Euler characteristics for Gaussian fields on manifolds

• 成果类型: Article
• 署名作者: Taylor, JE; Adler, RJ
• 署名单位: Stanford University; Technion Israel Institute of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• 发表日期: 2003
• 页码: 533-563
• 关键词: excursion sets; geometry; maxima; tube

摘要

We are interested in the geometric properties of real-valued Gaussian random fields defined on manifolds. Our manifolds, M, are of class C-3 and the random fields f are smooth. Our interest in these fields focuses on their excursion sets, f(-1)[u, + infinity), and their geometric properties. Specifically, we derive the expected Euler characteristic E[chi(f(-1)[u, + infinity))] of an excursion set of a smooth Gaussian random field. Part of the motivation for this comes from the fact that E[chi(f(-1)[u, + infinity))] relates global properties of M to a geometry related to the covariance structure of f. Of further interest is the relation between the expected Euler characteristic of an excursion set above a level u and P[sup(pis an element ofM) f (p) greater than or equal to u]. Our proofs rely on results from random fields on R-n as well as differential and Riemannian geometry.