Stochastic Calculus of Variations and Limit Theorems

随机变分和极限定理

• 批准号: 2054735
• 负责人: Mathew Johnson
• 金额: $ 27.24万
• 依托单位: University of Kansas Center for Research Inc
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-08-01 至 2024-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054735&HistoricalAwards=false
• 关键词: Stochastic; Calculus; Variations; Limit; Theorems

The goal of this project is to investigate a variety of problems in stochastic analysis, which is a part of probability theory that studies dynamical systems under the action of random impulses. A central objective is the analysis of stochastic partial differential equations, such as the heat and wave equations, perturbed by random noises. These equations provide mathematical models in a wide range of areas, such as growth models for interfaces, turbulence in fluid dynamics and polymer models. The proposed research will focus on the ergodicity and random fluctuations of spatial averages, which are related to observed characteristics in particular physical models. A second objective of the project is to broaden the range of applications of the stochastic calculus of variations, also called Malliavin calculus. The Malliavin calculus is a mathematical theory that extends the classical calculus of variations from functions to stochastic processes. It has proven to be a powerful tool in deriving rates of convergence in central limit theorems, which are of great relevance in statistical inference. Particular emphasis will be put in the analysis of random processes with long memory which are useful to handle data coming from finance, telecommunications and other areas. The project provides research training opportunities for graduate students. A first working block of the project consists in establishing quantitative central limit theorems for spatial averages of a wide class of stochastic partial differential equations driven by a Gaussian noise which has homogeneous covariance. Challenging problems are the case of the three dimensional wave equation driven by a noise which is white in time and it has a Riesz covariance in space, and also the case of noises which are rougher that the white noise. Establishing the rate for probability densities using techniques of Malliavin calculus is a central goal of the project. A second working block deals with deriving the asymptotic behavior of functionals of the fractional Brownian motion related to local times. An innovative methodology based on the Clark-Ocone formula will be developed. In a third working block we plan to address several open problems in the applications of the stochastic calculus of variation in limit problems including local asymptotic expansions of densities and rates of convergence for Euler approximations in stochastic Volterra equations.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.

该项目的目标是研究随机分析中的各种问题，随机分析是研究随机脉冲作用下动力系统的概率论的一部分。中心目标是分析受随机噪声扰动的随机偏微分方程，例如热方程和波动方程。这些方程提供了广泛领域的数学模型，例如界面的增长模型、流体动力学中的湍流和聚合物模型。拟议的研究将重点关注空间平均值的遍历性和随机波动，这与特定物理模型中观察到的特征有关。 该项目的第二个目标是扩大随机变分法（也称为 Malliavin 微积分）的应用范围。马利亚文微积分是一种数学理论，它将经典变分法从函数扩展到随机过程。事实证明，它是推导中心极限定理收敛速度的有力工具，这在统计推断中具有重要意义。将特别强调对具有长记忆力的随机过程的分析，这对于处理来自金融、电信和其他领域的数据非常有用。该项目为研究生提供研究培训机会。该项目的第一个工作块包括为由具有齐次协方差的高斯噪声驱动的一大类随机偏微分方程的空间平均值建立定量中心极限定理。具有挑战性的问题是由时间上为白噪声且空间上具有 Riesz 协方差的噪声驱动的三维波动方程的情况，以及比白噪声更粗糙的噪声的情况。使用 Malliavin 微积分技术建立概率密度率是该项目的中心目标。第二个工作块涉及导出与本地时间相关的分数布朗运动泛函的渐近行为。将开发一种基于克拉克-奥科恩公式的创新方法。在第三个工作块中，我们计划解决极限问题中随机变分计算应用中的几个未决问题，包括随机 Volterra 方程中欧拉近似的密度局部渐近展开和收敛率。该奖项反映了 NSF 的法定使命，并具有通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。