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\begin{document}

\heading{Levy Flights and Continuous Time Random Walks} 
\auname{Arthur Shulkin}

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Notation remark

$f(x)$ is a probability distribution function (PDF) of the random
variable $x$.

$f(t)$ is a PDF of the random variable $t$.

$f(x,t)$ is a common PDF of $x$ and $t$.

$\tilde{f}(k)$ is a Fourier transform of $f(x)$.

$\tilde{\phi}(s)$ is a Laplace transform of $f(t)$.

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\textbf{Levy flight} is a random walk in which the step lenghts are
distributed according to a Levy distribution (defined later).

\textbf{Continuous Time Random Walks} (CTRW) is a random walk in
which the step lenghts and the waiting times are distributed according
to a Levy distributions.

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\begin{center}\emph{\noun{Problem Definishion}}\end{center}

A. $\quad$Derive an exprassion for $f_{N}(x)$,the probability for
position $x$ after N steps, and $x(N)$ for a Levy flights motion.

B. $\quad$Derive an exprassion for $f(x,t)$,the probability for
position $x$ at time $t$, and $x(t)$ for an Continuous Time Random
Walks motion.

\textbf{Levy stable distribution}, $f(x;a,\alpha)$, is defined by
its characteristic function

\begin{equation}
\tilde{f}(k;a,\alpha)=<\exp(ikx)>=\exp(-|ak|^{\alpha})\end{equation}


Levy distribution is defined for $0<\alpha<2$, otherewise $f(x;a,\alpha)$
is not normalized $(\alpha<0)$ or gets negative values and cannot
be a pdf $(Var(f(x;a,\alpha))=0)$. here $a$ have the units of length.

For $f(t;a,\alpha)$ ($a$ have the units of time) which is defined
through its Laplace transform, $\tilde{\phi}(s;a,\alpha)$, $f(t;a,\alpha)$
is non negative for $0<\alpha<1$.

For $\alpha=1$ and $\alpha=2$ we get two well known special cases,
Lorentzian and Gaussian

\begin{equation}
f(x;a,1)=\frac{a}{\pi(a^{2}+x^{2})}\quad f(x;\frac{\sigma^{2}}{2},2)=\frac{1}{\sqrt{2\pi\sigma^{2}}}\exp(-\frac{x^{2}}{2\sigma^{2}})\end{equation}


\textbf{Width} the width of the PDF $f(\frac{x}{N};a,\alpha)$ defined
as $N$

\textbf{Stability} a random variable is stable if a linear combination
of two independent copies of the variable has the same distribution.
$f_{N}(\sum x_{i};a,\alpha)=f(x_{1};a,\alpha)*f(x_{2};a,\alpha)...*f(x_{N};a,\alpha)$,
$*$ stands for convolution, 

in Fourier space $\tilde{f_{N}}(k)=\prod\tilde{f}(k_{i};a,\alpha)$,
for Levy stable distribution one gets 

$\tilde{f_{N}}(k)=\exp(-|aN^{\frac{1}{\alpha}}k|^{\alpha})\rightarrow f_{N}(\sum x_{i};a,\alpha)=f(\frac{x}{N^{\frac{1}{\alpha}}};a,\alpha)$,
that is the generalized center limit theorem.

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in context of the Levy flight the width is the flight length 

\begin{equation}
x(N)\sim N^{\frac{1}{\alpha}}\end{equation}


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\begin{center}\emph{\noun{Continuous Time Random Walks}}\end{center}

An important equation that we will use is the Montroll-Weiss equation

\begin{equation}
\tilde{f}(k,s)=\frac{1-\tilde{f}(s)}{s}\frac{1}{1-\tilde{\phi}(s)\tilde{f}(k)}\end{equation}


{*} Montroll-Weiss Equation derivation is in the appendix at the end
of the lecture.

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For anomalous diffusion we assume the waiting times PDF $\tilde{\phi}(s;\tau,\beta)$,
and step length PDF $\tilde{f}(k;\lambda,\alpha)$.

Expand the PDFs for small $k$ and $s$ and put it into the Montroll-Weiss
equation and taking the first order terms gives

\begin{equation}
\tilde{f}(k,s)\thicksim\frac{\tau s{}^{\beta-1}}{\tau s^{\beta}+\lambda k^{\alpha}}\end{equation}


As an example one can verify the well known normal diffusion kernal,
i.e $\alpha=2,$$\beta=1$, $\lambda=\frac{\sigma^{2}}{2}$

\begin{equation}
f(x,t)=\frac{1}{\sqrt{\frac{2\pi\sigma^{2}t}{\tau}}}\exp(-\frac{\tau x^{2}}{2\sigma^{2}t})\;\rightarrow\;\tilde{f}(k,s)\thicksim\frac{1}{s+\frac{\sigma^{2}}{2\tau}k^{2}}\end{equation}


where $\frac{\sigma^{2}}{2\tau}$ is the diffusion constant.

transforming back to the space-time gives

\begin{equation}
f(x,t)\thicksim\int_{0}^{\infty}\int_{-\infty}^{\infty}dkdse^{-ikx-st}\frac{\tau s{}^{\beta-1}}{\tau s^{\beta}+\lambda k^{\alpha}}=\int_{-\infty}^{\infty}dke^{-ikx}E_{\beta}(-k^{\alpha}t^{\beta})\end{equation}


$E_{\beta}(-k^{\alpha}t^{\beta})$ (Mittag-Leffl{}er function) defi{}ned
by

\begin{equation}
E_{\beta}(z)=\sum\frac{z^{n}}{\Gamma(1+\beta n)}\end{equation}


One can see the Fourier function as $\tilde{f}(t^{\beta}k^{\alpha})$,
i.e the space coordinate function is $f(\frac{x^{\alpha}}{t^{\beta}})$.

From here one can extract the width time dependance

\begin{equation}
x(t)\sim t^{\frac{\beta}{\alpha}}\end{equation}


according to the power low of $x(t)$ we can classify the different
diffusion types.

$2\beta=\alpha$ is the normal diffusion case, $2\beta>\alpha$ is
the superdiffusion case, $2\beta<\alpha$ is the superdiffusion case.

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\begin{center}\emph{\noun{Appendix}}\end{center}

\begin{center}\emph{\noun{Montroll-Weiss Equation}}\end{center}

The equation is in the context of a random walker.

In order to derive the equation we will define few PDFs
\begin{enumerate}
\item $P(t)=1-\intop_{0}^{t}d\tilde{t}f(\tilde{t})$ - the probability that
no step is taken during a time t
\item $Q(x,t)$ - the probability that the walker arrived at position x
at time t
\item $f(x,t)=\int_{0}^{t}d\tilde{t}P(t-\tilde{t})Q(x,\tilde{t})$ - the
PDF that the walker at position x at time t.
\end{enumerate}
$Q(x,t)$ can be written as follow: $Q(x,t)=f(x-\tilde{x)f}(t-\tilde{t})Q(\tilde{x},\tilde{t})$
for a constant $(\tilde{x},\tilde{t})$ - the last step length and
time.

However we have to integrate over all posible last steps, i.e

\begin{equation}
Q(x,t)=\int_{-\infty}^{x}d\tilde{x}\int_{0}^{t}d\tilde{t}f(x-\tilde{x})f(t-\tilde{t})Q(\tilde{x},\tilde{t})+\delta(x)\delta(t)\end{equation}


where the last term takes into acount the initial condition $Q(x=0,t=0)=1$,
$f(\tilde{t})$=0 for $t\leq\tilde{t}$. 

note that $Q(x,t)$ is a convolution of $f(x),f(t)$ and $Q(x,t)$
with respect to both, $x$ and $t$

while $f(x,t)$ is a convolution of $F$ and $Q$ with respect to
$t$ only.

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Using the convolution theorem, we can fined Fourier-Laplace transform
of $Q(x,t)$ and $f(x,t)$

\begin{equation}
\tilde{Q}(k,s)=\tilde{f}(k)\tilde{\phi}(s)\tilde{Q}(k,s)+1\rightarrow\tilde{Q}(k,s)=\frac{1}{1-\tilde{f}(k)\tilde{\phi}(s)}\end{equation}


\begin{equation}
\tilde{f}(k,s)=\tilde{P}(s)\tilde{Q}(k,s)\end{equation}


\begin{equation}
\frac{dP(t)}{dt}=\delta(t)-f(t)\;\rightarrow\; s\tilde{P}(s)=1-\tilde{\phi}(s)\end{equation}


\begin{equation}
\tilde{P}(s)=\frac{1-\tilde{\phi}(s)}{s}\end{equation}


assign $\tilde{Q}(k,s)$ and $\tilde{P}(s)$ into $\tilde{f}(k,s)$
gives

\begin{equation}
\tilde{f}(k,s)=\frac{1-\tilde{\phi}(s)}{s}\frac{1}{1-\tilde{f}(k)\tilde{\phi}(s)}\end{equation}


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References

R. Metzer, J. Klafter, The random walk's guide to anomalous diffusion:
A fractional dynamics approach

M. Bazant, course 18.366 Random Walks and Diffusion, MIT

L. Vlahos, H. Isliker, Y. Kominis, K. Hizanidis, Normal and Anomalous
Diffusion: A Tutorial

Hughes B.D. Random Walks and Random Environments, Volume 1
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