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\AtBeginDocument{{\noindent\small
2006 International Conference in Honor of Jacqueline Fleckinger.
\newline {\em Electronic Journal of Differential Equations},
Conference 16, 2007, pp. 1--13.
\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or
http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{1}

\title[\hfilneg EJDE/Conf/16 \hfil   Fractional flux]
{Fractional flux and non-normal diffusion}

\author[A. Abdennadher, M.-C. Neel \hfil EJDE/Conf/16 \hfilneg]
{Ali Abdennadher, Marie-Christine Neel}  % in alphabetical order

\address{Ali Abdennadher \newline
Department of mathematics,
Institut National des Sciences Appliqu\'ees et de Technologie,
Centre Urbain Nord,
BP 676 Cedex 1080 Charguia Tunis, Tunisie}
\email{ali\_abdennadher@yahoo.fr}

\address{Marie-Christine Neel \newline
UMRA 1114 Climat Sol Environnement,
University of Avignon,
33, rue Pasteur, 84000 Avignon, France}
\email{marie-christine.neel@univ-avignon.fr}


\thanks{Published May 15, 2007.}
\subjclass[2000]{26A33, 45K05, 60E07, 60J60, 60J75, 83C40}
\keywords{Diffusion equations; fractional calculus; random walks;
\hfill\break\indent stable probability  distributions}

\dedicatory{Dedicated to Jacqueline Fleckinger on the occasion of\\
 an international conference in her honor}

\begin{abstract}
 Fractional diffusion equations are widely used for mass spreading in
 heterogeneous media. The correspondence between fractional equations and
 random walks  based upon stable L\'evy laws, keeps in analogy with that
 between heat equation and Brownian motion.
 Several definitions of fractional derivatives yield operators,
 which coincide on a wide domain and can be used
 in fractional partial differential equations.
 Then, the various definitions are useful in different purposes:
 they may be very close to some physics, or to numerical schemes,
 or be based upon important mathematical properties.
 Here we present a definition, which enables us to describe
 the flux of particles, performing  a random walk.
 We show that it is a left inverse to fractional integrals.
 Hence it coincides with Riemann-Liouville and Marchaud's derivatives
 when applied to functions, belonging to suitable domains.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction}

 Fick's law is a basic tool for the transport of dissolved matter.
When it holds, the concentration of solute evolves according to
heat equation. Nevertheless, there exist heterogeneous media where
experimental evidence indicates that dispersion does not obey
Fick's law and heat equation: data from electronics \cite{SchL1}
\cite{SchL2} and passive tracer experiments
\cite{Ben00}\cite{Ben01} show heavy tails, apparently connected
with statistics giving some importance to extreme events, in
similarity with densities of stable probability L\'evy laws.
Gaussian statistics are a limiting case of the latters, and
distribute successive jump lengths of Brownian motions, which
serve as small scale models for mass transport. On the macroscopic
level, the concentration of a cloud of particles performing
Brownian motions, satisfies  Fick's law and heat equation.

 Data with heavy tails correspond to L\'evy flights, which also are random walks, with jump lengths distributed according to stable  L\'evy laws.
The correspondence between small and large scales is obtained by
means of time and length references which we let tend to zero. If,
moreover, they satisfy some scaling relation, the concentration of
walkers tends to a limit,  evolving according to a variant of heat
equation, with Laplacean being replaced by derivatives of
fractional order. They  are non-local operators.

For unbounded domains, these results were derived from Generalized
Master  Equation via Fourier's transform \cite{Com} \cite{Gorsc2}.
They were extended to semi-infinite and bounded domains after some
adaptations, under hypotheses which we want to escape \cite{Kre}.
Here we present a novel definition of Riemann-Liouville or
Marchaud's derivatives. It seems to us that it makes it possible
to interpret fluxes directly for random walks observed from the
macroscopic point of view, without passing through Generalized
Master Equation, Fourier's transform and space-fractional heat
equation. Later, the gain in simplicity will allow us to tackle
problems mixing, for instance, boundary conditions and skewness.

\section{A new expression for Riemann-Liouville's and Marchaud's fractional
derivatives}

 Riemann-Liouville and Marchaud's fractional  derivatives interpolate between
integer orders of differentiation, and  generalize many aspects of
the notion. Real and complex orders of differentiation can be
defined in this context. Here we aim at introducing a new way of
presenting fractional calculus, in connection with particle
counting and random walks. In this purpose, we only need to
consider real valued orders of differentiation. And for the
moment, in a sake of simplicity, we focus on one-dimensional
problems, future works will be devoted to more general dimensions.

After having recalled essentials of the most widely used basic tools of
fractional calculus, we  show that a novel definition yields more or less
similar objects. Then, we outline the connection with fluxes of particles.

\subsection{Riemann-Liouville Fractional integrals and derivatives}

 Let $ \alpha$ be positive (the definition applies to complex numbers
with positive real part): the left-sided fractional integral of
order $\alpha$, computed over $[a,x]$ is
%
\begin{equation} \label{e1}
I_{a,+}^{\alpha}\varphi(x)=\frac{1}{\Gamma(\alpha)}\int_a^x(x-y)^ {\alpha-1}
 \varphi(y)dy
\end{equation}
according to \cite{Sa}. Here we will mainly consider the case
$a=-\infty$, with the simplified notation
$I_{+}^{\alpha}\varphi(x)=I_{-\infty,+}^{\alpha}\varphi(x)$ of
\cite{Rub}. Right-sided integrals are
\begin{equation} \label{e2}
I_{b,-}^{\alpha}\varphi(x)=\frac{1}{\Gamma(\alpha)}\int_x^b(y-x)^ {\alpha-1}
\varphi(y)dy
\end{equation}
with $I_{-}^{\alpha}\varphi(x)=I_{+\infty,-}^{\alpha}\varphi(x)$.

The corresponding left-sided Riemann-Liouville derivative  of
order $ \alpha$ is
\begin{equation} \label{e3}
\mathcal{D}_{+}^{\alpha}\varphi(x)=(\frac{d}{dx})^{[\alpha]+1}I_+
^{1-\{\alpha\}}=(\frac{d}{dx})^{[\alpha]+1}\frac{1}{\Gamma([\alpha]+1-\alpha)}
\int_{-\infty}^x(x-y)^ {-\{\alpha\}} \varphi(y)dy,
\end{equation}
where $[.]$ denotes integer part, while $\{.\}$ is defined by
$\alpha=[\alpha]+\{\alpha\}$. The right-sided Riemann-Liouville
derivative is
\begin{equation} \label{e4}
\mathcal{D}_{-}^{\alpha}\varphi(x)=(-\frac{d}{dx})^{[\alpha]+1}
I_-^{1-\{\alpha\}}=(-\frac{d}{dx})^{[\alpha]+1}
\frac{1}{\Gamma([\alpha]+1-\alpha)}\int_{x}^{+\infty}(y-x)^ {-\{\alpha\}}
\varphi(y)dy.
\end{equation}
When $\alpha$ is a positive integer, $\mathcal{D}_{-}^{\alpha}$
and $\mathcal{D}_{+}^{\alpha}$ are usual right and left-sided
derivatives or order $\alpha$. A natural question is of whether
fractional derivatives defined by \eqref{e3} and \eqref{e4} share with
derivatives of integer order the property of being left inverses
to the corresponding integrals. In fact, when $\varphi$ is in
$L^1_{\rm loc}( {\mathbb{R}})$, if moreover the integrals
$I_{\pm}^{[\alpha]+1}$ is absolutely convergent, we have
$(\mathcal{D}_{\pm}^{\alpha}I_{\pm}^\alpha\varphi)(x)=\varphi(x)$
a.e., due to  \cite[Lemma 4.7]{Rub}. When the above hypotheses are
satisfied,   $\mathcal{D}_{\pm}^{\alpha}$ can be thought of as
being a left inverse to  ${I}_{\pm}^{\alpha}$, which fails to hold
if $I_\pm^1\varphi$ and $I_\pm^{[\alpha]+1}\varphi$ do not belong
to  $L^1( {\mathbb{R}})$, even for $\varphi$ in  $L^p({\mathbb{R}})$ with
$1<p<1/\alpha$. Marchaud's definition seems to give a more
appropriate left inverse to fractional integrals.


\subsection{ Marchaud's Fractional  derivatives}

 Marchaud's definition combines generalized finite differences and
fractional integrals. A rather general definition of finite differences
was given by   \cite{Rub} in the form
$$
(\Delta_t^n f)(x)= \frac{1}{d_n}\left|
\begin{matrix} f(x-\lambda_0 t) &1& \lambda_0 & \dots
 & \lambda_0 ^{n-1}\\
 f(x-\lambda_1 t) &1& \lambda_1 & \dots  & \lambda_1 ^{n-1} \\
.& . & . & \dots  & . \\
 f(x-\lambda_n t) &1& \lambda_n & \dots  & \lambda_n  ^{n-1} \\
\end{matrix}
\right|,
$$
with
$$
d_n=\left|\begin{matrix}1&\lambda_1 & \dots  & \lambda_1^{n-1}\\
.&. & \dots  & .\\
1&\lambda_n & \dots  & \lambda_n^{n-1}\\
\end{matrix}\right|.
$$
For $\lambda_0=0$,  $\lambda_1=1$, \dots $\lambda_n=n$, and with
$T_t$ denoting the translation of amplitude $t$, the definition of
$\Delta^n_t$  just becomes
$(Id-T_t)^nf(x)=\Sigma_{j=0}^n(^n_j)(-1)^jf(x-jt)$, which is
$f(x-t)-f(x)$ for $n=1$. With these notations, Marchaud's
derivative $D_{\pm}^\alpha$ of function $f$ is the limit, when
$\varepsilon\to 0+$ of
\begin{equation} \label{e5}
 D_{ \pm,\varepsilon}^{\alpha}f(x)=\frac{1}
{\int_0^{+\infty}t^{-\alpha-1}(1-e^{-t})^n dt}
\int_{\varepsilon}^{+\infty}t^ {-\alpha-1} \Delta^n_{\pm t}f(x)dt,
\end{equation}
with $n>\alpha$. For $0<\alpha<1$, we have $n=1$ and (5) becomes
$$
D_{\pm,\varepsilon}^{\alpha}f(x)=\frac{-1}{\Gamma(-\alpha)}
\int_{\varepsilon}^{+\infty}t^ {-\alpha-1} [f(x)-f(x\mp t)]dt .
$$
In this definition, the relative freedom let to $\Delta^n_{t}$, is
useful when $\alpha$ is a complex number. In view  of our
objective, we will focus on real valued orders for integrals and
derivatives, hence  $\lambda_0=0$, $\lambda_1=1$, \dots
$\lambda_n=n$ with $ \Delta_{ t}^n f(x)=(Id-T_t)^nf(x)$, used by
\cite{Sa}   will be enough. For $\alpha=1$, we have to put $n=2$
in (5) if we want to use this expression, but we also can consider
that $ D_{ \pm}^{\alpha}$ is the usual left or right-sided
derivative of order $\alpha$ when $\alpha $ is a non-negative
integer.

We thus have a left inverse for $I_{\pm}^\alpha$ in a wider
domain, which in some sense is optimal, since it provides a
characterization of  $I_{\pm}^\alpha L^p$ for $1<p<1/\alpha$.
Indeed, for $0<\alpha<1$,   \cite[Theorem 6.2]{Sa} states the
following: If the $L^p({\mathbb{R}})$ limit of $ D_{
\pm,\varepsilon}^{\alpha}f$ exists  when $\varepsilon\to 0+$, or
if $sup_{\varepsilon>0}\Vert  D_{
\pm,\varepsilon}^{\alpha}f\Vert_{L^p(\mathbb{R})} $ is finite, if
moreover, $f$ belongs to $L^p(\mathbb{R})$ with $1\leq r<\infty$, then
$f$ belongs to  $I_{\pm}^\alpha L^p(\mathbb{R})$ and there exists
$\varphi$ s.t. $f(x)=I_{\pm}^\alpha\varphi(x)$ almost everywhere.
in $\mathbb{R}$. The theorem was stated in $ L^p(\mathbb{R})$, but the proof
adapts without any modification to $ L^p]- \infty,a]$ for
$D_+^\alpha$ and to $ L^p[a,+ \infty[$ for $D_-^\alpha$.
Derivatives  $D_\pm^\alpha$ and  $\mathcal{D}_\pm^\alpha$
coincide for functions of the form $I_\pm^\alpha\varphi$ with
$\varphi$ in $L^1_{\rm loc}$ such that  $I_\pm^{[\alpha]+1}$ converges
absolutely \cite{Rub}.

 Other expressions yield the left inverse of   $I_\pm^\alpha$. Among them,
 the Gr\" unwald-Letnikov fractional derivative \cite{Sa} or order
$\alpha$ of $f$ is the limit, when  mesh $h$ tends to zero, of
$h^{ -\alpha}$ times the series
$\Sigma_{k=0}^{\infty}(-1)^k(^\alpha_k)f(x-kh)$. It  provides
useful approximations to Riemann-Liouville or Marchaud's
derivatives, connected with finite differences numerical schemes.
Here we present a further expression for the left inverse of
${I}_\pm^\alpha$, not very different  from  Gr\" unwald-Letnikov
operator, since it contains an  integrals in place of the above
evocated  series. Then, we will discuss the physical meaning.

\subsection{ A new expression for the inverse of  $I_{\pm}^\alpha$   }

Here we consider $0<\alpha\leq 1$. With some modifications, the
following adapts to all positive values of $\alpha$.

{\bf Notation} Let $F$ be a function, while $l$ is positive. Set
\begin{equation} \label{e6}
 \mathcal{W}_{l,\pm}^{\alpha,F}f(x)=l^{-\alpha-1}
\int_{0}^{+\infty}f(x\mp t)F(t/l)dt.
\end{equation}
The limit of $ \mathcal{W}_{l,\pm}^{\alpha,F}f$ when $l$ tends to zero,
if it ever exists, will be denoted by  $ { W}_{\pm}^{\alpha,F}f$.
\begin{itemize}
\item We will say that $F$ satisfies Hypothesis (H1) if $F$ belongs
to $L^1({\mathbb{R}}^+)$ with $\int_0^{\infty}F(t)dt=0$.

\item We will say that $F$ satisfies Hypothesis (H2) if, in a
neighborhood $[A,+\infty[$ of $+\infty$, there exists a function $F_1$
such that  $\int_A^{+\infty}y^\alpha\vert F_1(y)\vert dy<\infty$ and
$F(x)=F_1(x)+\lambda x^{-\alpha-1}$,  for $0<\alpha<1$ but
$F(x)=F_1(x)+\lambda x^{-2- \varepsilon}$ with $\varepsilon>0$ for
$ \alpha=1$.
\end{itemize}
We will see that (H1) and (H2) imply that
$ { W}_{\pm}^{\alpha,F}$ is a left inverse to  $ { I}_{\pm}^{\alpha}$.
In this purpose, let us consider
$ { W}_{\pm}^{\alpha,F}\circ { I}_{\pm}^{\alpha}$.

Let $\varphi$ belong to $L^p[a,+\infty[$. We have
  $$
\mathcal{W}_{l,-}^{\alpha,F}\circ { I}_{-}^{\alpha}\varphi(x)
=\frac{l^{-\alpha-1}}{\Gamma(\alpha)}\int_0^{+\infty}F(t/l)
\int_{x+t}^{+\infty}\varphi(y)(y-x-t)^{\alpha-1}dydt.
$$
 Setting $t=lT$ yields
\begin{align*}
\mathcal{W}_{l,-}^{\alpha,F}\circ { I}_{-}^{\alpha}\varphi(x)
&=\frac{l^{-\alpha}}{\Gamma(\alpha)}\int_0^{+\infty}F(T)
\int_{x+lT}^{+\infty}\varphi(y)(y-x-lT)^{\alpha-1}dy\,dT\\
&=\frac{1}{\Gamma(\alpha)}\int_0^{+\infty}F(T)
\int_{T}^{+\infty}\varphi(x+l\theta)(\theta-T)^{\alpha-1}d\theta \,dT,
\end{align*}
 with $y=x+l \theta$.
Then, Fubini's theorem yields
\begin{equation} \label{e7}
\mathcal{W}_{l,-}^{\alpha,F}\circ { I}_{-}^{\alpha}\varphi(x)
=\frac{1}{\Gamma(\alpha)}\int_0^{+\infty}\varphi(x+l\theta)
\int_{0}^{\theta}F(T)(\theta-T)^{\alpha-1}\,dT\,d\theta,
\end{equation}
as soon as
$I_+^\alpha(HF)(\theta)=\int_{0}^{\theta}F(T)(\theta-T)^{\alpha-1}\,dT$
is integrable in ${\mathbb{R}}^+$. Let us use this point, which will be
stated in Lemma \ref{lem1} below. Let $H$ denote Heaviside's function: on
the right-hand side of \eqref{e7} we have $\int_{\mathbb{R}}\varphi(x+l
\theta)(I_+^\alpha(HF))(\theta)d\theta$ which,  by \cite[Theorem
1.3]{Sa} is an approximation of
$\int_0^{+\infty}I_+^{\alpha}(HF)(\theta)d\theta$ times Identity
in $L^p$.

For  $\varphi$ in $L^p]-\infty,a]$, instead of \eqref{e7} we have
\begin{equation} \label{e8}
\mathcal{W}_{l,+}^{\alpha,F}\circ { I}_{+}^{\alpha}\varphi(x)
=\frac{1}{\Gamma(\alpha)}\int_0^{+\infty}\varphi(x-l\theta)
\int_{0}^{\theta}F(T)(\theta-T)^{\alpha-1}\,dTd\theta.
\end{equation}
 Hence the following Theorem holds.

\begin{theorem} \label{thm1}
Suppose $F$ satisfies hypotheses (H1) and (H2),
with $0<\alpha\leq 1$.
\begin{itemize}
\item[(i)] For $\varphi$ in $L^p[a,+\infty[$,
$ \mathcal{W}_{l,-}^{\alpha}\circ { I}_{-}^{\alpha}\varphi$ tends
in $L^p[a,+\infty[$ to $\int_0^{+\infty}I_+^\alpha HF(t)dt\times\varphi$
when $l$ tends to zero, and pointwise everywhere $\varphi$ is right continuous.

\item[(ii)] For $\varphi$ in $L^p]-\infty,a]$,
$ \mathcal{W}_{l,+}^{\alpha}\circ { I}_{+}^{\alpha}\varphi$
tends in $L^p]-\infty,a]$ to $\int_0^{+\infty}I_+^\alpha HF(t)dt\times\varphi$
when $l$ tends to zero, and pointwise everywhere $\varphi$ is left continuous.
\end{itemize}
\end{theorem}

It remains to prove the following lemma.

\begin{lemma} \label{lem1}
 If $F$ satisfies (H1) and (H2), with $0<\alpha\leq 1$,
then $\int_{0}^{\theta}F(T)(\theta-T)^{\alpha-1}\,dT$ is integrable
in ${\mathbb{R}}^+$.
\end{lemma}

\begin{proof} If $F$ is as $F_1$  in hypothesis (H2)'s statement,
\cite[Lemma 4.12]{Rub} shows that
 $$
\Gamma(\alpha) I_-^\alpha(HF)(\theta)=\int_0^\theta F(T)
(\theta-T)^{\alpha-1}\,dT
$$
 is in $L^1$. It is enough to prove the lemma for
$F=-\frac{1}{\alpha}\chi_{[0,1]}+x^{-\alpha-1}\chi_{[1,+\infty[}$
if $\alpha$ is less than $1$,  for
$F=-\frac{1}{1+\varepsilon}\chi_{[0,1]}+x^{-2-\varepsilon}\chi_{[1,+\infty[}$
if $\alpha$ is equal to $1$, since modifying $F_1$ will immediately
lead to the general case. For $\alpha=1$, the result is obvious, for
$\alpha$ less than $1$, we have
 $$
\int_0^x(x-y)^{\alpha-1}\chi_{[0,1]}(y)dy=\frac{x^\alpha-(x-1)^\alpha}{\alpha}
$$
for $x>1$, and
 $$
\int_0^x(x-y)^{\alpha-1}y^{-\alpha-1}\chi_{[1,+\infty[}(y)dy
=x^{-1}(G(1)-G(1/x)+\frac{x^\alpha-1}{\alpha})
$$
 when $x$ is large enough, with $G$ being defined by
$G(X)=\int_0^X[(1-z)^{\alpha-1}-1]z^{-\alpha-1}dz$. From this we deduce
\begin{equation} \label{e9}
\begin{aligned}
&\int_0^xF(t)(x-t)^{\alpha-1}dt\\
&=\alpha^{-1}(x^{\alpha-1}-\alpha^{-1}x^\alpha(1-(1-1/x)^\alpha))
+x^{-1}(G(1)-\alpha^{-1})-x^{-1}G(1/x).
\end{aligned}
\end{equation}
 Function $\frac{(1-t)^{\alpha-1}-1}{t}$ is continuous and integrable
in $[0,1[$. In the neighborhood of $0$,
$\frac{(1-t)^{\alpha-1}-1}{t}t^{-\alpha}$ is equivalent to
$(1-\alpha)t^{-\alpha}$, hence $G(1/x)$ is equivalent to
$x^{\alpha-1}$ when $x$ is large. Hence $x^{-1}G(1/x)$ is integrable
in a neighborhood of $+\infty$. It also is the case for
$\alpha^{-1}[x^{\alpha-1}-\alpha^{-1}x^\alpha(1-(1-1/x) ^\alpha)$.
We now will check that $G(1)-\alpha^{-1}$ is zero. To see this,
set $g(p,q)=\int_0^1((1-t)^{q-1}-1)t^{p-1}dt$. For  complex valued $p$ and $q$ satisfying $Re(p)>0$ and $Re(q)>0$,  $\int_0^1(1-t)^{q-1}t^{p-1}dt$ is a beta function \cite{AbSt} and we have
\begin{equation} \label{e10}
g(p,q)=\frac{\Gamma(p)\Gamma(q)}{\Gamma(p+q)}-\frac{1}{p}.
\end{equation}
 Let us fix $q=\alpha$, and vary the complex number $p$: $t^p$ is a function
 of $p$, whose derivative $t^pLn(t)$ is dominated by the $L ^1]0,1[$
function  $t^p \vert Ln(t)\vert $ for $Re(p)\geq p_0>-1$, so that,
by dominated convergence, $g(p,\alpha)$ is  derivable with respect to $p$.
Hence it is analytic  for $Re(p)\geq p_0>-1$. Since
$\frac{\Gamma(q)}{\Gamma(p+q)}$ is also analytic in the neighborhood of
$0$ while $\Gamma(p)$ has a simple pole with residuum $1$, the right-hand
side of \eqref{e10} is holomorphic  for $Re(p)\geq p_0>-1$.
Hence  relation \eqref{e10}
holds for $p=-\alpha$, and Lemma \ref{lem1} is proved.
\end{proof}

Therefore Theorem \ref{thm1} holds. It states that the operator
$\mathcal{ W}_{l,-}^{\alpha,F}$, which is defined on
$I_-^{\alpha}L^p[a;+\infty[$, has a limit in  $L^p[a;+\infty[$
when $l$ tends to zero. Up to multiplication by a function of $F$
and $\alpha$, the limit is a left inverse to $I_-^{\alpha}$, hence
it coincides with $D_-^{\alpha}$. Similarly,
$\mathcal{W}_{l,+}^{\alpha,F}$, defined in
$I_+^{\alpha}L^p]-\infty,a]$, tends to  $D_+^{\alpha}$, times a
function of $F$ and $\alpha$. Theorem \ref{thm1} adapts to higher
values of $\alpha$, provided Hypothesis 1 is made stronger.

 For values of $\alpha$ between $0$ and $1$,  will see that Theorem
\ref{thm1} allows us to represent the flux of particles within the
frame work of a
 wide class of Random Walks.


\section{Particles  flux for L\'evy Flights, in the macroscopic limit}

  Brownian Motion is a particular case of  L\'evy Flights. The latters
are Continuous Time Random Walks: a large number of particles
perform a succession of independent jumps, whose lengths $X_i$ are
identically  distributed. To be more precise, with $l$ being a
length scale, the density of  $X_i/l$ is the normalized
$\alpha$stable L\'evy density $L_{\alpha,\theta}$  of exponent
$\alpha$ between $1$ and $2$ and with skewness parameter $\theta$
(see Appendix A). Waiting times $T_i$ between successive jumps are
such that the independent random variables $T_i/\tau$ have density
$\psi$, whose average is $1$. Here, for definiteness, we set $
\psi(t)=e^{-t}$. Looking at the cloud of particles from the
macroscopic point of view means that we let length and time scales
$l$ and $\tau$ tend to zero. Then, if the scaling relation
$l^\alpha/\tau=K$  holds \cite{Com} \cite{Gorsc2}, the probability
of finding a particle in a given interval tends to  a limit, which
has a density satisfying a space-fractional diffusion equation
such as \eqref{e17}. This  implies that the flux of particles satisfies a
fractional generalization of Fick's law \cite{MainPara}. All these
results are based upon Generalsized Master Equation and Fourier's
analysis.

In fact, we will see that Theorem \ref{thm1} connects more
directly particle  flux and fractional derivatives.


\subsection{Computing the flux for L\'evy flights  with length scale
$l$ and mean waiting time $\tau$ satisfying $l^\alpha/\tau=K$ }

 For a given particle, the location after $n^{th}$ jump is
$\Sigma_{i=0}^n X_i$, and it happens at time $\Sigma_{i=0}^n T_i$.
Let us denote by $\mu(.,t)$ the  measure giving the probability
$\mu(I,t)$  that the particle be in interval $I$ at time $t$. With
this notation, the balance of particles crossing abscissa $x$
during $[t,t+dt[$ is the difference of two expressions. The first
one is the probability
$\int_{-\infty}^xF_{\alpha,\theta}^d(\frac{x-y}{l})d\mu(y,t)\frac{\psi(0)}{
\tau}dt$ of crossing $x$ to the right, with
$F_{\alpha,\theta}^d(y/l)$ being the probability
$\int_{y}^{+\infty}\frac{1}{l}L_{\alpha,\theta}(z/l)dz
=\int_{y/l}^{+\infty}L_{\alpha,\theta}(z)dz $ for a jump to have
an amplitude of more than $y$.
 The second  is the probability
$\int_x^{+\infty}F_{\alpha,\theta}^g(\frac{x-y}{l})d\mu(y,t)
\frac{\psi(0)}{ \tau}dt$ of crossing $x$ to the left, with
$F_{\alpha,\theta}^g(-y/l)$ being the probability
  $\int_{-\infty}^{-y/l}\frac{1}{l}L_{\alpha,\theta}(z/l)dz$
for a jump to have an amplitude of more than $y$, but to the left.
The flux  is the probability rate, hence the  following difference:
$$
Kl^{-\alpha}\Big[\int_{-\infty}^xF_{\alpha,\theta}^d(\frac{x-y}{l})d\mu(y,t)
-\int_x^{+\infty}F_{\alpha,\theta}^g(\frac{x-y}{l})d\mu(y,t)\Big].
$$
When $\mu(.,t)$ has density $C(.,t)$, the flux
$Q_l^{\alpha,\theta}C(.,t)(x)$ is given by
\begin{equation} \label{e11}
Q_l^{\alpha,\theta}C(.,t)(x)=Kl^{-\alpha}\Big[
\int_0^{+\infty}C(x-y,t)F_{\alpha,\theta}^d(\frac{y}{l})dy
-\int_0^{+\infty}C(x+y)F_{\alpha,\theta}^g(\frac{-y}{l})dy\Big].
\end{equation}
Both integrals are similar to \eqref{e6}, except that  $F_{\alpha,\theta}^d$ and
$F_{\alpha,\theta}^g(-.)$  satisfy (H2) with $\alpha-1$ instead of $\alpha$, according to Appendix A,
 but of course not (H1).  Appendix B shows that
 $\int_0^{+\infty}F_{\alpha,\theta}^d(y)dy
=\int_0^{+\infty}F_{\alpha,\theta}^g(-y)dy
=\mathcal{I}_{\alpha,\theta}$. Hence, with $f _{\alpha,\theta}$
being a compactly supported function of class $L^1$ s.t.
$\int_0^{+\infty}f _{\alpha,\theta}(y)dy=\mathcal{I}_{\alpha,\theta}$,
 setting  $\tilde{F}_{\alpha,\theta}^d=F_{\alpha,\theta}^d-f _{\alpha,\theta}$
and  $\tilde{F}_{\alpha,\theta}^g(-y) =F_{\alpha,\theta}^g(-y)-f
_{\alpha,\theta}(y)$ yields functions, satisfying (H1) and (H2)
with $\alpha-1$ instead of $\alpha$. Then, we have
$Q_l^{\alpha,\theta}C(.,t)(x)=Q_{+,l}^{\alpha,\theta}
C(.,t)(x)-Q_{-,l}^{\alpha,\theta}C(.,t)(x)$,
with
\begin{equation} \label{e12}
\begin{aligned}
Q_{+,l}^{\alpha,\theta}f(x)
&=Kl^{-\alpha}\int_0^{+\infty} {F}_{\alpha,\theta}^d(y/l)(f(x-y)-f(x))dy\\
&=K\Big[ (\mathcal{W}_{l,+}^{\alpha-1,\tilde{F}_{\alpha,\theta}^d}f)(x)
 +l^{-\alpha}\int_0^{+\infty}f_{\alpha,\theta}(y/l)(f(x-y)-f(x))dy\Big]
\end{aligned}
\end{equation}
at the left of $x$, and
\begin{equation} \label{e13}
\begin{aligned}
Q_{-,l}^{\alpha,\theta}f(x)
&=Kl^{-\alpha}\int_0^{+\infty}{F}_{\alpha,\theta}^g(-y/l)(f(x+y)-f(x))dy \\
&=K\Big[ (\mathcal{W}_{l,-}^{\alpha-1,\tilde{F}_{\alpha,\theta}^g(-.)}f))(x)
+l^{-\alpha}\int_0^{+\infty}f_{\alpha,\theta}(y/l)(f(x+y)-f(x))dy\Big]
\end{aligned}
\end{equation}
at the right. Since  $\tilde{F}_{\alpha,\theta}^d$ and
$\tilde{F}_{\alpha,\theta}^g(-.)$  satisfy (H1) and  (H2) with $\alpha-1$ instead of $\alpha$, hence
$(\mathcal{W}_{l,+}^{\alpha-1,\tilde{F}_{\alpha,\theta}^d}f)(x)$  tends to
 $\int_0^{+\infty}I^{\alpha-1}_+(H\tilde{F}_{\alpha,\theta}^d)
(y)dyD^{\alpha-1}_+(f)(x)$  in $L^p]-\infty,a]$ when $f$ belongs to
 $I_+^{\alpha-1}L^p]-\infty,a]$ and
\begin{equation*}
(\mathcal{W}_{l,-}^{\alpha-1,\tilde{F}_{\alpha,\theta}^g(-.)}f)(x)\mbox{
  tends
  to}\;\int_0^{+\infty}I^{\alpha-1}_+(H\tilde{F}_{\alpha,\theta}^g(-.))(y)dyD^{\alpha-1}_-(f)(x)
\end{equation*}
in $L^p[a,+\infty[$  when $f$ belongs to $I_-^{\alpha-1}L^p[a,+\infty[$.
We will see that appropriately choosing $f_{\alpha,\theta}$ allows us
to see on the right-hand sides of
\eqref{e12}  and \eqref{e13} expressions which are
``local fractional derivatives'', in the sense of Kolwankar and Gangal.

\subsection{Kolwankar and Gangal's local fractional derivatives}

  The notion of ``a local fractional derivative'' was  introduced
\cite{KolG} in view of building a tool, designed for the study of
continuous but nowhere differentiable functions  frequently occurring
in Nature and economics. Those fractional derivatives share some
properties with previously defined ones, such as chain rule or
generalized Leibniz rule  \cite{BaVa}. They are very useful for
to compute fractal dimensions of graphs. In fact, they vanish for
smooth enough functions, and hence can become ``invisible''.

For $q$ between $0$ and $1$, the right-sided Kolwankar and Gangal's
\cite{KolG} fractional derivative of order $q$ of function $f$, computed
at $x$, will be denoted by
 $$
D^{KG,q}_+f(x)=\lim_{h\to 0+}\frac{d}{dh}I^{1-q}_{x,+}(f(.)-f(x))(x+h).
$$
Let us suppose that $f$ is continuous is $[x,x+\varepsilon]$, with
positive $\varepsilon$. When the limit exists, it is equal to the
limit, when $h$ tends to $0+$, of
$h^{-1}I^{1-q}_{x,+}(f(.)-f(x))(x+h)$, due to l'H\^ opital's rule
and to $ \lim_{h\to 0}(I^{1-q}_{x,+}(f(.)-f(x))(x+h))=0$ .
Moreover, we have
$h^{-1}I^{1-q}_{x,+}(f(.)-f(x))(x+h)
=\frac{h^{-q}}{\Gamma(1-q)}\int_0^{1}(1-t)^{-q}(f(x+th)-f(x))dt$.

At the left, we have
 $$
D^{KG,q}_-f(x)=\lim_{h\to 0+}\frac{d}{dh}I^{1-q}_{x,-}(f(x)-f(.))(x-h),
$$
also equal to the limit, when $h$ tends to $0+$, of
$h^{-1}I^{1-q}_{x,-}(f(x)-f(.))(x-h)$.
If, for positive and finite $b-a$ and with $q<q+\varepsilon<1$,
function $f$ belongs to  H\" older space $H^{q+\varepsilon}[a,b]$,
then $sup_{x,y\in [a,b]}(\frac{\vert f(y)-f(x)\vert}
{\vert y-x\vert^{q+\varepsilon}})$
is finite, hence Kolwankar and Gangal's local derivatives of order
$q$ are zero in $[a,b]$.

 For $\alpha<2$, an appropriate choice of $f_{\alpha,\theta}$ yields that
the second expressions on the right-hand sides of \eqref{e12} and
\eqref{e13} tend to right and left-sided local fractional  derivatives
of order $\alpha-1$. The choice is
$f_{\alpha,\theta}(t)=\mathcal{I}_{\alpha,\theta}(2-\alpha)
(1-t)^{1-\alpha}\chi_{[0,1]}$; then, we have
$l^{-\alpha}\int_0^{+\infty}f_{\alpha,\theta}(y/l)(f(x+y)
-f(x))dy=\mathcal{I}_{\alpha,\theta}l^{1-\alpha}
(2-\alpha)\int_0^{1}(1-t)^{1-\alpha}(f(x+lt)-f(x))dt$.
Consequently, when $f$ has a local derivative of order $\alpha-1$
at point $x+$,
$l^{-\alpha}\int_0^{+\infty}f_{\alpha,\theta}(y/l)(f(x+y)-f(x))dy$
has a limit when $l$ tends to zero, and the limit is
$\mathcal{I}_{\alpha,\theta}\Gamma(3-\alpha)$ times the right-sided
local derivative of order $1-\alpha$. The same holds at the left
of $x$: $l^{-\alpha}\int_0^{+\infty}f_{\alpha,\theta}(y/l)(f(x-y)-f(x))dy$
tends to  $\mathcal{I}_{\alpha,\theta}\Gamma(3-\alpha)$ times the
left-sided local derivative of order $1-\alpha$. When $f$ is differentiable
 at $x$, the limit is zero.

For  $\alpha=2$, we have the usual derivative instead of
local Kolwankar and Gangal's derivatives,  the above choice of
 $f_{\alpha,\theta}$ is no longer relevant, but the end of next
subsection will show that the method yields Fick's law simply and directly.

We now are ready for looking at the limit of operator flux
$Q^{\alpha,\theta}_{l}$ when $l$ tends to zero.

\subsection{The flux, in the $l\to 0$ limit}

 We know from \eqref{e1} that the flux of particles, passing trough $x$
at time $t$, is
$Q^{\alpha,\theta}_{l}C(.,t)(x)=Q^{\alpha,\theta}_{+,l}C(.,t)(x)
-Q^{\alpha,\theta}_{-,l}C(.,t)(x)$. Moreover, according to \eqref{e12}
and \eqref{e13}, $Q^{\alpha,\theta}_{\pm,l}f(x)$ splits into two terms.
For $0<\alpha<2$, with moreover $D^{\alpha-1}_{\pm}f\in L^p$, the first
one, $K\mathcal{W}_{l,\pm}^{\alpha-1,\tilde{F}_{\alpha,\theta}^d}f$,
 tends to $K\lambda_\pm D_\pm^{\alpha-1}f$ in $L^p$, according to
Theorem \ref{thm1}, with
 \begin{gather} \label{e14}
\lambda_+=\int_0^{+\infty}I_+^{\alpha-1}H\tilde{F}^d_{\alpha,\theta}(y)dy,
\\ \label{e15}
\lambda_-=\int_0^{+\infty}I_+^{\alpha-1}H\tilde{F}^g_{\alpha,\theta}(-y)dy.
\end{gather}
The second term $Kl^{-\alpha}\int_0^{+\infty}f_{\alpha,\theta}(y/l)(f(x\mp y)
-f(x))dy$ in \eqref{e12} and \eqref{e13} tends to
 $K\mathcal{I}_{\alpha,\theta}D_\pm^{KG,\alpha-1}f$ at points where the
local fractional derivative exists. Parameter
$\mathcal{I}_{\alpha,\theta}$ can be computed numerically, with the help
of integral expressions \cite{RWeron} for stable L\'evy distributions.
It also can be deduced from Appendix B. Oppositely, direct computation of
$\lambda_\pm$ from
 \eqref{e14}   and \eqref{e15} is not easy, but can be avoided and
replaced by directly checking
$W_\pm^{\alpha-1,\tilde{F}_{\alpha\theta}^{d,g}}f$ and $D_\pm^{\alpha-1}f$,
 with particular functions $f$ in $I_+^{\alpha-1}]-\infty,a]$ or
$I_-^{\alpha-1}[a,+\infty[$ such that the local derivative is zero.

Let us show that considering $f= \chi_{[1,2[}$ yields $\lambda_-$. Indeed,
for $x\in ]1,2[$, the local derivative exists and  is equal to zero, while
we have
\begin{equation*}
Q_{-,l}^{\alpha,\theta}\chi_{[1,2[}(x)=l^{-\alpha}\int_{2-x}^{+\infty}{F}^g_{\alpha,\theta}(-y/l)dy
= C_{\alpha,-\theta}\frac{(2-x)^{1-\alpha}}{\alpha-1}+O(l)
\end{equation*}
 for $1<\alpha<2$, with $C_{\alpha,-\theta}$ being defined by \eqref{e17}. But
\begin{equation*}
D_-^{\alpha-1}\chi_{[1,2[}(x)=\frac{-1}{1-\alpha}
\int_{2-x}^{+\infty}y^{-\alpha}dy=-\frac{(2-x)^{1-\alpha}}{\Gamma(2-\alpha)}.
\end{equation*}
For $x\geq 2$, we have $D_-^{\alpha-1}\chi_{[1,2[}(x)=0$, hence
$D_-^{\alpha-1}\chi_{[1,2[}$ belongs to $L^p[1,+\infty[$ for
$1\leq p<\frac{1}{\alpha-1}$, so that $\chi_{[1,2[}$ belongs to
$I^{\alpha-1}_-L^p[1,+\infty[$.  Hence, according to
\cite[Theorem 6.2]{Sa}, we have
$\lambda_-=-\frac{\Gamma(2-\alpha)}{\alpha-1}C_{\alpha,-\theta}
=\frac{\sin{\frac{\pi}{2}(\alpha+\theta)}}{\sin{\pi\alpha}}$,
and similarly
$\lambda_+=\frac{\sin{\frac{\pi}{2}(\alpha-\theta)}}{\sin{\pi\alpha}}$.
Hence for sufficiently well-behaved functions (in $L^p({\mathbb{R}})\cap
H^{\alpha-1+ \varepsilon}({\mathbb{R}})$) the limit of the operator,
giving the flux is
 \begin{equation} \label{e16}
f\mapsto K(\frac{\sin{\frac{\pi}{2}(\alpha-\theta)}}{\sin{\pi\alpha}}
D_+^{\alpha-1}f-\frac{\sin{\frac{\pi}{2}(\alpha+\theta)}}{\sin{\pi\alpha}}
D_-^{\alpha-1}f),
\end{equation}
which is a fractional variant of Ficks law. In fact, for $\alpha=2$,
the method has to be slightly adapted but yields Fick's law itself.

In subsection 2.2, we pointed out that case $\alpha=2$ has to be
considered  separately. To do this, take $A(l)$ a function,
tending to $+\infty$ when $l$ tends to zero, with  $lA(l)$ tending
to zero: for instance, we can choose $A(l)=l^{-1/2}$. Parameter
$\theta$ is equal to zero, $L_\alpha^\theta$ is even and
superscripts $d$ and $g$ in ${F}_{2,0}^{g,d}$ are of no use:
instead we put  ${F}_{2,0}$. We have
\begin{align*}
Q_{-,l}^{2,0}f(x)&=Kl^{-2}\int_0^{+\infty}{F}_{2,0}(y/l)(f(x+y)-f(x))dy \\
&=Kl^{-1}\int_0^{+\infty}{F}_{2,0}(y)(f(x+ly)-f(x))dy \\
&=K\int_0^{A(l)}{F}_{2,0}(y)\frac{f(x+ly)-f(x)}{ly}dy\\
&\quad  +l^{-1}K\int_{A(l)}^{+\infty}{F}_{2,0}(y)(f(x+ly)-f(x))dy.
\end{align*}
If $f$ is differentiable at point $x$,
$$
\int_0^{A(l)}{F}_{2,0}(y)\frac{f(x+ly)-f(x)}{ly}dy
$$
 tends to the usual derivative $f'(x)$, times
$\int_0^{+\infty}{F}_{2,0}(y)ydy$, itself equal to $1/2$ due to
${F}_{2,0}(x)=\int_x^{+\infty}\frac{1}{2\sqrt{\pi}}e^{-y^2/4}dy$. And
$l^{-1}\vert \int_{A(l)}^{+\infty}{F}_{2,0}(y)(f(x+ly)-f(x))dy\vert$ is less
than $l^{-2}{F}_{2,0}(A(l))\int_{lA(l)}^{+\infty}\vert f(x+y)-f(x)\vert dy$,
which tends to $0$ when $f$ is fixed in $L^1$. Similar results are obtained
on the left side of $x$, hence for $\alpha=2$, in the limit ``$l$ tends to
zero'' operator flux tends to $K\frac{d}{dx}f(x)$ which satisfies Fick's
law.

 The fractional version \eqref{e16} implies a space-fractional variant
of heat equation.

\subsection{Space-fractional heat equation}

For functions $f$ of the form $I_\pm^{\alpha-1}\varphi$ with $\varphi$ in
$L^p({\mathbb{R}})\cap L^1({\mathbb{R}})$, if, moreover, $f$ belongs to
$H^{\alpha-1+\varepsilon}$, \eqref{e16} is of the form
\begin{align*}
&\frac{K}{\Gamma(2-\alpha)}\frac {d}{dx}
\Big[\frac{\sin{\frac{\pi}{2}(\alpha-\theta)}}{\sin{\pi\alpha}}
\int_{-\infty}^x (x-y)^{1-\alpha}f(y)dy \\
&+\frac{\sin{\frac{\pi}{2}(\alpha+\theta)}}{\sin{\pi\alpha}}
\int_x^{+\infty} (y-x)^{1-\alpha}f(y)dy \Big],
\end{align*}
which yields  operator flux $Q^*$ for particles performing L\'evy Flights
in the diffusive limit ($l$ and $\tau$ tend to zero with $l^\alpha/\tau=K$),
when $f$ is the concentration at time $t$. Casting the expression for
$Q^*C^*$ into mass conservation law $\partial_t C^*=-\partial_x Q^*C^*$
yields for the evolution of the concentration $C^*(x,t)$
  \begin{equation} \label{e17}
\partial_t C^*(x,t)=K\nabla_{x,\alpha}^\theta C^*(x,t)
 \end{equation}
with $\nabla_{x,\alpha}^\theta$ being the Riesz-Feller fractional
derivative of order $\alpha$ and skewness parameter $\theta$ \cite{GoMa99},
defined by
\begin{equation} \label{e18}
\begin{aligned}
(\nabla_{x,\alpha}^\theta f)(x)
&= \frac{-1}{\Gamma(2-\alpha)}\frac {d^2}{dx^2}
 \Big[\frac{\sin{\frac{\pi}{2}(\alpha-\theta)}}{\sin{\pi\alpha}}
 \int_{-\infty}^x (x-y)^{1-\alpha}f(y)dy\\
&\quad +\frac{\sin{\frac{\pi}{2}(\alpha+\theta)}}{\sin{\pi\alpha}}
\int_x^{+\infty} (y-x)^{1-\alpha}f(y)dy \Big],
\end{aligned}
 \end{equation}
in agreement with \cite{Gorsc2}.

Two points were essential in the reasoning, leading to \eqref{e16} and
\eqref{e18}. The first one is the asymptotic behavior, for
$x\to +\infty$, of the cumulated probabilities $P\{X>x\}$ and
$P\{X<-x\}$ for a jump to have an amplitude $X$ larger than $x$,
and directed to the right or to the left. Both probabilities have
to satisfy (H2) with $\alpha-1$ instead of $\alpha$.
An other property  of stable laws was
used, in view of (H1): it is the fact that the integrals, over
${\mathbb{R}}^+$, of  $P\{X>x\}$ and $P\{X<-x\}$, are equal. This allowed
us to subtract this integral times $f(x)$ from both sides of the
difference, giving the flux, without any net change. In fact, any
Continuous Time Random Walk made of successive independent jumps,
identically distributed according to a random variable $lX$
satisfying both conditions, has a flux whose limit is \eqref{e16} when
$l$ tends to zero, provided mean waiting time $\tau$ exists  with
also $l^\alpha/\tau=K$.

\section*{Conclusion}
 Among many objects, interpolating between derivatives of integer orders,
several tools termed fractional derivatives, were designed for
various  purposes. Some  of them are connected with the idea that
integration and derivation are inverses of each other.

 Within this frame work, there are several ways for to define fractional
derivatives, which are more or less similar to each other. They
are more or  less interesting, according to the sets of functions,
which we want them to operate on. Among them,  Gr\"
unwald-Letnikov derivatives led to performing numerical schemes.
Theorem \ref{thm1} indicates a novel definition of fractional derivatives,
not so far from  Gr\" unwald-Letnikov's: an integral replaces a
series. It seems to be appropriate for to represent fluxes of
particles performing Continuous Time Random Walks satisfying some
hypotheses. Among them,  L\'evy flights play an important role,
since stable laws are ubiquitous in Nature. We developped this
point for random walks in a free one-dimensional space, also using
the local derivatives  invented by Kolwankar and Gangal for
fractal graphs.

Combining those objects also applies to situations with boundary
conditions, for instance in a half space $\{x\in {\mathbb{R}}/x>0\}$
limited by a wall at $x=0$.  There are several possibilities for
the interaction between wall and particles. For instance, we can
imagine that they do not exchange any energy: particles bouncing
on the wall continue the distance, they had to fly if there were
no wall, but they stay on the same side. Then, when writing down
the balance of particles crossing abscissa $x$, we have to take
into account that, among random walkers flying to the left (in the
direction of the wall) some of them bounce and come back to the
right of $x$: they have to be excluded from balance
$\mathcal{W}_{l,-}$. If a particle located in $y\in]0,x[$ has to
jump a length larger than $\vert y-2x\vert$ to the left, it
arrives at the right of the wall and has to be taken into account
in  $\mathcal{W}_{l,+}$. By doing this, we obtain that the flux
through $x$ is the sum of two terms. The first one is the flux
corresponding to a concentration profile equal to the even
extension of the actual one, to a free space without any wall. The
second term is proportional to the left-sided Riemann-Liouville or
Marchaud's derivative of order $\alpha-1$, if $\alpha$ denotes the
stability exponent of the jump length distribution of the L\'evy
flight. It contains a factor, which becomes zero when the
distribution is symmetric, in agreement with results \cite{Kre},
previously obtained by an other method.


\section*{Appendix A: Densities of alpha stable L\'evy laws}

 Stable laws are a generalization of Gaussian statistics.
In many occasions, and here also, the word ``stable'' refers to
some property,  invariant under a definite set of transformations,
as in the following definition.

\noindent{\bf Definition:} Let $X$ be a random variable,
distributed according to the probability law $F$. Random variable
$X$ and law $F$  are said to be stable if \cite{Lev} for every
$(a_1,a_2) \in {\mathbb{R}}^{+2}$ and $(b_1,b_2) \in {\mathbb{R}}^2$, there
exist $a\in {\mathbb{R}}^+$ and    $b\in {\mathbb{R}}$ such that
$F(a_1x+b_1)*F(a_2x+b_2)$ (the law of the random variable
$a_1X_1+b_1+a_2X_2+b_2$, with $X_1$ and $X_2$ being independent
and  distributed according to $F$) be equal to $F(aX+b)$.

When $F$ is as in the above definition, for any sequence of
independent random variables $X_i$ identically distributed
according to $F$, there exists a sequence $c_n$ of positive
numbers such that $\frac{X_1+\dots X_n}{c_n}$ be distributed
according to $F$ itself for any positive integer $n$ \cite{Fel}
\cite{GneK}. Moreover,  $c_n$ is a power of $n$, and the inverse
$\alpha$ of the exponent belongs to $]0,2]$ and serves as a label
for the law: it is called the stability exponent of the law, which
is said to be $\alpha$ stable. For $\alpha=2$ we have normal law,
which is symmetric. For  $\alpha \in ]0,2[$, stable laws may be
symmetric or skewed. Stable laws play an important role in Nature
because they are attractors, which are defined below.

\noindent{\bf Definition:} Let $F$  be the probability of a
sequence of independent random variables $X_n$. The probability
law $G$ is an attractor for $F$ if there exists sequences $A_n$
and $B_n$, with $B_n>0$, such that the law of $\frac{X_1+\dots
+X_n}{B_n}-A_n$ tends to $G$ when $n$ tends to  $\infty$
\cite{Fel}.

Loosely speaking, $\alpha$ stable laws are attractors for
probability laws whose density behaves asymptotically as
$x^{-\alpha-1}$ if $\alpha$ belongs  to $]0,2[$, normal law (with
$\alpha=2$)is  an attractor for  probability laws whose
asymptotics is $x^{-\alpha'-1}$ with $\alpha'\geq 2$  \cite{Fel}
\cite{GneK}.

Except for some values (e.g. $\alpha=1$ or $2$), the density of a
stable law cannot be given in closed form. But, up to translations
and dilatations, the  Fourier transform is $e^{-\vert
k\vert^\alpha e^{isign(k)\pi\theta/2}}$. The corresponding density
$L_\alpha^\theta$ satisfies
$L_\alpha^\theta(-x)=L_\alpha^{-\theta}(x)$. Up to dilatations and
translations, two labels determine stable densities: the stability
exponent $\alpha$, and the skewness parameter $ \theta$, which
belongs to $[\alpha-2,2-\alpha]$.

In neighborhoods of $\infty$, except for $\alpha=2$,
$L_\alpha^\theta$ behaves as a negative power of the variable
\cite{Schn} \cite{MaiLuPa}.  For $1<\alpha<2$,
$\alpha-2<\theta\leq 2-\alpha $ and $x>A>0$, we have
\begin{equation} \label{e19}
L_\alpha^\theta(x)=\frac{1}{\pi x}\Sigma_{n=1}^{+\infty}(-x^{-\alpha})^n
\frac{\Gamma(1+n\alpha)}{n!}\sin{\frac{n\pi}{2}(\theta-\alpha)}.
\end{equation}
We will denote by $C_\alpha^\theta=\frac{-1}{\pi }\Gamma(1+\alpha)
\sin{\frac{\pi}{2}(\theta-\alpha)}$ the coefficient of the leading
term in expansion \eqref{e19}.

\section*{Appendix B: Integrals of cumulated alpha stable L\'evy laws}

Due to symmetry, the integrals
$\int_0^{+\infty}F_{\alpha,\theta}^d(y)dy$ and
$\int_0^{+\infty}F_{\alpha,\theta}^g(-y)dy$ are equal for
$\theta=0$. In fact, and this point is important for us, this
equality holds for all admissible values of $\theta$. Let us prove
the claim.

First, notice that
$F_{\alpha,\theta}^g(-x)=\int_{-\infty}^{-x}L_\alpha^\theta(y)dy
=\int_x^{+\infty}L_\alpha^{-\theta}(-y)dy=F_{\alpha,-\theta}^d(x)$.
Then, we will uses Mellin's transform, defined by
$\mathcal{M}\omega(z)=\int_0^{+\infty}t^{z-1}\omega(t)dt$ for
function $\omega$. With $z=1$ we see that
$\int_0^{+\infty}F_{\alpha,\theta}^d(y)dy=\mathcal{M}F_{\alpha,\theta}^d(1)$,
while we have $F_{\alpha,\theta}^d(x)=I_-^1L_\alpha^{-\theta}(x)$,
hence
$\int_0^{+\infty}F_{\alpha,\theta}^d(y)dy
=(\mathcal{M}I_-^1L_\alpha^{-\theta})(1)$.

For $z>1$ and sufficiently good-behaved functions in neighborhoods of $\infty$,
such as $L_\alpha^\theta$, we have
 $$
(\mathcal{M}I_-^1\omega)(z)=\frac{\Gamma(z)}{\Gamma(z+1)}
(\mathcal{M}\omega)(z+1),
$$
for $z<\alpha$ according to \cite{Rub} page 44.  From this, due to
$F_{\alpha,\theta}^d(x)=\int_x^{+\infty}L_\alpha^\theta(y)dy
=I_-^1L_\alpha^\theta(x)$, we deduce
 $$
(\mathcal{M}F_{\alpha,\theta}^d)(z)=\frac{\Gamma(z)}{\Gamma(z+1)}
(\mathcal{M}L_\alpha^\theta)(z+1).
$$
 The Mellin transform $\mathcal{M}L_\alpha^\theta$ is given in \cite{Schn}:
$$
(\mathcal{M}L_\alpha^\theta)(z)=\frac{1}{\alpha}\frac{\Gamma(z)
\Gamma((1-z)\alpha^{-1})}{\Gamma((1-z)\frac{\alpha-\theta}{2\alpha})
\Gamma(1-(1-z)\frac{\alpha-\theta}{2\alpha})},
$$
which is of the form
\begin{equation} \label{e20}
(\mathcal{M}L_\alpha^\theta)(z)=\frac{1}{\pi\alpha}\Gamma(z)
\Gamma(\frac{1-z}{\alpha})\sin{((1-z)\pi\frac{\alpha-\theta}{2\alpha})}
\end{equation}
due to complements formula for Gamma functions \cite{AbSt}.
In fact, \cite{Schn} proved (20) for $0<Re(z)<1$. Nevertheless,
$\mathcal{M}L_\alpha^\theta(z)$, as a function of  $z$, is holomorphic
for  $0<Re(z)<\alpha+1$, due to the behavior of $L_\alpha^\theta(x)$
 for large real values of $x$. On the right-hand side of \eqref{e20},
$\Gamma(z)\Gamma((1-z)\alpha^{-1})$ is holomorphic also except at
poles of $\Gamma((1-z)\alpha^{-1})$, which means that we have to
exclude $1$ from $\{z\in{\mathbb{C}}/0<Re(z)<\alpha+1\}$. Then, analytic
continuation extends \eqref{e20} to
$\{z\in{\mathbb{C}}/0<Re(z)<\alpha+1\}-\{1\}$.

>From this we deduce
\begin{align*}
\int_0^{+\infty}F_{\alpha,\theta}^d(y)dy
&=(\mathcal{M}F_{\alpha,\theta}^d)(2)\\
&=\frac{\Gamma(2)\Gamma(-1/\alpha)}{\alpha\pi}
 \sin{\pi\frac{\theta-\alpha}{2\alpha}}\\
&=-\frac{\Gamma(-1/\alpha)}{\alpha\pi}\cos{\pi\frac{\theta}{2\alpha}}.
\end{align*}
 We see that  $\int_0^{+\infty}F_{\alpha,\theta}^d(y)dy$ is
an even function of $\theta$, hence the claimed result.

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