This talk was Dedicated to Einstein's miracle year at his 26 以此次讲座，致敬当年爱因斯坦26岁时的几篇牛文之一，对布朗运动的研究。 对随机游走的研究，已经取得了很深入的进展，本次讲座从布朗运动模型入手，逐步深入，引入分数阶布朗运动，levy随机飞行等概念 这些模型在各种复杂系统中非常常见，比如金融市场，网络交通流量等等， 会简略介绍这些模型在金融系统的应用，以及分析基于布朗运动随机游走的金融模型的弊端 给大家一个随机游走世界的全景

1. Above and Under
Brownian Motion
Brownian Motion , Fractional Brownian
Motion , Levy Flight, and beyond
Seminar Talk at Beijing Normal University
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong 1

2. Outline
 Discrete Time Random walks
Ordinary random walks
Lévy flights
 Generalized central limit
theorem
Stable distribution
 Continuous time random walks
Ordinary Diffusion Lévy Flights
Fractional Brownian motion
(subdiffusion) Ambivalent processes 2

3. Part 1
Discrete Time Random walks
3

4. Ordinary random walks
4

5. Central limit theorem
5

6. Lévy flights

7. Lévy flight scales
superdiffusively

9. Part 2
Generalized central limit
theorem
9

10. Generalized central limit
theorem
 A generalization due to Gnedenko and
Kolmogorov states that the sum of a number
of random variables with power-law tail
distributions decreasing as 1 / | x | α + 1 where
0 < α < 2 (and therefore having infinite
variance) will tend to a stable distribution
f(x;α,0,c,0) as the number of variables grows.
10

11. Stable distribution
 In probability theory, a random variable is
said to be stable (or to have a stable
distribution) if it has the property that a linear
combination of two independent copies of the
variable has the same distribution, up to
location and scale parameters.
 The stable distribution family is also
sometimes referred to as the Lévy alpha-
stable distribution.
11

12.  Such distributions form a four-parameter
family of continuous probability distributions
parametrized by location and scale
parameters μ and c, respectively, and two
shape parameters β and α, roughly
corresponding to measures of asymmetry
and concentration, respectively (see the
figures).
 C:chaosTalklevyStableDensityFunction.cdf

13. Characteristic function of
Stable distribution
 A random variable X is called stable if its
characteristic function is given by
13

14. Symmetric α-stable distributions
with unit scale factor
14

15. Skewed centered stable
distributions with different β
15

16. Unified normal and power law
 For α = 2 the distribution reduces to a Gaussian
distribution with variance σ2 = 2c2 and mean μ; the
skewness parameter β has no effect
 The asymptotic behavior is described, for α < 2
16

17. Log-log plot of skewed centered stable distribution PDF's showing the
power law behavior for large x. Again the slope of the linear portions
is equal to -(α+1)

18. Part 1
Continuous time random walks
18

19. spatial displacement ∆x and a
temporal increment ∆t

21. Ordinary Diffusion

22. Lévy Flights

23. Fractional Brownian motion
(subdiffusion)

24. 1d Fractional Brownian motion

25. 2d Fractional Brownian motion

26. Ambivalent processes

28. Concluding Remarks
The ratio of the exponents α/β resembles the
interplay between sub- and superdiffusion.
For β < 2α the ambivalent CTRW is effectively
superdiffusive,
for β > 2α effectively subdiffusive.
For β = 2α the process exhibits the same
scaling as ordinary Brownian motion, despite
the crucial difference of infinite moments and a
non-Gaussian shape of the pdf W(x, t).
28

29. Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email: wangxiong8686@gmail.com
29