论文标题
随机的黎曼几何学的发现之旅
A Discovery Tour in Random Riemannian Geometry
论文作者
论文摘要
我们通过所谓的分数高斯领域来研究riemannian歧管$（\ mathsf {m}，\ mathsf {g}）$的随机扰动。字段$ h^\ bullet：ω\ mapsto h^ω$将通过共形转换$ \ Mathsf {g} \ mapsto \ Mathsf {g}^ω\ colon \！我们的重点将放在常规案例上，赫斯特参数$ h> 0 $，著名的liouville几何形状在二维为边界。我们想了解基本的几何和功能分析量，例如直径，体积，热内核，布朗运动，光谱结合或光谱差距在噪声的影响下会发生变化。如果是这样，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的里曼尼亚语歧管的分数高斯领域，这是一个引人入胜的独立兴趣的对象。
We study random perturbations of Riemannian manifolds $(\mathsf{M},\mathsf{g})$ by means of so-called Fractional Gaussian Fields, which are defined intrinsically by the given manifold. The fields $h^\bullet: ω\mapsto h^ω$ will act on the manifolds via conformal transformation $\mathsf{g}\mapsto \mathsf{g}^ω\colon\!\!= e^{2h^ω}\,\mathsf{g}$. Our focus will be on the regular case with Hurst parameter $H>0$, the celebrated Liouville geometry in two dimensions being borderline. We want to understand how basic geometric and functional analytic quantities like diameter, volume, heat kernel, Brownian motion, spectral bound, or spectral gap will 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.
