The Annals of Probability

Euler characteristics for Gaussian fields on manifolds

Jonathan E. Taylor and Robert J. Adler

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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.

Article information

Ann. Probab., Volume 31, Number 2 (2003), 533-563.

First available in Project Euclid: 24 March 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G60: Random fields 53A17: Kinematics 58A05: Differentiable manifolds, foundations
Secondary: 60G17: Sample path properties 62M40: Random fields; image analysis 60G70: Extreme value theory; extremal processes

Random fields Gaussian processes manifolds Euler characteristic excursions Riemannian geometry


Taylor, Jonathan E.; Adler, Robert J. Euler characteristics for Gaussian fields on manifolds. Ann. Probab. 31 (2003), no. 2, 533--563. doi:10.1214/aop/1048516527.

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