Stochastic Analysis and Asymptotic Problems

随机分析和渐近问题

基本信息

批准号：
1811181
负责人：
David Nualart
金额：
$ 33万
依托单位：
University of Kansas Center for Research Inc
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2022-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1811181&HistoricalAwards=false
关键词：
Stochastic Analysis Asymptotic Problems

项目摘要

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.

该研究项目研究了随机分析中的问题，这是概率理论的一部分，该理论是在随机冲动的作用下研究动态系统的。该项目的一个核心目的是研究偏微分方程的研究被粗糙的噪声扰动。这样的方程为广泛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有关的解决方案的间歇性和混乱性。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扩展到随机过程。事实证明，它是推导中央限制定理中收敛速率的强大工具，在统计推断中具有很大相关性。重点将放在对长期内存的随机过程的分析上，这对于来自金融，电信和其他领域的数据的分析很有用。该项目解决了三个主题领域的问题。第一个主题是对随时间白色且具有同质空间协方差的高斯噪声驱动的随机热方程的研究。一个具有挑战性的目标是为解决方案功能的功能提供可变化的公式，并调查该公式在一系列问题中的应用，包括大型渐进性和间歇性属性。还计划进一步开发由粗糙噪声和时间无关的噪声驱动的随机部分差分方程的分析。第二个主题涉及Malliavin微积分在各种开放问题中的应用，包括收敛速度和密度的渐近扩张，当目标分布是高斯法律的混合体并限制了几何高斯功能的定理。第三个主题在分析布朗尼运动和相关的自相似过程的分析中解决了问题，包括矩阵值分数布朗尼动作的特征值的动态，以及自我交往本地时代的指数整合性。该奖项颁发了NSF的法定任务，并通过评估范围来审查概念和众所周知的知识群体的支持。