# 隨機漫步

t
) = (X
1
, X
2
, ...)。但是，也可以定義在隨機時間採取步驟的隨機遊走，在這種情況下，必須定義X
t

## 點陣隨機漫步

### 一維隨機漫步

5次擲硬幣后所有可能的結果

${\displaystyle E(S_{n})=\sum _{j=1}^{n}E(Z_{j})=0.}$

${\displaystyle E(S_{n}^{2})=\sum _{i=1}^{n}\sum _{j=1}^{n}E(Z_{j}Z_{i})=n.}$

${\displaystyle \lim _{n\to \infty }{\frac {E(|S_{n}|)}{\sqrt {n}}}={\sqrt {\frac {2}{\pi }}}.}$

ab為正整數。在一維綫上從0開始一個隨機漫步過程，那麽從0到第一次碰到b或-a時的期待時間是ab。先到達b后到達a的幾率為${\displaystyle a/(a+b)}$ ，因爲简单随机游走是

k −5 −4 −3 −2 −1 0 1 2 3 4 5
${\displaystyle P[S_{0}=k]}$  1
${\displaystyle 2P[S_{1}=k]}$  1 1
${\displaystyle 2^{2}P[S_{2}=k]}$  1 2 1
${\displaystyle 2^{3}P[S_{3}=k]}$  1 3 3 1
${\displaystyle 2^{4}P[S_{4}=k]}$  1 4 6 4 1
${\displaystyle 2^{5}P[S_{5}=k]}$  1 5 10 10 5 1

#### 作爲馬爾可夫鏈

${\displaystyle \,P_{i,i+1}=p=1-P_{i,i-1}.}$

### 在更高的維度上

${\displaystyle P(r)={\frac {2r}{N}}e^{-r^{2}/N}}$

### 与维纳过程的关系

${\displaystyle \sigma ^{2}={\frac {t}{\delta t}}\,\varepsilon ^{2},}$

${\displaystyle \sigma ^{2}=6\,D\,t.}$

${\displaystyle D={\frac {\varepsilon ^{2}}{6\delta t}}}$  (仅在三維空間中有效).

${\displaystyle D={\frac {\varepsilon ^{2}}{4\delta t}}.}$

${\displaystyle D={\frac {\varepsilon ^{2}}{2\delta t}}.}$

## 参考文献

1. Wirth, E.; Szabó, G.; Czinkóczky, A. MEASURE LANDSCAPE DIVERSITY WITH LOGICAL SCOUT AGENTS. ISPRS – International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. 2016-06-08, XLI–B2: 491–495. Bibcode:2016ISPAr49B2..491W. doi:10.5194/isprs-archives-xli-b2-491-2016.
2. ^ Wirth E. (2015). Pi from agent border crossings by NetLogo package页面存档备份，存于互联网档案馆）. Wolfram Library Archive
3. ^ Pearson, K. The Problem of the Random Walk. Nature. 1905, 72 (1865): 294. Bibcode:1905Natur..72..294P. doi:10.1038/072294b0.
4. ^ Pal, Révész (1990) Random walk in random and nonrandom environments, World Scientific
5. ^ Kohls (2016), Expected Coverage of Random Walk Mobility Algorithm页面存档备份，存于互联网档案馆）, arxiv.org.
6. ^ Random Walk-1-Dimensional – from Wolfram MathWorld. Mathworld.wolfram.com. 2000-04-26 [2016-11-02]. （原始内容存档于2016-11-18）.
7. ^ Edward A. Colding et al, Random walk models in biology, Journal of the Royal Society Interface, 2008
8. ^ Kotani, M. and Sunada, T. Spectral geometry of crystal lattices. Contemporary. Math. Contemporary Mathematics. 2003, 338: 271–305. ISBN 9780821833834. doi:10.1090/conm/338/06077.
9. ^ Kotani, M. and Sunada, T. Large deviation and the tangent cone at infinity of a crystal lattice. Math. Z. 2006, 254 (4): 837–870. doi:10.1007/s00209-006-0951-9.
10. ^ Pólya's Random Walk Constants. Mathworld.wolfram.com. [2016-11-02]. （原始内容存档于2021-05-09）.
11. ^ MacKenzie, D. MATHEMATICS: Taking the Measure of the Wildest Dance on Earth. Science. 1883, 290 (5498): 1883–4. PMID 17742050. doi:10.1126/science.290.5498.1883.
12. ^ Chapter 2 DIFFUSION页面存档备份，存于互联网档案馆）. dartmouth.edu.
13. ^ Diffusion equation for the random walk页面存档备份，存于互联网档案馆）. physics.uakron.edu.