Stochastic Processes in non-Euclidean spaces

该奖项是根据2009年《美国复苏和再投资法案》(公法111-5)提供资金的。本文对有限维黎曼流形上的布朗运动进行了深入的研究，并深入了解了Laplace-Beltrami算子与空间几何之间的深层联系，以及该空间上的布朗运动及其热核测度的性质。本文致力于研究这一范式的某些推广，包括研究次黎曼流形和某些无限维李群上的扩散，李群上的Levy过程，以及‘树空间’上的布朗运动。‘树空间’是一个具有几何和组合结构的连续空间，具有生物学应用。这些研究涉及分析、几何和概率的交叉，随机微分方程解及其生成元的研究是许多所考虑问题的统一框架。概率为分析和几何提供了强大的工具，而随机过程为许多物理和生物现象提供了易于处理的模型。例如，布朗运动提供了一种理解空间上热流的方法。PI将研究在某些物理或生物应用中自然发生的空间上的几个随机过程的性质。次黎曼流形出现在经典和量子力学中，某些几何量在这个背景下得到了最好的理解。无限维空间出现在量子场论和弦理论中的物理学中。征税过程最近一直是密集研究的主题，部分原因是在金融领域的新应用。“树空间”模拟了所有系统发育树的空间。在这一授予期内的研究应该涉及不同的数学学科，如调和分析、泛函分析和数学物理，并应在其他科学领域找到应用，如物理和生物学。

This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). Brownian motion on finite dimensional Riemannian manifolds is well studied, and the deep relationship between the Laplace-Beltrami operator and the geometry of a space and the properties of Brownian motion and its heat kernel measure on that space are well understood.This proposal is devoted to the study of certain generalizations of this paradigm, including the study of diffusions on sub-Riemannian manifolds and certain infinite dimensional Lie groups, Levy processes on Lie groups, and Brownian motion on ``tree space,'' a continuous space with geometric and combinatorial structures which has biological applications. These studies lie in the intersection of analysis, geometry, and probability, and the study of solutions to stochastic differential equations and their generators is a uniting framework of many of the problems considered.Probability provides a powerful tool in analysis and geometry, and stochastic processes give tractable models for many physical and biological phenomena. For example, Brownian motion gives a way of understanding heat flow on a space. The PI will investigate properties of several stochastic processes on spaces which occur naturally in some physical or biological applications. Sub-Riemannian manifolds arise in classical and quantum mechanics, and certain geometric quantities are best understood in this setting.Infinite dimensional spaces appear in physics in quantum field theory and string theory. Levy processes have recently been a subject of intense research, due in part to new applications in finance. ``Tree space'' models the space of all phylogenetic trees.The research during this grant period should have implications in various mathematical disciplines, such as harmonic analysis, functional analysis, and mathematical physics, and should find applications in other scientific fields, such as physics and biology.