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.

该项目的目的是研究随机分析中的各种问题，这是在随机冲动的作用下研究动态系统的概率理论的一部分。一个中心目标是分析随机噪声的随机部分微分方程，例如热和波方程。这些方程式提供了许多领域的数学模型，例如界面的增长模型，流体动力学和聚合物模型中的湍流。拟议的研究将重点介绍空间平均值的终身性和随机波动，这些平均值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变化的随机演算的应用范围，也称为Malliavin conculus。 Malliavin演算是一种数学理论，将变化的经典演算从函数到随机过程扩展。事实证明，它是推导中央限制定理收敛速率的强大工具，在统计推断中具有很大相关性。特别重点将放在对具有较长内存的随机过程的分析中，这对于处理来自金融，电信和其他领域的数据很有用。该项目为研究生提供了研究培训机会。该项目的第一个工作块包括建立定量中央限制定理，以通过具有同质协方差的高斯噪声驱动的一类宽类随机偏微分方程的空间平均值。具有挑战性的问题是三维波方程的情况，该方程是由时间为白色的噪声驱动的，它在太空中具有riesz的协方差，而且噪声的情况比白噪声更粗糙。使用Malliavin微积分技术确定概率密度的速率是该项目的核心目标。第二个工作区块涉及得出与当地时代有关的布朗运动功能的渐近行为。将开发基于Clark-Ocone公式的创新方法。在第三个工作区域中，我们计划在限制问题的随机计算的应用中解决几个开放问题，包括局部渐近扩展的密度差异和随机Volterra方程中Euler近似值的收敛速率。这项奖项反映了NSF的法定任务，并通过使用基金会的Merit和Broadial and Intfactial和Broadia和Broadia and Intfactial和Broadia and Intfactial and Intfactia和Broadia and Intfactia和Broadia and tocria and tocriatial and Broadia and tocria and tocria and tocria and tobleit。