一步发展粗糙噪声和时间无关噪声驱动的随机偏微分方程的分析。第二个主题是关于Malliavin演算在各种开放问题中的应用，包括收敛速率和密度的渐近展开，当目标分布是高斯定律和几何高斯泛函的极限定理的混合物时。第三个主题解决了分数阶布朗运动和相关自相似高斯过程的分析问题，包括矩阵值分数阶布朗运动的特征值动力学和自交局部时间的指数可积性。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。

This research project investigates questions in stochastic analysis, a part of probability theory that studies dynamical systems under the action of random impulses. A central objective of the project is the study of partial differential equations perturbed by rough noises; such equations provide mathematical models for a wide range of phenomena, including interface growth, turbulence in fluid dynamics, and polymer structure. The research will investigate, among other topics, the intermittency and chaotic properties of solutions, which are related to important characteristics of physical systems. A second objective of the project is to broaden the range of applications of the stochastic calculus of variations, also called Malliavin calculus. Malliavin calculus is a mathematical theory that extends the classical calculus of variations from functions to stochastic processes. It has proved to be a powerful tool in deriving rates of convergence in central limit theorems, which are of great relevance in statistical inference. Emphasis will be placed on analysis of random processes with long memory, which are useful for analysis of data coming from finance, telecommunications, and other areas.This project addresses questions in three topical areas. The first topic is the study of the stochastic heat equation driven by a Gaussian noise that is white in time and has homogenous spatial covariance. A challenging goal is to derive a change-of-variable formula for functionals of the solution and investigate applications of this formula to a range of questions, including large-time asymptotics and intermittency properties. It is also planned to further develop the analysis of stochastic partial differential equations driven by rough noises and time-independent noises. A second topic concerns applications of Malliavin calculus to a variety of open questions, including rates of convergence and asymptotic expansions of densities, when the target distribution is a mixture of Gaussian laws and limit theorems for geometric Gaussian functionals. A third topic addresses questions in the analysis of fractional Brownian motion and related self-similar Gaussian processes, including the dynamics of eigenvalues of matrix-valued fractional Brownian motions and the exponential integrability of self-intersection local times.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.