RANDOM WALK’S MODELS FOR FRACTIONAL DIFFUSION EQUATION

. Fractional diﬀusion equations are used for mass spreading in in-homogeneous media. They are applied to model anomalous diﬀusion, where a cloud of particles spreads in a diﬀerent manner than the classical diﬀusion equation predicts. Thus, they involve fractional derivatives. Here we present a continuous variant of Gr¨unwald-Letnikov’s formula, which is useful to compute the ﬂux of particles performing random walks, allowing for heavy-tailed jump distributions. In fact, we set a deﬁnition of fractional derivatives yielding the operators which enable us to retrieve the space fractional variant of Fick’s law, for enhanced diﬀusion in disordered media, without passing through any partial diﬀerential equation for the space and time evolution of the concentration.


1.
Introduction. Fractional calculus has been attracting the attention of scientists and engineers from long time ago. It is three centuries old as the conventional calculus, but not very popular among science and/or engineering community. The beauty of this subject is that fractional derivatives (and integrals) are not a local (or point) property (or quantity). Thereby this considers the history and non-local distributed effects. In other worlds, perhaps this subject translates the reality of nature better! In the last years, it has found use in studies of viscoelastic materials, as well as in many fields of science and engineering including viscoelasticity, bubbles dynamics [3] [6], fluid flow, rheology, diffusive transport [7], electrical networks, electromagnetic theory and probability [1].
In this paper, we will focus on mapping inverting (at the left) fractional integrals. In fact, left inverses to such mappings can be given by explicit formulas [9] [11] [10] [5], such as Riemann's and Liouville's. Marchaud's method is more general, combines convolution and finite differences. It coincides with Riemann-Liouville's formulas for a broad class of functions, and also with Grünwald-Letnikove's definition, which is at the basis of numerical approximations to fractional derivatives. Besides, in this paper, we will establish a new expression for the left inverse of I α ± , defined by the limit, when l tends to 0+, of the function for which to f we associate This point was proved in [8], for values of α belonging to R * + . Here, we show that it holds for α ∈ C such that Re α > 0.
So, we will deal with the case of a complex order such that its real part is positive. This continuous variant involves convolution kernels which mimic essential properties of Grünwald Letnikov's weights, but are more general. In fact, for α ∈ R * + , the weights w α k of the discrete convolution form defined sequences, proportional to k −α−1 near infinity, and all moments of integer order r < α are equal to zero provided α is not an integer. The continuous convolution kernels satisfy the hypothesis A 1 (α) and A 2 (α) [9]. A first application consists in computing the flux of particles spreading according to a random walk, consisting of successive jumps, independent of each other but possibly depending on the location of the point they start from, as when boundary conditions are applied, e.g. An important physical issue is in the space fractional variant of Fick's law for enhanced diffusion in disordered media.

Left inverse of
For α ∈ C such that Re α > 0, the left and right-sided fractional integrals of the order of α of function f are [9] [11] (I α a,+ f ) (x) := with −∞ ≤ a < x < b ≤ +∞. I α ± is the Riemann-Liouville's fractional integral of the order of α. Riemann-Liouville's left and right-sided derivatives of the order of α are with n = [Re α] + 1. D α ± is the Riemann-Liouville's derivative of the order of α.

Left inverse of
We assume that f = I α ± ϕ exists. Then, we can define Marchaud's derivatives according to We note that D α ± is a left inverse of I α ± , over the following set: {ϕ ∈ (L ∓ p )\I α ± ϕ exist as a Lebesgue or as an improper one}. Remark 1. Let N ∈ R. The limit above exists in If ϕ ∈ L p (R), then the limit above exists in L p (R). Moreover, the limit exists point-wise almost everywhere.

2.3.
Grünwald-Letnikov's formulas. Grünwald-Letnikov's method yields approximations to the inverse of a fractional integrals for α ∈ C such that Re α > 0. It is defined according to is the Grunwald-Letnikov's derivative of the order of α. The following kernel behaves as k −α−1 when k is large, provided α is not an integer [2] and +∞ k=0 (−1) k α k = 0 holds for Re α > 0, and implies +∞ k=0 when r is an integer satisfying 0 ≤ r < α. Grünwald-Letnikov's derivative of the order of α yields a left inverse to I α ± . Remark 2. Let 1 < p < 1 Re α , i.e., for 0 < Re α < 1 and N ∈ R. So, we have Remark 3. For 1 ≤ p < +∞ and ϕ ∈ (L ∓ p L 1 loc (R)), such that f = I α ± ϕ exists and I 3. Left inverse of a fractional integrals by a new tool. We present a continuous variant of Grünwald-Letnikov's formulas, with integrals instead of series. It involves a convolution kernel F which mimics the above mentioned features of Grünwald-Letnikov's weights w α k by matching the following hypotheses.
• If α ∈ N, the points (i), (ii) et (iii) persist provided F satisfies the condition: Proof. Non-integer values of α will be considered first.
For α not an integer. Proving (i) is enough for (i) and (ii), and consists in comparing ϕ against the limit of h −α +∞ 0 f (· + hy) F (y) dy under hypothesis A 1 (α)-A 2 (α). We have such that But, from the beginning, we set ϕ ∈ L + r , so that, Φ ∈ L r (R). Moreover, according to 3.2 below, we have I α + (HF ) ∈ L 1 (R + ), hence Υ ∈ L 1 (R). Due to Young's inequality, it implies that Φ * Υ ∈ L r (R), ie almost everywhere. So that, we can apply Fubini's theorem which implies that Setting We remark that K ∈ L 1 (R) and R K (t) dt = 1. As Φ ∈ L r (R), so, the theorem of approximation to identity implies that Let now α be a positive integer. When F (x) x α is integrable, the Lemma 4.12 [9] implies the Theorem.
Before showing the proposition, above, we aim to prove 3.3 and 3.4 below, which are involved in its proof.
And let 0 ≤ m < α < m + 1, with m ∈ N. Then, for A, a real large enough, the function Proof. For α ∈ C and N ∈ N * and | x |< 1, we have Hence, Since is integrable in a neighborhood of +∞ by the criterion for Riemann integrability.
Proof. Notice that with We have G(1) = g(−α , α) and β(p , q) For (p , q) ∈ C 2 , such that Re p > 0 and Re q > 0 we have β(p , q) = Γ(p) Γ(q) Γ(p+q) . Hence for Re p > 0 and Re q > 0. Fixing q = α, this equality extends to complex valued p which are not negative integers and satisfy Re(p) > −m − 1. Let the function F α defined by Let showing by the dominated convergence's theorem that 1 0 F α (· , t) dt is holomorphic function over a connected domain which we will set it later. Hence, we will deduce the analyticity of this function in this domain. In fact, for fixed t ∈]0, 1[, we have F α (·, t) is a function of p, also it is holomorphic whose derivative is log(t) F α (·, t). And due to the proof of 3.3, we have Consequently, the dominated convergence's theorem implies that p → g(p , α) is holomorphic on Ω = {z ∈ C\Re z ≥ p 1 > −m − 1}. It yields also that this function is analytic in the previous domain.
Moreover, the function defined by and which coincide with p → g(p , α) for Re p > 0, is also holomorphic or analytic in the set Π = {Ω \ {0 , −1 , · · · , −m}}. Indeed, p → Γ(p) is holomorphic on Ω, except at poles 0 , −1 , · · · , −m of Γ. And p → 1 Γ(p+α) is always holomorphic on C, in particular on Π. Also, the map defined by is holomorphic on Π because p = 0 and p + k = 0 . Since Π is a connected subset of {z ∈ C \ Re z > 0} where it has an accumulation point, the theorem of analitic continuation implies that both functions which we studied their analiticity, coincide in Π, i.e., We note that i.e., (−α) ∈ Π. Thus, for p = −α, we have Proof of Proposition 1. Since g * ∈ L 1 loc (R), I α + (Hg * ) ∈ L 1 loc (R) and it suffices to check whether I α + (Hg * ) is integrable in a neighborhood of +∞. Three preliminary steps will prepare the proof.
First, for x > A, we have Then, And We set f (y) = (x − y) α−1 , then, we apply the expansion of Taylor-Young at 0 of ordre m. Thus, we obtain We have Thus, Third, such that Consequently, we have We now looking at the integrability of the following map near +∞ : x . For this aim we consider that And Thus, we obtain = 0, thanks to 3.4 Such that .
This proves the Proposition.

Porous and heterogeneous medium and fractional Fick's law.
For particles performing a Markovian random walk, the flux depends on the concentration and on the transition probability density function. It also depends on the geometry of the medium and on the physical properties of the boundaries, if there are, as we will see. After having set these points, we will focus on what happens to the flux when time and length scales of the random walk are made small, compared with those, characterizing the variations of the density of the cloud of particles.
4.1. The flux of particles performing Lévy flights. For a given particle, the location after n th jump is n i=0 X i , and it happens at time n i=0 T i . • Jump amplitudes are independent variables, with density ϕ l (x) = 1 l L θ α ( x l ). Here, L θ α denotes the density of a normalized Lévy law of stability index α (1 < α < 2) and skewness parameter θ. l is a length scale.
• Waiting times are also independent, and for the sake of simplicity we assume that they are distributed according to the Poisson density ψ τ (t) = 1 τ e − t τ , such that τ is the mean waiting time.
The length scale l and mean waiting time τ satisfy l α τ = K.

a) Flux of particles performing Lévy flights in an infinite medium.
Let us denote by ± x ∓∞ F ± α , θ ( x−y l ) dµ(y, t) dt τ the probability of crossing x to the right or to the left, such that µ(·, t) is the measure giving the probability µ(I, t) that the particle be in interval I at time t. Let P (·, t) be the density of the previous measure. And F ± α,θ (± y l ) = ± ±∞ ± y l L θ α (z) dz is the probability for a jump to have an amplitude of more than y to the right or to the left. Also, we set that the probability of making one jump during infinitesimal time interval [t, t + dt[ is dt t . So, the flux is the probability rate, hence the following difference: 2520 WAFA HAMROUNI AND ALI ABDENNADHER (10) b) Flux of particles performing Léevy flights in semi-infinite domain; limited by a reflecting boundary.
The expression giving the flux may be modified more or less deeply by the presence of a boundary. In fact, imagine that each particle hitting a purely reflecting wall located at x = 0 bounces and finally flies the length of the jump, which had been assigned to it before the shock. In this case, we have to take account of two points. First, if the amplitude of the jumps directed to the left and starting from x + y (y > 0) is larger than 2x + y, then arrives at the right of x hence do not enter the balance. Second, jumps directed to the left and starting from x − y, with 0 < y < x, may cross x to the right if the amplitude is of more than 2x − y. Therefore, the flux is given by this equation: Let P * (x) = P (x), for x > 0 and P * (x) = P (−x), for x < 0, we obtain In fact, the previous equation is given by the following variable exchange: The integral transform P → l −α +∞ 0 P (x ± y, t) F ∓ α,θ ( ∓ y l ) dy are present on the right-hand sides of (11) and (10) giving the flux. Nevertheless, cumulated probabilities F ± α,θ (±·) satisfy A 2 (α − 1), but of course not A 1 (α − 1). In fact, let X a random variable α− stable. And we denote σ the scale parameter, θ the skewness parameter and P the measure of probability. Then, we have This result is obtained due to the asymptotic behavior of the density of a stable law given for x > A > 0, (with A large enough). In fact, we have The coefficient of the leading term is Hence, with f α,θ being any integrable function, compactly supported, such that We will deal with an other kernel of the integral transform and defined by the following equation:F ± α,θ (y) = F ± α,θ (y) − f α,θ (y). Then, we have c) Involving the new expression for left inverse of Riemann-Liouville's integral in computing the microscopic flux. F ± α,θ satisfies both A 1 (α − 1) and A 2 (α − 1). So, we can apply the previous theorem witch yields the following equations: t) . besides, an appropriate choice of f α,θ yields that the third expression in (15) tend to right and left-sided local fractional derivatives of order α − 1. The choice is Then, we have such that D KG, q ± denote the Kalwankar and Gangal's fractional derivatives of the order of q. In fact, for 0 < q < 1, we have For sufficiently well-behaved functions (in L p (R) ∩ H α−1+ε ) the Kolwankar and Gangal's derivative exists and is identically zero.
The value of λ ± can be given by comparing D α−1 ± f against the limit of for some particular function f. The comparising will be simpler with functions f whose local derivative of order α − 1 is identically zero in neighborhood of infinity.
In fact, set f (x) = χ [1, 2[ (x) and x ∈]1, 2[, then we have We have As we know that It yields this equation: .
On one hand, the compute of the numerator gives for On the other hand, the compute of the denominator gives for x ∈]1, 2[, Then, we have .

4.2.
Fractional Fick's law. When τ and l tend to zero while satisfying the scaling l α τ = K, the limit ∞ Q (x, t) of mapping ∞ W α, θ l, τ (x, t) witch denote the flux through x for random walks in unbounded domains, is such that To compute the diffusive limit of the flux in a semi-infinite domain limited by a reflective barrier, we have to rewrite the third term of (11) in an improved form. With According to the Theorem, we have . Then, the diffusive limit of the flux in semi-infinite medium; limited by a reflecting boundary, is such that 4.3. Space-fractional diffusion equation. When the density of particles P and the macroscopic flux Q are derivable, mass conservation without sources implies Moreover, we have Also, the local Kolwankar-Gangal derivatives with order of less than 1 are identically zero. Hence, in an infinite medium, we deduce from (16) the space-fractional diffusion equation in an infinite domain given by this equation such that ∇ α,θ x is the Riesz-Feller derivative of order α and skewness parameter θ. In a medium, limited by a reflective barrier, we deduce from (18) that P evolves according to 5. The fractional dispersion equation via Laplace-Fourier analysis and the generalized master equation in a medium limited by a reflective barrier. In the theory of CTRWs (Continuous Time Random Walks), it is assumed that waiting times are independent and identically distributed random variables with density ψ. Let , and E β = E β, 1 is the Mittag-Leffler function of order β ∈]0, 1]. We denote , α > 0 , β ∈ C and z ∈ C.
To proceed further we assume that the transition probability follows an α− stable 1 < α < 2 law. Let C (x, t) be the density of the probability of finding a walker in [x, x + dx] at time t. And let C satisfies the initial condition C(x, 0) = σ(x − x 0 ), with x 0 > 0. Also, we may imagine a purely reflecting wall located at x = 0.
Let Λ(x, x ) the probability density that a walker go through x, given that the particle is at position x . We assume that with ρ d (x, x ) is the direct transition probability and ρ r (x, x ) is the transition probability after reflection by the elastic wall. We denote ρ an even function and f the density of the α− stable law such that (due to the interaction between Lévy flights and reflecting wall) [4]. The Fourier transform of f can be written asf Then, we haveρ (k) =f (kl) = e −l α |k| α .