Reflecting time-Space Gaussian random field on compact Riemannian manifold and excursion probability

Abstract

Motivated from measuring the performance of quantum computing and storage systems together with nanorheology over different shapes of devices, we introduce a reflecting time-space Gaussian random field (RGRF) on a general -parameter compact Riemannian manifold to model the quantum particle movement dynamics. The main task in studying the RGRF is to estimate an asymptotic upper bound of its excursion probability, which asymptotically converges to zero. In doing so, we establish a relationship between the RGRF and its netput Gaussian random field (GRF) via a Skorohod mapping. Then, as an intermediate step, we approximate the excursion probability of the GRF by constructing a chart oriented homeomorphism mapping between a flat manifold and a general compact Riemannian manifold. The netput GRF can be either isotropic or anisotropic. Case studies in terms of anisotropic GRFs over a flat manifold such as normalized fractional Brownian sheets (FBSs) and those over a sphere such as normalized anisotropic fractional spherical Brownian motions (FSBMs) are also presented, where an α-FSBM is introduced and its existence is proved for some constant .