Stochastic Calculus of Variations and Limit Theorems

该项目的目标是研究随机分析中的各种问题，这是概率论的一部分，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，如热和波动方程，随机噪声扰动。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的应用范围，也称为Malliavin演算。Malliavin演算是一种数学理论，它将经典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，在推导收敛速度的中心极限定理，这是非常相关的统计推断。特别强调将放在随机过程的分析与长记忆，这是有用的，以处理来自金融，电信和其他领域的数据。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建立定量中心极限定理的空间平均值的一类随机偏微分方程驱动的高斯噪声，具有齐次协方差。随机问题是由时间上为白色且空间上具有Riesz协方差的噪声驱动的三维波动方程的情况，以及噪声比白色噪声更粗糙的情况。使用Malliavin演算技术建立概率密度率是该项目的中心目标。第二个工作块处理导出局部时间相关的分数布朗运动的泛函的渐近行为。将根据克拉克-奥康公式制定一种创新方法。在第三个工作块中，我们计划解决几个开放的问题，在应用中的随机变分法的极限问题，包括局部渐近展开的密度和速度的收敛欧拉近似随机沃尔泰拉equations.This award反映了NSF的法定使命，并已被认为是值得支持通过评估使用基金会的智力价值和更广泛的影响审查标准。

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.