Mathematical Sciences: Intrinsic Stochastic Analysis on Path and Loop Spaces

PI将研究黎曼流形上的路径和回路空间的分析和几何性质。分析是基于这些空间上的维纳测度，它在有限维流形的分析中起着类似于勒贝格测度的作用。由于Wiener测度引起了布朗运动和布朗桥，概率方法（随机微分方程，扩散理论等）将被广泛用于我们的分析。路径和回路空间的几何和分析性质将通过所谓的Ornstein-Uhlenbeck过程的概率性质来研究。 PI将在各种几何设置（主要是有边界的流形）中发现梯度算子的分部积分公式的推广，并计算熟悉的几何对象，如路径和回路空间（如希尔伯特流形）的挠率和曲率张量，这些几何对象涉及基本黎曼流形的通常挠率和曲率张量的随机积分。长期的目标是开发一个内在的，几何Malliavin演算和调查hypercontractivity，对数Sobolev不等式，迈耶的等价性在我们的新的几何设置和他们的相互作用与黎曼结构的基础流形。 跨学科研究是当今科学研究的趋势。概率论是数学的一个分支，它研究集体现象的随机行为。 在过去的二十年中，概率论已经成功地应用于经典数学问题，如偏微分方程和几何。 这种新的概率观点不仅刺激了这些经典学科的研究，而且开辟了新的研究途径，如随机微分几何和扩散理论。PI将使用概率方法来研究一类重要的几何对象的性质，称为循环空间（例如，球体上的闭合路径的集合），这在现代物理学中刚刚开始获得重要性。他将展示基空间的曲率（在上述情况下的球体）如何影响与循环空间相关的某些随机过程的行为，并对空间几何与潜在随机过程之间的相互作用给出定性和定量的描述。

The PI will investigate analytical and geometrical properties of the path and loop spaces over a Riemannian manifold. The analysis is based on the Wiener measure on these spaces, which plays a similar role as the Lebesgue measure in the analysis of finite dimensional manifolds. Since the Wiener measure gives rise to Brownian motion and Brownian bridge, the probabilistic methods (stochastic differential equations, diffusion theory, etc.) will be used extensively in our analysis. The geometric and analytical properties of path and loop spaces will be studied through the probabilistic properties of the so-called Ornstein-Uhlenbeck process. The PI will discover generalizations of integration by parts formula for the gradient operator in various geometric settings (mainly for manifolds with boundary) and compute the familiar geometric objects such as torsion and curvature tensors of the path and loop spaces (as Hilbert manifolds) in terms of stochastic integrals involving the usual torsion and curvature tensors of the underlying Riemannian manifold. The long-term goal is to develop an intrinsic, geometric Malliavin calculus and to investigate hypercontractivity, logarithmic Sobolev inequality, and Meyer's equivalence in our new geometric setting and their interaction with the Riemannian structure of the base manifold. Interdisciplinary research is the current trend of scientific research. Probability theory is a branch of mathematics which studies random behavior of collective phenomena. In the last two decades probability theory has been applied with great success to problems from classical mathematical subjects such as partial differential equations and geometry. This new probabilistic point of view not only stimulated research in these classical subjects but also opened new avenues of research such as stochastic differential geometry and diffusion theory. The PI will use probabilistic methods to study properties of an important class of geometric objects called loop spaces (for example, the collection of closed paths on a sphere), which just began to gain importance in modern physics. He will show how the curvature of the base space (the sphere in the above case) affects the behavior of certain random processes associated with loop spaces and give both qualitative and quantitative descriptions of the interaction between the geometry of the space and the underlying random processes.