Stochastic Calculus of Variations and Stochastic Analysis with Fractal Noises

该项目旨在在三个不同的随机分析主题中建立新的结果。首先，提出了一种估计抛物型线性随机偏微分方程解的负矩的新方法。这些估计将使我们能够利用马利万微积分的技术推导出在有限个数点处解的概率律的规律性。第二个目标是进一步发展关于分数布朗运动和相关过程的随机演算，同时使用Malliavin演算和路径技术。我们的第三个目标是为高斯过程的不同类型泛函的渐近行为建立混沌中心极限定理。这些泛函的例子包括分数阶布朗运动的自交局部时间，以及随机积分的幂变函数和相关泛函。随机分析是一个现代数学领域，旨在研究受随机噪声干扰的常微分方程和偏微分方程。这些方程在物理学和经济学的许多领域中作为模型发挥着核心作用。为了推导出解的重要性质，比如计算解在某个区间取值的概率，我们需要应用合适的数学技术，比如伊藤微积分和马利文微积分。我们的目标是为这些技术的发展及其在随机偏微分方程中的应用做出实质性的贡献。另一方面，虽然经典输入噪声具有独立的增量，但由于在水文，电信和数学金融中的一些应用，最近人们对具有长记忆特性的输入噪声（如分数布朗运动）感兴趣。关于这些长记忆过程的随机演算的发展也是这个项目的目标之一。

The project aims to establish new results in three diffeent topics of stochastic analysis. First, a new approach for estimating the negative moments of the solutions to linear stochastic partial differential equations of parabolic type will be developed. These estimates will allow us to derive the regularity of the probability law of the solution at a finite number of points using the techniques of Malliavin Calculus. A second objective is to further develop the stochastic calculus with respect to the fractional Brownian motion and related processes using both Malliavin Calculus and path-wise techniques. Our third goal is to establish chaotical central limit theorems for the asymptotic behavior of different types of functionals of a Gaussian process. Examples of these functionals include the self-intersection local time of the fractional Brownian motion, and power variation and related functionals of stochastic integrals.Stochastic analysis is a modern area in mathematics which aims to study ordinary and partial differential equations perturbed by a random noise. These equations play a central role as models in many areas in physics and economics. In order to derive important properties of the solutions, like to compute the probability that the solution takes values in some interval, one needs to apply suitable mathematical techniques like the Ito Calculus and the Malliavin Calculus. We aim to make substantial contributions to the development of these thecniques and their applications to stochastic partial differential equations. On the other hand, while the classical input noise used has independent increments, motivated by some applications in hydrology, telecommunications and mathematical finance, there has been a recent interest in input noises possessing a long memory property like fractional Brownian motion. The development of a stochastic calculus with respect to these long memory processes is also one of the aims of this project.