Euler characteristics for Gaussian fields on manifolds (Q1394517)

The authors investigate geometric properties of real-valued Gaussian random fields defined on manifolds. These manifolds \(M\) are of class \(C^3\) and the random fields \(f\) are smooth. The interest in these fields focuses on their excursion sets, \(f^{-1}[u,\infty)\), and their geometric properties. Specifically the authors derive the expected Euler characteristic \(E[\chi(f^{-1}[u,\infty))]\) of an excursion set of a smooth Gaussian random field. Part of the motivation for this comes from the fact that \(E[\chi(f^{-1}[u,\infty))]\) relates global properties of \(M\) to a geometry related to the covariance structure of \(f\). The relation between the expected Euler characteristic of an excursion set above a level \(u\) and \(P[\sup_{p\in M}f(p)\geq u]\) is of further interest. The proofs rely on results from random fields on \(R^n\) as well as differential and Riemannian geometry.