Stochastic Analysis and Asymptotic Problems

本研究课题探讨随机分析中的问题，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。该项目的一个中心目标是研究受粗糙噪声扰动的偏微分方程;这些方程为各种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。 该研究将研究与物理系统的重要特征相关的解的非线性和混沌特性等主题。 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Malliavin演算。 Malliavin演算是将经典变分法从函数扩展到随机过程的数学理论。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，这是非常相关的统计推断。本研究的重点是对金融、电信等领域的数据分析有用的长记忆随机过程的分析。 第一个主题是研究由时间上为白色且空间上具有齐次协方差的高斯噪声驱动的随机热方程。 一个具有挑战性的目标是推导出一个变量的变化公式的解决方案的泛函和调查应用这个公式的一系列问题，包括大时间的渐近性和不确定性。还计划进一步发展对粗糙噪声和与时间无关的噪声驱动的随机偏微分方程的分析。 第二个主题涉及应用Malliavin演算的各种开放的问题，包括收敛速度和渐近展开的密度，当目标分布是一个混合物的高斯法律和极限定理的几何高斯泛函。第三个主题解决分数布朗运动和相关的自相似高斯过程的分析问题，包括矩阵值分数布朗运动的特征值动力学和自相交局部时间的指数可积性。该奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.