Minkowski空间中布朗运动在伪球面上的边缘到达分布

发布时间：2018-02-26 21:11

本文关键词： 布朗运动到达边缘分布 Minkowski空间 Laplace-Beltrami算子 Dirich-let问题变量分离法　出处：《吉林大学》2017年博士论文　论文类型：学位论文

【摘要】：自19世纪英国植物学家Rorbert Brown观察到花粉微粒在液体中做不规则运动以来,关于这个被命名为布朗运动的物理现象的研究就从未停止.先是物理学家A.Einstein与Marian von Smoluchowski等得到了欧氏空间中布朗运动转移概率密度所满足的扩散方程,荷兰科学家Adriaan Daniel Fokker与德国物理学家Max Planck对一般扩散过程的扩散方程做了研究.在构造了布朗运动的数学模型之后,数学家们在研究布朗运动时发现:布朗粒子所在的空间几何结构不同,会影响扩散过程的常返性.在德国数学家P.Koebe与法国数学家H.Poincare证明了著名的单值化定理之后,数学家们通过位势理论彻底将布朗运动与Riemannian几何联系在一起.而随着A.Einstein搭建了狭义相对论的理论框架,越来越多的物理学家、数学家将传统的热力统计学理论推广到相对论的框架中来,由此得到了布朗粒子在Minkowski空间中,概率密度函数ρ所满足的扩散方程(?)ρ/(?)t=1/2△gρ,以及布朗运动的边缘到达分布u满足的稳态方程1/2△gρu = 0,其中△g是Minkowski空间中的Laplace-Beltrami算子.由此,计算布朗粒子边缘到达分布的问题就转化为求解一个Dirichlet问题.我们在第一章中介绍了以上的背景与理论基础.在此基础上,本文旨在探讨布朗运动在不同维数的Minkowski空间中伪球面上到达特定纬度的边缘到达分布.由于Minkowski空间中的伪球面就是双曲面模型,因此,我们尝试使用文献[11]中的方法与思路,通篇使用偏微分方程的变量分离法.在第二章中,我们给出了 Minkowski空间中类空间、类时间、类光区域以及类空间、类时间伪球面的基本概念,给出了此空间中的内积、距离、度量公式,以及直角坐标下Laplace-Beltrami算子的表达形式.我们还探讨了一下Minkowski空间中Laplace-Beltrami算子谱的非负性,详细的计算了三维及高维情形中使用伪球极坐标时Laplace-Beltrami算子的表达式.我们介绍了几个求解方程时会遇到的常微分方程、超几何函数、Gegenbauer多项式和为了简便所使用的符号.接下来,在第三章中,我们将三维Minkowski空间分成R13与R23两种情况.对空间R13,我们进行了详细的计算,分别得到了类空间与类时间伪球面上边缘到达分布的公式,归结为定理3.1.1与定理3.2.1,并对解的概率性质进行了验证.对于空间R23,由于问题的等价性,我们直接得到了定理3.3.1.最后,在第四章中,我们对一般的高维情形进行了讨论.根据l与v不同导致的单位球面Laplace算子退化程度不同,将问题划分为四种情况.重复三维情形的计算过程,使用变量分离法得到方程的通解,再利用边值条件以及三角函数系与Gegenbauer多项式的正交性,得到我们想要的边缘到达分布,由此得到定理4.1.1与定理4.3.1.并且,对于类空间情形,我们详细验证了所得到的每一个边缘到达分布的概率性质.
[Abstract]:Since 19th century, when British botanist Rorbert Brown observed the irregular movement of pollen particles in liquids, The study of the physical phenomenon named Brownian motion has never stopped. Firstly, physicists A. Einstein and Marian von Smoluchowski have obtained the diffusion equation satisfied by the probability density of the transfer of Brownian motion in Euclidean space. Adriaan Daniel Fokker, a Dutch scientist, and Max Planck, a German physicist, studied the diffusion equations of general diffusion processes. When studying the Brownian motion, mathematicians found that the spatial geometric structure of the Brownian particle would affect the recurrence of diffusion process. After the famous singularization theorem was proved by the German mathematician P. Kobey and the French mathematician H. Poincare, Mathematicians connected the Brownian motion to Riemannian geometry completely through the potential theory. And as A. Einstein built the theoretical framework of special relativity, more and more physicists, Mathematicians extended the traditional theory of thermal statistics to the framework of relativity, and obtained the diffusion equation of Brownian particle in Minkowski space, which is satisfied by the probability density function 蟻? ) 蟻 / R? 1 / 2 g 蟻 and 1/2 g 蟻 u, where g is the Laplace-Beltrami operator in Minkowski space, and the stable equation of the edge arrival distribution u of the Brownian motion is 1/2 g 蟻 u = 0, where g is the Laplace-Beltrami operator in Minkowski space. The problem of calculating Brownian particle edge arrival distribution is transformed into solving a Dirichlet problem. In the first chapter, we introduce the above background and theoretical basis. The purpose of this paper is to investigate the edge arrival distribution of Brownian motion on pseudo-sphere in Minkowski space with different dimensions. Since the pseudo-sphere in Minkowski space is a hyperboloid model, we try to use the method and train of thought in reference [11]. In the second chapter, we give the basic concepts of class space, class time, quasi-optical region, quasi-space and quasi-time pseudo-sphere in Minkowski space, and give the inner product and distance in this space. We also discuss the nonnegativity of the spectrum of Laplace-Beltrami operators in Minkowski spaces. The expressions of the Laplace-Beltrami operator using pseudo spherical polar coordinates in three dimensional and high dimensional cases are calculated in detail. We introduce several ordinary differential equations, hypergeometric functions, Gegenbauer polynomials and symbols for convenience. In the third chapter, we divide the three-dimensional Minkowski space into two cases, R13 and R23. For the space R13, we calculate it in detail, and obtain the formulas of the arrival distribution of the edges on the pseudo-sphere of quasi-space and quasi-time, respectively. Theorem 3.1.1 and theorem 3.2.1 are reduced to theorem 3.1.1, and the probabilistic properties of the solution are verified. For the space R23, due to the equivalence of the problem, we directly obtain Theorem 3.3.1.Finally, in Chapter 4th, In this paper, we discuss the general high dimensional case. According to the degeneracy degree of the unit spherical Laplace operator caused by the difference of l and v, the problem is divided into four cases. The general solution of the equation is obtained by using the method of variable separation, and the boundary value condition and the orthogonality between trigonometric function system and Gegenbauer polynomial are used to obtain the desired edge arrival distribution, and the theorems 4.1.1 and 4.3.1 are obtained. We verify the probabilistic properties of each edge arrival distribution in detail.
【学位授予单位】：吉林大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O552.1

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