Excursion Probabilities of Isotropic and Locally Isotropic Gaussian Random Fields on Manifolds

TL;DR

This paper derives asymptotic approximations for the probability that a Gaussian random field on a Riemannian manifold exceeds a high threshold, extending known Euclidean results to more general manifolds.

Contribution

It generalizes excursion probability approximations for Gaussian fields from Euclidean spaces to arbitrary Riemannian manifolds, including both smooth isotropic and non-smooth locally isotropic cases.

Findings

Explicit Euler characteristic approximation for smooth isotropic fields.

Asymptotic similarity to Pickands' approximation for non-smooth locally isotropic fields.

Extension of previous results from spheres to general Riemannian manifolds.

Abstract

Let $X= \{X(p), p\in M\}$ be a centered Gaussian random field, where $M$ is a smooth Riemannian manifold. For a suitable compact subset $D\subset M$, we obtain the approximations to excursion probability $\mathbb{P}\{\sup_{p\in D} X(p) \ge u \}$, as $u\to \infty$, for two cases: (i) $X$ is smooth and isotropic; (ii) $X$ is non-smooth and locally isotropic. For case (i), the expected Euler characteristic approximation is formulated explicitly; while for case (ii), it is shown that the asymptotics is similar to Pickands' approximation on Euclidean space which involves Pickands' constant and the volume of $D$. These extend the results in \citep{Cheng:2014} from sphere to general Riemannian manifolds.