Stochastic Partial Differential Equations, Fractional Noises and Limit Theorems

这个建议涉及随机分析中的不同主题，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。一个基本目标是研究粗糙噪声驱动的随机偏微分方程，粗糙噪声比经典的白色噪声更奇异。这些方程提供的随机场模型在应用数学和数学物理的广泛领域中发挥着重要作用，例如界面的生长模型，流体动力学中的湍流和聚合物模型。拟议的研究将集中在，除其他主题外，在解决方案，这是与特定的物理模型中观察到的特性的不稳定性和混沌特性。 该提案的第二个目标是扩大随机变分法（也称为Malliavin演算）的应用范围。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。 在这个方向上，该建议的目的是建立新的渐近结果，在各种框架，包括数值计划的分数扩散，其中的噪声表现出长程依赖。该方法的第一个工作模块是研究受时间上为白色的高斯噪声扰动的随机热波动方程，该噪声在空间上为分数布朗运动，Hurst参数小于1/2. 噪声的粗糙度在这些方程的分析中产生了新的困难和重要的挑战。我们计划在广泛的方向，包括解的存在性和唯一性，在空间和时间的连续模，粗糙的初始数据，解的密度的规律性，矩估计，尖尾估计，不连续性，混沌行为和渐近分析，当空间或时间参数很大时，发展这样的方程的深入研究。 第二个工作块处理Malliavin演算在各种公开问题中的应用，包括在固定数量的空间点上的一般类随机偏微分方程的解的联合密度的光滑性，分数Bessel过程的分析和矩阵值分数布朗运动的特征值，以及关于分数布朗运动的散度积分的p-变差的计算。研究将集中在Hurst参数小于1/2的粗糙情况下，新的挑战性的困难出现。第三个工作块的建议是建立新的中央极限定理的限制是一个混合的高斯法律和获得速度的收敛和渐近误差分布，使用技术Malliavin演算。特别是渐近问题进行调查，包括近似计划随机微分方程驱动的分数布朗运动，加权q-变化的分数布朗运动和极限定理的交叉当地时间。

This proposal deals with different topics in stochastic analysis, which is a part of probability theory that studies dynamical systems under the action of random impulses. A basic objective is the study of stochastic partial differential equations driven by rough noises, which are more singular than the classical white noise. These equations provide random field models that play a fundamental role in a wide range of areas of applied mathematics and mathematical physics, such as growth models for interfaces, turbulence in fluid dynamics and polymer models. The proposed research will focus, among other topics, on intermittency and chaotic properties of the solutions, which are related to observed characteristics in particular physical models. A second objective of the proposal is to broaden the range of the 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 proved to be a powerful tool in deriving rates of convergence in central limit theorems. In this direction, the proposal aims to establish new asymptotic results in a variety of frameworks including numerical schemes for fractional diffusions, where the noise exhibits long-range dependence. Getting exact rates of convergence in this context is of great relevance in practical applications of these models.A first working block of the proposal is the study of the stochastic heat and wave equations perturbed by a Gaussian noise which is white in time and it is a fractional Brownian motion in space with Hurst parameter less than 1/2. The roughness of the noise creates new difficulties and important challenges in the analysis of these equations. We plan to develop an in-depth study of such equations in a wide range of directions including existence and uniqueness of solutions, modulus of continuity in space and time, rough initial data, regularity of the density of the solution, moment estimates, sharp tail estimates, intermittency properties, chaotic behavior and asymptotic analysis when the space or time parameters are large. A second working block deals with the applications of Malliavin calculus in variety of open problems, including the smoothness of the joint density for the solution to a general class of stochastic partial differential equations at a fixed number of spatial points, the analysis of the fractional Bessel processes and the eigenvalues of a matrix-valued fractional Brownian motion, and the computation of the p-variation of divergence integrals with respect to the fractional Brownian motion. The research will focus in the rough case where the Hurst parameter is less than 1/2, and new challenging difficulties appear. A third working block of the proposal is to establish new central limit theorems where the limit is a mixture of Gaussian laws and to obtain rates of convergence and asymptotic error distributions, using techniques of Malliavin calculus. Particular asymptotic problems to be investigated include approximation schemes for stochastic differential equations driven by a fractional Brownian motion, weighted q-variations of the fractional Brownian motion and limit theorems for intersection local times.