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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=613857
• 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.

无穷维概率测度已经在各种背景下被引入和研究，从扩散过程和扩散方程的经典理论，到最近的几何和统计物理中的概率模型。本研究计划旨在探讨这些背景下产生的一些问题。 高斯自由场（Gaussian free fields，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、共形几何等领域的发展中起着重要的作用。本文将继续研究高维GFF的几何性质，并对已有的对数相关GFF（即，GFF的协方差在空间变量中具有对数奇异性），包括理论和技术，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，经典理论与随机模型的比较产生了一些有趣的问题;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）理论提供了GFF的严格数学解释，它涉及一般无限维高斯测度的构造和性质。我将通过研究AWS上各种函数方程的测度理论扩展来回顾AWS的结构性质。我希望系统地识别AWS上表征齐次混沌的函数方程的概率适应性。潜在的发现可以揭示AWS的精细结构特征，提供高斯测度的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是经典维纳空间的抽象，布朗运动的空间，长期以来一直是研究扩散过程的基础，更一般地说，是偏微分方程中的概率方法。Wright-Fisher（WF）方程是在单纯形区域边界退化的扩散方程，常用于群体遗传学中对某些等位基因的流行率进行建模。我计划使用概率方法来研究高维WF方程，旨在提供边界/角附近的基本解的严格估计。这些估计可能是有用的，在模拟的解决方案在实践中。这项工作也将导致一般的边界/角落的退化对规则性的基本解决方案在更高的维度上的影响进行研究。