随机Lipschitz-Killing曲率:约简原理,分部积分和Wiener混沌

IF 0.4 Q4 STATISTICS & PROBABILITY
Anna Vidotto
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引用次数: 2

摘要

本文收集了二维标准平面(算术随机波)和二维单位球(随机球谐波)上随机特征函数漂移集的Lipschitz-Killing曲率的一些最新结果。特别地,本调查的目的是强调分部积分公式的关键作用,以便对随机LKCs有一个非常整洁的表达式。实际上,研究流形上随机波的局部几何泛函的主要工具是利用它们的维纳混沌分解,并表明(通常)在所谓的高能极限下,一个单一的混沌分量支配着它们的行为。此外,约简原理表明,在阈值水平u≠0 u\ ne0处随机波漂移集LKCs的优势Wiener混沌分量与h2 (f) H_2(f)的积分成正比,其中f为感兴趣的随机场,h2 H_2为第二个Hermite多项式。这将通过分部积分公式来展示。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Random Lipschitz–Killing curvatures: Reduction Principles, Integration by Parts and Wiener chaos
In this survey we collect some recent results regarding the Lipschitz–Killing curvatures (LKCs) of the excursion sets of random eigenfunctions on the two-dimensional standard flat torus (arithmetic random waves) and on the two-dimensional unit sphere (random spherical harmonics). In particular, the aim of the present survey is to highlight the key role of integration by parts formulae in order to have an extremely neat expression for the random LKCs. Indeed, the main tool to study local geometric functionals of random waves on manifold is to exploit their Wiener chaos decomposition and show that (often), in the so-called high-energy limit, a single chaotic component dominates their behavior. Moreover, reduction principles show that the dominant Wiener chaotic component of LKCs of random waves’ excursion sets at threshold level u ≠ 0 u\ne 0 is proportional to the integral of H 2 ( f ) H_2(f) , f f being the random field of interest and H 2 H_2 the second Hermite polynomial. This will be shown via integration by parts formulae.
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来源期刊
CiteScore
1.30
自引率
0.00%
发文量
22
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