黎曼流形上反射高斯场与漂移概率论文介绍
- 论文题目
- 英文摘要
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 alpha-FSBM is introduced and its existence is proved for some constant bar-alpha in (0,1].
- 中文摘要
由于受到不同形状装置上量子计算与存储系统及纳米流变学的启迪，我引进了一种拥有(1+d)时空参数的一般紧黎曼流形上反射高斯场来建立量子
移动动力学的模型。研究该高维反射随机场的主要目的在于估计其漂移概率的一个渐近上界并且该上界也同时渐近地趋于零。为了达到这样研究的目
标，我们首先通过Skorohod映射在该反射高斯场与其净输入高斯场之间建立起了一个关联。然后，作为证明过程中的一个中间步骤，我们通过在平流
形与一般紧黎曼流形之间建立一个基于图的同胚影射来逼近那个高斯场的漂移概率。所涉及净输入高斯场可以是各向同性的也可以是各向异性的。进
而，我们研究了关于平流形上各向异性高斯场（比如规范化分数布朗单）和球面上各项异性高斯场（比如规范化各向异性分数球面布朗运动）的多个
案例。在此过程中，我引进了具有常参数的分数布朗运动且在给定介于0和1之间某常数的条件下证明了其存在性。
- 关键词与关键技术
- Reflecting time-space Gaussian random-field; Riemannian manifold; excursion probability; anisotropic fractional spherical Brownian
motion; quantum computing and storage; Pickands' constant
- 黎曼流形上的反射高斯场本质上是黎曼流形上排队系统扩散逼近的极限形式，它是量子计算机存储处理球面流形上排队系统（见戴万阳下列相关
论文）的直接发展形式，在其它不同领域也有潜在的应用。用量子物理的通俗语言来讲就是 "薛定鄂猫在黎曼流形上跑来跑去，同时还撞来撞去"。
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