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，随机分析是研究随机脉冲作用下的动力系统的概率理论的一部分。一个中心目标是分析受随机噪声干扰的随机偏微分方程组，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，如界面生长模型、流体动力学中的湍流和聚合物模型。拟议的研究将集中在空间平均的遍历性和随机波动，这与特定物理模型中的观测特征有关。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Malliavin微积分。Malliavin演算是一种数学理论，它将经典的变分法从函数扩展到随机过程。它已被证明是一个强大的工具来推导中心极限定理的收敛速度，这些定理在统计推断中具有重要的相关性。重点将放在具有长记忆的随机过程的分析上，这些随机过程对于处理来自金融、电信和其他领域的数据是有用的。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在于建立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组的空间平均的定量中心极限定理。具有挑战性的问题是由时间上是白的、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的比率是该项目的中心目标。第二个工作块涉及得到分数布朗运动的泛函关于当地时间的渐近行为。将开发一种以克拉克-奥科内公式为基础的创新方法。在第三个工作区，我们计划解决极限问题中随机变分应用中的几个公开问题，包括随机Volterra方程中欧拉近似密度的局部渐近展开和收敛速度。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.