Some Properties of Dipolar SLE

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卓冠宇(Kuan-Yu Cho) 查詢紙本館藏

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論文名稱

(Some Properties of Dipolar SLE)

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摘要(中)

在這篇論文中，我們研讀了一系列定義在複數平面上的曲線（或這曲線空間的等價類）的測度，包括了溢出測度、布朗氣泡測度、布朗環測度。我們研讀了帶形上的 Loewner 鍊。我們證明了帶有驅動點 0+ 的帶型 SLE?(?) 的共形限制性。當 ? ∈ (0, 8/3] 時，我們得到了大小區域的帶型 SLE? 之間的 Radon-Nikodym 導數的表達式。

本論文的大綱如下：在第 1 章，我們簡要地介紹 SLE 的發展。在第 2 章，我們介紹了一些關於共形變換的性質。我們還介紹了布朗測度和邊界泊松核。我們給出了帶形情況的一些布朗測度的計算。在第 3 章，我們首先介紹（弦） Loewner 微分方程，並給出帶形情形的證明。我們給出了帶形 Loewner 鏈經由共形變換後的一些計算。最後我們介紹 SLE 過程。在第 4 章，我們給了主要結論的證明。

摘要(英)

In this thesis, we study a series of measures defined on the space of curves in the complex plane (or the equivalence classes of the curves space), including excursion measures, Boundary bubble measures, and Loop measures. We study the Loewner chain in the strip {? = ? + ?? ∈ ℂ ∶ ? ∈ (0, ?)}. We show the conformal restriction property for the dipolar SLE?(?) with a force point 0+, and we obtain an expression for the Radon-Nikodym derivative of dipolar SLE? in a domain with respect to dipolar SLE? in a subdomain for ? ∈ (0, 8/3].

The outline of this thesis is as follows: In Chapter 1, we briefly introduce the development of SLE. In Chapter 2, we introduce some properties about conformal transformations. We also introduce Brownian measures and the boundary Poisson kernels. Some calculations of Brownian measures for the dipolar case will be given. In Chapter 3, we first introduce the (chordal) Loewner differential equation and give the proofs for the dipolar case, and we show some calculations for the Loewner chain mapped by conformal transformations, and then introduce SLE process. In Chapter 4, we give the proofs of our main results.

關鍵字(中)

★ 帶型 SLE

關鍵字(英)

論文目次

中文摘要v
Abstract vi
List of Figures viii
1 Introduction 1
1.1 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Conformal Transformation 17
2.1 Harmonic Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.1 Half-Plane Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2.2 Strip Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Brownian Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3 Schramm-Loewner Evolution 62
3.1 Loewner Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.1.1 Chordal Loewner Equation . . . . . . . . . . . . . . . . . . . . . . 64
3.1.2 Dipolar Loewner Equation . . . . . . . . . . . . . . . . . . . . . . . 68
3.1.3 Loewner Chain Mapped by a Conformal Transformation . . . . . . 74
3.2 Schramm-Loewner Evolution (SLE) . . . . . . . . . . . . . . . . . . . . . . 81
4 Proof of Theorem 86
4.1 The locality property for dipolar SLE6 . . . . . . . . . . . . . . . . . . . . 87
4.2 The Conformal Restriction Property for Dipolar SLE8/3 . . . . . . . . . . . 88
4.3 The Radon-Nikodym Derivative of Dipolar SLE . . . . . . . . . . . . . . . 101
Bibliography 107

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審核日期

2022-8-8

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