在整數晶格 Z d 上的隨機漫步 S x n = x + Pn k=1 Xk，其中每一個隨機向量 Xi , i = 1, 2, · · · , n 皆獨立且具有相同分佈 D(x)。此論文，我們假設 D(x) 在 Zd 空間情形下擁有對稱性且當 |x| → ∞ 時，遞減速率為 |x|−d−α，其中 α ∈ (0, ∞) \ {2} 且 d > α ∧ 2。本文主要是探討此具長域隨機漫步下的一些 漸近行為。第一個主要結果在於獲得此模型之格林函數的漸近行為，此外 我們還得到主要項係數及其收斂速度;第二個主要結果在討論容積的漸近 行為，並且進一步得到在長域隨機漫步下的 Wiener’s Criterion。

Let S x n = x + Pn k=1 Xk be the n-step random walk on Z d starting at x, where X′ i s are independent identically distributed random vectors with distribution D(x). In the thesis, we suppose that the distribution D(x) is symmetric on Zd and the rate of decayisoforder|x|−d−α as|x|→∞withα∈(0,∞)\{2}andd>α∧2,where a ∧ b = min {a, b}. The purpose of the thesis is to investigate asymptotic behaviors of the long-range random walk. First of all, we get the asymptotic behavior of the Green function. Moreover, we obtain the coefficient of the main term and its rates of convergence. Secondly, we discuss the asymptotic behavior of the capacity for the long-range random walk. Moreover, we derive the Wiener’s Criterion for the long-range random walk.

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