Above under and beyond brownian motion talk
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The seminar addresses various types of random walks, including ordinary random walks, Lévy flights, and fractional Brownian motion, highlighting their characteristics and behaviors. It discusses the generalized central limit theorem and stable distributions, emphasizing the significance of power-law tail distributions and their implications on asymptotic behavior. The concluding remarks focus on the interplay between subdiffusion and superdiffusion, underscoring differences in scaling and distribution shapes.
Above under and beyond brownian motion talk
- 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
- 6. Lévy flights
- 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
- 22. Lévy Flights
- 24. 1d Fractional Brownian motion
- 25. 2d Fractional Brownian motion
- 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