A Discovery Tour in Random Riemannian Geometry

IF 1 3区数学 Q1 MATHEMATICS

Potential Analysis Pub Date : 2024-01-26 DOI:10.1007/s11118-023-10118-0

引用次数: 0

Abstract

We study random perturbations of a Riemannian manifold \((\textsf{M},\textsf{g})\) by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields \(h^\bullet : \omega \mapsto h^\omega \) will act on the manifold via the conformal transformation \(\textsf{g}\mapsto \textsf{g}^\omega := e^{2h^\omega }\,\textsf{g}\) . Our focus will be on the regular case with Hurst parameter \(H>0\) , the critical case \(H=0\) being the celebrated Liouville geometry in two dimensions. We want to understand how basic geometric and functional-analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap change under the influence of the noise. And if so, is it possible to quantify these dependencies in terms of key parameters of the noise? Another goal is to define and analyze in detail the Fractional Gaussian Fields on a general Riemannian manifold, a fascinating object of independent interest.

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随机黎曼几何探索之旅

摘要我们通过所谓的分数高斯场（Fractional Gaussian Fields）来研究黎曼流形 \((\textsf{M},\textsf{g})\)的随机扰动，这些场是由给定流形内在定义的。场（h^/bullet : \omega \mapsto h^\omega \）将通过保角变换作用于流形（\textsf{g}\mapsto \textsf{g}^\omega := e^{2h^\omega }\,\textsf{g}\) 。我们的重点是具有赫斯特参数（H>0\）的正则情况，临界情况（H=0\）是二维中著名的柳维尔几何。我们想了解直径、体积、热核、布朗运动、频谱约束或频谱间隙等基本几何和函数分析量在噪声影响下是如何变化的。如果是这样，是否有可能根据噪声的关键参数对这些依赖性进行量化？另一个目标是详细定义和分析一般黎曼流形上的分数高斯场，这是一个令人着迷的独立兴趣对象。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Potential Analysis 数学-数学

CiteScore

2.20

自引率

9.10%

发文量

审稿时长

>12 weeks

期刊介绍： The journal publishes original papers dealing with potential theory and its applications, probability theory, geometry and functional analysis and in particular estimations of the solutions of elliptic and parabolic equations; analysis of semi-groups, resolvent kernels, harmonic spaces and Dirichlet forms; Markov processes, Markov kernels, stochastic differential equations, diffusion processes and Levy processes; analysis of diffusions, heat kernels and resolvent kernels on fractals; infinite dimensional analysis, Gaussian analysis, analysis of infinite particle systems, of interacting particle systems, of Gibbs measures, of path and loop spaces; connections with global geometry, linear and non-linear analysis on Riemannian manifolds, Lie groups, graphs, and other geometric structures; non-linear or semilinear generalizations of elliptic or parabolic equations and operators; harmonic analysis, ergodic theory, dynamical systems; boundary value problems, Martin boundaries, Poisson boundaries, etc.