Probability measures in infinite dimensional spaces: random paths, random fields and random geometry

无限维空间中的概率度量：随机路径、随机场和随机几何

• 批准号: RGPIN-2015-05968
• 负责人: Chen, Linan
• 金额: $ 1.17万
• 依托单位: McGill University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=708729
• 关键词: Probability; measures; infinite; dimensional; spaces

Infinite dimensional probability measures have been introduced and studied in various contexts, from the classical theory of diffusion processes and diffusion equations, to the more recent probabilistic models in geometry and statistical physics. This research program aims to explore some problems arising from these contexts. Gaussian free fields (GFF), which can be viewed as certain function (or distribution) spaces equipped with Gaussian measures, have played important roles in recent developments of various fields, such as the quantum field theory, statistical mechanics, conformal geometry and etc. I will continue studying geometrical properties of GFF in higher dimensions, and extend part of the existing literature on log-correlated GFF (i.e., the covariance of the GFF has log singularity in spatial variables), including both the theories and the techniques, to a more general class of Gaussian random fields. I will analyze various aspects of the geometry associated with these random fields, and explore potential applications of random geometry models in other subjects. On one hand, there are interesting problems arising from the comparison between the classical theory and its counterpart under the randomization; on the other hand, the study of random models provides new perspectives on problems in the classical setting. A rigorous mathematical interpretation of GFF is provided by the theory of Abstract Wiener Space (AWS), which concerns constructions and properties of general infinite dimensional Gaussian measures. I will review the structural properties of AWS through studying measure theoretical extensions of various functional equations on AWS. I hope to systematically identify the probabilistic adaptations of the functional equations which characterize homogeneous chaos on AWS. Potential findings could shed light on the fine structural features of AWS, provide further details of possible supporting spaces of the Gaussian measures, and reflect the distinct natures of the infinite dimensional Gaussian measures from those in finite dimensions. The AWS is the abstraction of the classical Wiener space, the space of the Brownian motion, which has long been the basis of the study of diffusion processes and more generally, probabilistic methods in partial differential equations. The Wright-Fisher (WF) equation - a diffusion equation degenerate at the boundary of a simplex domain - is commonly used in the population genetics to model the prevalence of certain alleles. I plan to use a probabilistic method to study the higher-dimensional WF equation, aiming at providing rigorous estimates for the fundamental solution near the boundary/corners. These estimates could be useful in simulating the solutions in practice. This work will also lead to studying in general the impact of the degeneracy at the boundary/corners on the regularity of the fundamental solution in higher dimensions.

从经典的扩散过程理论和扩散方程，到最近的几何和统计物理中的概率模型，无限维概率度量已经在不同的背景下被引入和研究。本研究计划旨在探讨这些背景下出现的一些问题。 高斯自由场(GFF)可以看作是具有高斯测度的函数(或分布)空间，在量子场论、统计力学、保角几何等各个领域的最新发展中扮演着重要的角色。我将继续研究GFF的高维几何性质，并将现有关于对数相关GFF(即GFF的协方差在空间变量中具有对数奇异性)的部分文献扩展到更一般的一类高斯随机场。我将分析与这些随机场相关的几何学的各个方面，并探索随机几何模型在其他学科中的潜在应用。一方面，在随机化下经典理论与经典理论的比较中出现了一些有趣的问题；另一方面，随机模型的研究为经典背景下的问题提供了新的视角。 抽象维纳空间(AWS)理论给出了GFF的严格数学解释，它涉及到一般无限维高斯测度的构造和性质。我将通过研究AWS上各种函数方程的度量理论扩展来回顾AWS的结构性质。我希望系统地确定描述AWS上齐次混沌的函数方程的概率适应性。潜在的发现可能会揭示AWS的精细结构特征，提供高斯测度可能支持空间的进一步细节，并反映无限维高斯测度与有限维高斯测度的不同性质。 AWS是经典的Wiener空间--布朗运动空间的抽象，长期以来一直是研究扩散过程的基础，更广泛地说，是偏微分方程中的概率方法。Wright-Fisher(WF)方程是一种在单纯形域边界退化的扩散方程，在群体遗传学中常用于模拟某些等位基因的流行情况。我计划使用概率方法来研究高维WF方程，目的是为边界/角点附近的基本解提供严格的估计。这些估计可能有助于在实践中模拟解决方案。这项工作还将导致一般地研究边界/角点处的简并性对高维基本解的正则性的影响。