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

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.