Stochastic Calculus of Variations and Limit Theorems

这个项目的目标是研究随机分析中的各种问题，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。一个中心目标是分析随机偏微分方程，如热和波方程，受随机噪声的干扰。这些方程提供了广泛领域的数学模型，如界面生长模型、流体动力学中的湍流和聚合物模型。拟议的研究将侧重于空间平均的遍历性和随机波动，这与特定物理模式中观测到的特征有关。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，在导出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。将特别强调对具有长记忆的随机过程的分析，这对处理来自金融、电信和其他领域的数据很有用。本项目为研究生提供研究训练机会。该项目的第一个工作模块包括建立由具有齐次协方差的高斯噪声驱动的广泛类别的随机偏微分方程的空间平均的定量中心极限定理。具有挑战性的问题是由白噪声驱动的三维波动方程它在时间上是白的并且在空间上有Riesz协方差，还有比白噪声更粗糙的噪声。利用马利文演算技术建立概率密度的比率是该项目的中心目标。第二个工作块是推导分数布朗运动泛函与局部时间的渐近行为。将开发一种基于克拉克-奥康公式的创新方法。在第三个工作块中，我们计划解决在极限问题中随机变分法应用中的几个开放问题，包括密度的局部渐近展开和随机Volterra方程中欧拉近似的收敛速度。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。

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.