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

No / The motion of an artificial micro-scale swimmer that uses a chemical reaction catalyzed on its own surface to achieve autonomous propulsion is fully characterized experimentally. It is shown that at short times, it has a substantial component of directed motion, with a velocity that depends on the concentration of fuel molecules. At longer times, the motion reverts to a random walk with a substantially enhanced diffusion coefficient. Our results suggest strategies for designing artificial chemotactic systems.

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Schramm-Loewner evolution (SLE(kappa)) is an important contemporary tool for identifying critical scaling limits of two-dimensional statistical systems. The SLE(kappa) one-parameter family of processes can be viewed as a special case of a more general, two-parameter family of processes we denote SLE(kappa, mu). The SLE(kappa, mu) process is defined for kappa>0 and real numbers mu; it represents the solution of the chordal Loewner equations under special conditions on the driving function parameter which require that it is a Brownian motion with drift mu and variance kappa. We derive properties of this process by use of methods applied to SLE(kappa) and application of Girsanov's Theorem. In contrast to SLE(kappa), we identify stationary asymptotic behavior of SLE(kappa, mu). For kappa in (0,4] and mu > 0, we present a pathwise construction of a process with stationary temporal increments and stationary imaginary component and relate it to the limiting behavior of the SLE(kappa, mu) generating curve. Our main result is a spatial invariance property of this process achieved by defining a top-crossing probability for points in the upper half plane with respect to the generating curve.

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Thesis advisor: Nancy Rallis / The theory of Brownian motion is an integral part of statistics and probability, and it also has some of the most diverse applications found in any topic in mathematics. With extensions into fields as vast and different as economics, physics, and management science, Brownian motion has become one of the most studied mathematical phenomena of the late twentieth and early twenty-first centuries. Today, Brownian motion is mostly understood as a type of mathematical process called a stochastic process. The word "stochastic" actually stems from the Greek word for "I guess," implying that stochastic processes tend to produce uncertain results, and Brownian motion is no exception to this, though with the right models, probabilities can be assigned to certain outcomes and we can begin to understand these complicated processes. This work reaches to attain this goal with regard to Brownian motion, and in addition it explores several applications found in the aforementioned fields and beyond. / Thesis (BA) — Boston College, 2007. / Submitted to: Boston College. College of Arts and Sciences. / Discipline: Mathematics. / Discipline: College Honors Program.

The main object of study in this thesis is branching Brownian motion, in which each particle moves like a Brownian motion and gives birth to new particles at some rate. In particular we are interested in where particles are located in this model at large times T : so, for a function f up to time T , we want to know how many particles have paths that look like f. Additive spine martingales are central to the study, and we also investigate some simple general properties of changes of measure related to such martingales.

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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

Ho, Yuk Kwan = 粒子鏈的布朗運動 : 相互關係時間之探討 / 何煜坤. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2008. / Includes bibliographical references (p. 82-84). / Abstracts in English and Chinese. / Ho, Yuk Kwan = Li zi lian de Bulang yun dong : xiang hu guan xi shi jian zhi tan tao / He Yukun. / Chapter 1 --- Introduction --- p.1 / Chapter 1.1 --- Historical background --- p.1 / Chapter 1.2 --- Motivation --- p.4 / Chapter 2 --- Modelling of the system of the particle chain --- p.6 / Chapter 2.1 --- Interactions between the particles --- p.7 / Chapter 2.2 --- Assumptions of the Brownian force --- p.10 / Chapter 3 --- Time evolution of the probability distribution --- p.14 / Chapter 3.1 --- Diffusion under a uniform external force field --- p.14 / Chapter 3.2 --- Multi-dimensional Fokker-Planck equation --- p.18 / Chapter 3.3 --- Fundamental solution to the Fokker-Planck equation --- p.21 / Chapter 3.3.1 --- Fulfillment of the Fokker-Planck equation by the stochas- tic process described by the Langevin equation --- p.21 / Chapter 3.3.2 --- Gaussian process of the stochastic process in the system --- p.24 / Chapter 3.4 --- Relaxation of the fluctuations and the variances of the system --- p.27 / Chapter 3.4.1 --- Dependence of system parameters - study of a two-body system --- p.27 / Chapter 3.4.2 --- Dependence of system size --- p.33 / Chapter 4 --- Time evolution of the correlation function --- p.36 / Chapter 4.1 --- Method of Rice - harmonic analysis --- p.38 / Chapter 4.1.1 --- Natural mode expansion of the correlation functions --- p.41 / Chapter 4.1.2 --- Satisfaction of the equipartition principle --- p.44 / Chapter 4.2 --- Relaxation of the correlation functions --- p.45 / Chapter 4.2.1 --- Dependence of system parameters - study of a two body system --- p.46 / Chapter 4.2.2 --- Dependence of system size --- p.50 / Chapter 4.3 --- Connection with relaxation modes of fluctuations and variances --- p.53 / Chapter 5 --- Coloured Brownian force --- p.58 / Chapter 5.1 --- Fluctuation-dissipation theorem --- p.59 / Chapter 5.2 --- The system of a large particle with a particle chain --- p.64 / Chapter 5.2.1 --- Equivalent heat bath with which the large particleis interacting --- p.67 / Chapter 5.2.2 --- Retarded friction from its underlying physical origin --- p.71 / Chapter 5.2.3 --- Effective random force of the heat bath and its underly- ing physical origin --- p.73 / Chapter 5.2.4 --- Displacement correlation function for the large particle interacting with the heat bath --- p.77 / Chapter 6 --- Conclusion --- p.81 / Bibliography --- p.82 / Chapter A --- Magnetic force between two magnetic dipoles --- p.85 / Chapter B --- Hydrodynamic interaction --- p.88 / Chapter B.l --- Faxen´ةs Law --- p.90 / Chapter B.2 --- Method of reflection --- p.92 / Chapter B.3 --- Interactions between three translating identical spheres --- p.94 / Chapter C --- Proof of the cross-correlation theorem and Wiener-Kintchine theorem --- p.97 / Chapter D --- Proof of the relation between θ(t) and β(t) in Eq. 5.42 --- p.99 / Chapter E --- Proof of the zero-value of k in Eq. 5.60 --- p.101