above under and beyond brownian motion talk
DESCRIPTION
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TRANSCRIPT
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 Leacutevy flights
Generalized central limit theorem
Stable distribution
Continuous time random walks Ordinary Diffusion Leacutevy Flights Fractional Brownian motion (subdiffusion) Ambivalent processes
2
Discrete Time Random walks
Part 1
3
4
Ordinary random walks
5
Central limit theorem
Leacutevy flights
Leacutevy flight scales
superdiffusively
Generalized central limit
theorem
Part 2
9
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 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
10
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 Leacutevy alpha-
stable distribution
11
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
2
Outline
Discrete Time Random walks Ordinary random walks Leacutevy flights
Generalized central limit theorem
Stable distribution
Continuous time random walks Ordinary Diffusion Leacutevy Flights Fractional Brownian motion (subdiffusion) Ambivalent processes
2
Discrete Time Random walks
Part 1
3
4
Ordinary random walks
5
Central limit theorem
Leacutevy flights
Leacutevy flight scales
superdiffusively
Generalized central limit
theorem
Part 2
9
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 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
10
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 Leacutevy alpha-
stable distribution
11
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
Discrete Time Random walks
Part 1
3
4
Ordinary random walks
5
Central limit theorem
Leacutevy flights
Leacutevy flight scales
superdiffusively
Generalized central limit
theorem
Part 2
9
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 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
10
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 Leacutevy alpha-
stable distribution
11
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
4
Ordinary random walks
5
Central limit theorem
Leacutevy flights
Leacutevy flight scales
superdiffusively
Generalized central limit
theorem
Part 2
9
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 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
10
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 Leacutevy alpha-
stable distribution
11
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
5
Central limit theorem
Leacutevy flights
Leacutevy flight scales
superdiffusively
Generalized central limit
theorem
Part 2
9
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 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
10
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 Leacutevy alpha-
stable distribution
11
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
Leacutevy flights
Leacutevy flight scales
superdiffusively
Generalized central limit
theorem
Part 2
9
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 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
10
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 Leacutevy alpha-
stable distribution
11
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
Leacutevy flight scales
superdiffusively
Generalized central limit
theorem
Part 2
9
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 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
10
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 Leacutevy alpha-
stable distribution
11
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
Generalized central limit
theorem
Part 2
9
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 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
10
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 Leacutevy alpha-
stable distribution
11
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
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 lt α lt 2 (and therefore having infinite
variance) will tend to a stable distribution
f(xα0c0) as the number of variables grows
10
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 Leacutevy alpha-
stable distribution
11
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
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 Leacutevy alpha-
stable distribution
11
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
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)
CchaosTalklevyStableDensityFunctioncdf
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
A random variable X is called stable if its
characteristic function is given by
Characteristic function of
Stable distribution
13
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
14
Symmetric α-stable distributions
with unit scale factor
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
15
Skewed centered stable
distributions with different β
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
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 lt 2
Unified normal and power law
16
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
Log-log plot of skewed centered stable distribution PDFs showing the power law behavior for large x Again the slope of the linear portions is equal to -(α+1)
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
Continuous time random walks
Part 1
18
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
spatial displacement ∆x and a
temporal increment ∆t
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
Ordinary Diffusion
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
Leacutevy Flights
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
Fractional Brownian motion
(subdiffusion)
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
1d Fractional Brownian motion
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
2d Fractional Brownian motion
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
Ambivalent processes
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
28 28
Concluding Remarks
The ratio of the exponents αβ resembles the
interplay between sub- and superdiffusion
For β lt 2α the ambivalent CTRW is effectively
superdiffusive
for β gt 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 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29
29 29
Xiong Wang 王雄
Centre for Chaos and Complex Networks
City University of Hong Kong
Email wangxiong8686gmailcom
29