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 $(1+d)$-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 $\overline{\alpha }\in (0,1]$.

从测量量子计算和存储系统的性能以及纳米流变学在不同形状的设备上的动机，我们引入了一个反射时空高斯随机场 (RGRF)$(1+d)$-参数紧凑黎曼流形来模拟量子粒子运动动力学。研究 RGRF 的主要任务是估计其偏移概率的渐近上界，该上界渐近收敛于零。为此，我们通过 Skorohod 映射在 RGRF 与其净输入高斯随机场 (GRF) 之间建立关系。然后，作为中间步骤，我们通过在平面流形和一般紧致黎曼流形之间构建面向图表的同胚映射来近似 GRF 的偏移概率。净输入 GRF 可以是各向同性的或各向异性的。还介绍了平面流形上各向异性 GRF 的案例研究，例如归一化分数布朗片 (FBS) 和球体上的各向异性 GRF，例如归一化各向异性分数球形布朗运动 (FSBM)，α -FSBM 被引入并证明了它的存在性对于一些常数$\overline{\alpha }\in (0,1]$.