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.

中文翻译：

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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来研究黎曼流形\((\textsf{M},\textsf{g})\)的随机扰动，分数高斯场本质上由给定流形定义。场\(h^\bullet : \omega \mapsto h^\omega \)将通过保形变换作用于流形\(\textsf{g}\mapsto \textsf{g}^\omega := e^{ 2h^\omega}\,\textsf{g}\)。我们的重点将放在赫斯特参数\(H>0\)的常规情况上，关键情况 \(H=0\)是著名的二维刘维尔几何。我们想要了解直径、体积、热核、布朗运动、光谱界限或光谱间隙等基本几何和功能分析量在噪声的影响下如何变化。如果是这样，是否可以根据噪声的关键参数来量化这些依赖性？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。