臺灣博碩士論文加值系統

本論文的目的主要是探討在Z^d 空間上一個具長域範圍之隨機漫步, 其機率分佈D(x) 具有對稱性以及當|x| → ∞ 其|x|^(−α−d) 的遞減速率跟它的次方有關係, 對一些α > 0 和r > 0。第一個結果, 我們討論迴轉半徑r階的漸進行為。此外, 我們還取得主要係數為固定一個xj 軸上的方向, 對於j = 1, 2, ..., d。在第二個結果中, 我們討論recurrent 或transient 具長域隨機漫步與α > 0 和維度d > 0 有關係。這個證明是用傅立葉轉換
以及最基本的分數導數。

The purpose of the thesis is to investigate some properties of a long-range random walk on Z^d which is 1-step distribution D(x) has symmetry and the rate of decay is order |x|^(−α−d) as |x| → ∞ for some α > 0 and d > 0. The first result, we discuss the asymptotic behavior of the gyration radius of order r where 0 < r < α and α > 2. Furthermore we obtain the main coefficient as fixed one xj-axis direction for j = 1, 2, ..., d. The second result, we discuss the recurrence or transient for the long-range random walk with α > 0 and dimension d > 0. The proof is basic on Fourier transform and fractional derivative.

Contents
1 Introduction 1
2 Main results (Theorem 1 and Theorem 3) 6
3 Proof of Theorem 1 (where r is even) 9
4 Proof of Theorem 1 (for 0 < r < 2 and α > 2) 10
5 Proof of Theorem 3 13

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