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April 2003 Euler characteristics for Gaussian fields on manifolds
Jonathan E. Taylor, Robert J. Adler
Ann. Probab. 31(2): 533-563 (April 2003). DOI: 10.1214/aop/1048516527

Abstract

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, $f^{-1}[u, +\infty)$, and their geometric properties. Specifically, we derive the expected Euler characteristic $\Ee[\chi(f^{-1}[u, +\infty))]$ of an excursion set of a smooth Gaussian random field. Part of the motivation for this comes from the fact that $\Ee[\chi(f^{-1}[u,+\infty))]$ 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 $\Pp[ \sup_{p \in M} f(p) \geq u ]$. Our proofs rely on results from random fields on $\Rr^n$ as well as differential and Riemannian geometry.

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Jonathan E. Taylor. Robert J. Adler. "Euler characteristics for Gaussian fields on manifolds." Ann. Probab. 31 (2) 533 - 563, April 2003. https://doi.org/10.1214/aop/1048516527

Information

Published: April 2003
First available in Project Euclid: 24 March 2003

zbMATH: 1026.60039
MathSciNet: MR1964940
Digital Object Identifier: 10.1214/aop/1048516527

Subjects:
Primary: 53A17 , 58A05 , 60G15 , 60G60
Secondary: 60G17 , 60G70 , 62M40

Keywords: Euler characteristic , Excursions , Gaussian processes , Manifolds , Random fields , Riemannian geometry

Rights: Copyright © 2003 Institute of Mathematical Statistics

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Vol.31 • No. 2 • April 2003
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