Stochastic Analysis of Gaussian Fractional Noises

本项目旨在建立随机分析的原创成果，开辟新的研究方向。第一个目标是在各种情况下，包括分数阶布朗运动的加性泛函，布朗局部时间增量的空间矩，复分数阶布朗运动的圈数，以及时空高斯速度场中的扩散近似，应用马利文微积分技术推导出新的中心极限定理。在这些问题中，还将研究Kolmogorov和相关距离的收敛速度，以及概率密度的收敛速度。该项目的第二个目标是研究由高斯噪声驱动的随机偏微分方程，该高斯噪声在时间上是白色的，并且在空间变量中具有分数布朗运动的协方差。一个解的存在性和唯一性，它的轨迹和概率分布的规律性，以及弯矩的费曼-卡茨公式，是该项目的具体目标。该项目还旨在利用分数阶微积分技术和分数阶布朗运动的变分性质，建立分数阶布朗运动驱动的随机微分方程的欧拉型数值近似格式的收敛速度。随机分析为研究受随机噪声干扰的常微分方程和偏微分方程提供了合适的数学工具，这些模型在物理学、电信学和经济学的许多领域都是有用的模型。这些方程的应用需要有效的数值近似格式，以及对解的概率分布的方便估计。这个项目将通过发展强大的数学技术，如马利文微积分，在这些主题上取得重大进展。另一方面，该项目将拓宽Malliavin微积分的应用范围，在概率和统计中建立新的基本渐近结果。受电信和数学金融应用的启发，该项目将专注于具有长记忆特性的输入噪声，如分数布朗运动。

The aim of this project is to establish original results and open new research directions in stochastic analysis. A first objective is to derive new central limit theorems applying techniques of Malliavin calculus in a variety of settings including additive functionals of the fractional Brownian motion, spacial moments of the Brownian local time increments, winding number of the complex fractional Brownian motion, and diffusion approximations in a space-time Gaussian velocity field. Rates of convergence of the Kolmogorov and related distances, and convergence of probability densities will be also investigated in these problems. A second objective of the project is to study stochastic partial differential equations driven by a Gaussian noise which is white in time and it has the covariance of a fractional Brownian motion in the space variable. The existence and uniqueness of a solution, the regularity of its trajectories and its probability distribution, and Feynman-Kac formulas for the moments, are concrete goals of the project. The project also aims to establish the rate of convergence of Euler-type numerical approximation schemes for stochastic differential equations driven by a fractional Brownian motion using techniques of fractional calculus and variational properties of the fractional Brownian motion.Stochastic analysis provides suitable mathematical tools to study ordinary and partial differential equations perturbed by a random noise, which are useful models in many areas of physics, telecommunications and economics. The application of these equations requires efficient numerical approximation schemes, and convenient estimates for the probability distribution of the solution. This project will make a significant progress in these topics by developing powerful mathematical techniques like the Malliavin calculus. On the other hand, the project will broaden the range of applications of Malliavin calculus, establishing new fundamental asymptotic results in probability and statistics. Motivated by applications in telecommunications and mathematical finance, the project will focus on input noises possessing long memory property such as the fractional Brownian motion.