Integral representations provide a useful framework of study and simulation of fractional Browian motion, which has been used in modeling of many natural situations. In this thesis we extend an integral representation of fractional Brownian motion that is supported on a bounded interval of ℝ to integral representation that is supported on bounded subset of ℝ[superscript d]. These in turn can be used to give new series representations of fractional Brownian motion. / Graduation date: 2013

Thesis (M.S. in Electrical Engineering)--Naval Postgraduate School, March 2004. / Thesis advisor(s): Murali Tummala, Roberto Cristi. Includes bibliographical references (p. 127-129). Also available online.

Thesis (Ph. D.)--University of Wisconsin--Madison, 1984. / Typescript. Vita. Includes bibliographical references.

Thesis--University of Wisconsin--Madison. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 126-128).

Thesis (Ph. D.)--University of Wisconsin--Madison, 1984. / Typescript. Vita. eContent provider-neutral record in process. Description based on print version record. Includes bibliographical references (leaves 191-197).

M.Sc. (Mathematical Statistics) / Please refer to full text to view abstract

Thesis (doctoral)--Rheinische Friedrich-Wilhelms-Universität Bonn, 1991. / Includes bibliographical references (p. 120-125).

No description available.

令B{U+209C}表示一個平面布朗運動。我們把C \B[0, 1] 的無界連通分支的邊界稱爲B[0; 1] 的外邊界。在本文中，我們將討論如何計算B[0,1] 的外邊界的Hausdorff 維數。 / 我們將在第二章討論Lawler早期的工作[7]。他定義了一個常數ζ(所謂的不聯通指數） 。利用能量的方法， 他證明了 B[0,1]的外邊界的Hausdorff維數是2(1 - ζ)概率大於零， 然後0-1律可以明這個概率就是1。但是用他的方法我們不能算出ζ的準確值。 / Lawler, Schramm and Werner 在一系列文章[10]，[11] 和[13] 中研究了SLE{U+2096}和excursion 測度。利用SLE6 和excursion 測度的共形不變性，他們可以計算出了布朗運動的相交指數ξ (j; λ )。因此ζ = ξ (2; 0)/2 = 1/3，由此可以知道B[0, 1] 的外邊界的Hausdorff 維數就是4/3。從而可以說完全證明了著名的Mandelbrot 猜想。 / Let B{U+209C} be a Brownian motion on the complex plane. The frontier of B[0; 1] is defined to be the boundary of the unbounded connected component of C\B[0; 1].In this thesis, we will review the calculation of the Hausdorff dimension of the frontier of B[0; 1]. / We first dissuss the earlier work of Lawler [7] in Chapter 2. He defined a constant ζ (so called the dimension of disconnection exponent). By using the energy method, he proved that with positive probability the Hausdorff dimension of the frontier of B[0; 1] is 2(1 -ζ ), then zero-one law show that the probability is one. But we can not calculate the exact value of ζ in this way. / In the series of papers by Lawler, Schramm and Werner [10], [11] and [13], they studied the SLE{U+2096} and excursion measure. By using the conformal invariance of SLE₆ and excursion measure, they can calculate the exact value of the Brownian intersection exponents ξ(j, λ). Consequently, ζ = ξ(2, 0)/2 = 1/3, and the Hausdorff dimension of the frontier of B [0,1] is 4/3 almost surely. This answers the well known conjecture by Mandelbrot positively. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Zhang, Pengfei. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2012. / Includes bibliographical references (leaves 53-55). / Abstracts also in Chinese. / Chapter 1 --- Introduction --- p.6 / Chapter 2 --- Hausdorff dimension of the frontier of Brownian motion --- p.11 / Chapter 2.1 --- Preliminaries --- p.11 / Chapter 2.2 --- Hausdorff dimension of Brownian frontier --- p.13 / Chapter 3 --- Stochastic Loewner Evolution --- p.24 / Chapter 3.1 --- Definitions --- p.24 / Chapter 3.2 --- Continuity and Transience --- p.26 / Chapter 3.3 --- Locality property of SLE₆ --- p.30 / Chapter 3.4 --- Crossing exponent for SLE₆ --- p.32 / Chapter 4 --- Brownian intersection exponents --- p.37 / Chapter 4.1 --- Half-plane exponent --- p.37 / Chapter 4.2 --- Whole-plane exponent --- p.41 / Chapter 4.3 --- Proof of Theorem 4.6 and Theorem 4.7 --- p.44 / Chapter 4.4 --- Proof of Theorem 1.2 --- p.47 / Chapter A --- Excursion measure --- p.48 / Chapter A.1 --- Metric space of curves --- p.48 / Chapter A.2 --- Measures on metric space --- p.49 / Chapter A.3 --- Excursion measure on K --- p.49 / Bibliography --- p.53

An important object of study in harmonic analysis is the heat equation. On a Euclidean space, the fundamental solution of the associated semigroup is known as the heat kernel, which is also the law of Brownian motion. Similar statements also hold in the case of a Lie group. By using the wrapping map of Dooley and ildberger, we show how to wrap a Brownian motion to a compact Lie group from its Lie algebra (viewed as a Euclidean space) and find the heat kernel. This is achieved by considering It??o type stochastic differential equations and applying the Feynman-Ka??c theorem. We also consider wrapping Brownian motion to various symmetric spaces, where a global generalisation of Rouvi`ere???s formula and the e-function are considered. Additionally, we extend some of our results to complex Lie groups, and certain non-compact symmetric spaces.