EXPLICIT SOLUTIONS OF SOME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA STABLE SUBORDINATORS

The aim of this work is to represent the solutions of one-dimensional fractional partial differential equations (FPDEs) of order (α∈R+\N) in both quasi-probabilistic and probabilistic ways. The canonical processes used are generalizations of stable Lévy processes. The fundamental solutions of the fractional equations are given as functionals of stable subordinators. The functions used generalize the functions given by the Airy integral of Sirovich (1971). As a consequence of this representation, an explicit form is given to the density of the 3/2-stable law and to the density of escaping island vicinity in vortex medium. Other connected FPDEs are also considered.


Introduction
The fractional differential equations describe physical phenomena in inhomogeneous medium, such as diffusion in porous medium or with fractal geometry or with turbulence, kinematics in viscoelastic medium, relaxation processes in complex systems (viscoelastic materials, glassy materials, synthetic polymers, biopolymer), propagation of seismic waves, pollution and transport of data across the internet, for more information on this topics see [1,12,17,18,32], and the references therein.The diffusion packet width, in these mediums, grows proportional to t ν , ν = 1/2.When ν < 1/2, it is called subdiffusion, it is the case, for example, of the porous medium, and when ν > 1/2, it is called superdiffusion, it is the case of chaotic flows generated by vortices [1,12,32].It is known that the case t 1/2 is the normal diffusion connected to Gaussian process.Two approaches are used to describe these diffusion processes, the first one is to consider them as a superposition of ordinary diffusions, the second one is to consider the anomaly in the microscopical scale and describe the diffusion in terms of certain random walks.This last approach gives processes generalizing the Brownian motion (Lévy processes and their 2 Explicit solutions of some FPDEs via subordinators generalizations).Some properties of the anomaly are explained by the interplay of both the diffusive and ballistic behavior.
The high-order fractional partial differential equations are generalizations of the integer high-order partial differential equations.These ladders appear in the literature in connection with many physical applications, for example, heat-type equations of order three is used in the analysis of trimolecular reactions and that of order four is used in the study of the grooves developed on surfaces when obstacles are met [10].The aim of several works on these equations is to make explicit the link between them and stochastic processes [3,9,13,14,22,23,[25][26][27] and the references therein.The major difficulty in this study is that the fundamental solution is not everywhere positive, so the stochastic representation of the solution is not always easily obtained, as it is the case for the heat equation and the Brownian motion.Krylov [16] has introduced the initial value problem ∂u ∂t = (−1) n+1 ∂ 2n u ∂x 2n , t > 0, x ∈ R, u(0,x) = f (x), (1.1) for n > 1 with f ∈ L 1 (R).
By Fourier's calculus, it is easy to see that the solution of (1.1) is given by where p 2n (t,x) is the fundamental solution; that means p 2n (t,x) is the solution of (1.1) when the initial condition is the Dirac distribution (u(0,x) = δ 0 (x)).p 2n (t,x) can be expressed as It is proven in [13,16] that the function p 2n (t,x) is not everywhere positive when n > 1.As a consequence, two approaches are then created.The first one is a formal probabilistic analogy with the Brownian case [9,13,14,16,22,23,[25][26][27] and others.Krylov [16] has used an additive signed measure to construct a "signed probability space."The solution is then represented as u(t,x) = E(x + X t ) where E is the expectation and {X t , t ≥ 0} is the canonical process on this space.This process is called Krylov motion or pseudoprocess and it has p 2n (t,x) as transition density [27].After that, the question how to generalize the stochastic calculus to pseudoprocesses gave rise to several papers [4,13,22,[25][26][27], and so forth, where particular questions such as stochastic integral, It ô formula, first hitting time and first hitting place, Girsanov formula, stochastic differential equations, have been studied.Another formal way is to use the subordination analogy theory as in [9,23].
The disadvantage of this approach is that the measure used is only finitely additive so many standard probabilistic tools cannot be applied, further the "signed probability measure" has no clear physical significance.But on the other hand, the formal stochastic calculus elaborated by this approach is useful in physics and characterizes more or less the particle and its trajectories.In [27], Nishioka represents the biharmonic pseudoprocess Latifa Debbi 3 (n = 2 in (1.1)) as a composition, in certain sense, of two different particles, a monopole and a dipole.
The second approach uses only probabilistic tools.In [4], Burdzy and Ma ¸drecki gave the process connected to the fourth-order equation as a "wide limit" of a sequence of C N -valued random processes.In this construction, they used three independent Brownian motions on a product of two probabilistic spaces, and they mentioned that they can replace two of them by stable processes.Their calculus can be applied to the fractional heat equations with differentiation order between 0 and 4: unfortunately at the end of the paper, they pointed out that this calculus leads to a paradox, further, one cannot interpret the process as the trajectories of a moving particle.
Benachour et al. [3] have used the iterated Brownian motion in the study of the fourthorder heat equation.They associated to the operator (∂/∂x) 4 the process {B |wt | , t ≥ 0}, where {B t , t ≥ 0} and {w t , t ≥ 0} are two independent Brownian motions.
In this work, we are interested in the two approaches for the initial value problem of the fractional equations where α ∈ R + \N, ∂ α /∂x α is a fractional differential operator, c is a real constant, and κ is a real constant to be given later according to the differential operator taken.We note that, by a simple change of the function u, we can extend the study to the equation The paper is organized as follows: after giving some basic definitions and properties in Section 2, we give in Section 3, the resolution of (1.4), the principal properties of the fundamental solution and the quasi-probabilistic approach.In Section 4, we give the probabilistic representation of the solution.To this aim, we introduce a class of functions given by Airy integral.Two subsections, at the end, are devoted to particular cases of mathematical or physical interest and to fractional equations which generalize (1.4); fractional equations in high dimension and fractional equations with more than one differential operator.

Preliminaries
In the literature, various fractional differential operators are defined, see [21,24,28].The results in this paper apply to several of them, such as Riemann-Liouville operator, Nishimoto operator, and the non-selfadjoint fractional operator introduced in [5] and used in [7] to the study of stochastic fractional differential equations.Definition 2.1.Let f be a real function.The fractional derivative of order α of the function f in Riemann-Liouville sense, when it exists, is given by where m = [α] + 1, and [α] is the integer part of α and Γ is the Gamma function.
Definition 2.2.Let f be an analytic function and let z 0 be a point of its domain of definition.Draw the curve C + along the cut joining the points z 0 and +∞ + i z 0 as {x + i z 0 , z 0 + r ≤ x < +∞} * ∨ {z 0 + re iθ , 0 < θ < 2π} ∨ {x + i z 0 , z 0 + r ≤ x < +∞} where the symbol ∨ means "followed by" and * means that the curve is taken in the opposite direction.By the same way we can introduce the curve C − along the cut joining the points z 0 and −∞ + i z 0 .Suppose that the function f has nonbranch point inside and on the curve C ∈ {C − ,C + }.The α-fractional derivative of f in the point z 0 in Nishimoto sense, if it exists, is the complex number In [5], the author introduced a fractional differential operator D α δ as a generalization of the inverse of the generalized Riesz-Feller potential [8,15] for α > 0. where ) is the largest even integer less than or equal to α (even part of α), and δ = 0 when α ∈ 2N + 1, and Ᏺ (resp., Ᏺ −1 ) is the Fourier (resp., Fourier inverse) transform.
The Fourier transform and its inverse are given by (2.5) The operator D α δ is a non-selfadjoint, closed, densely defined operator on L 2 (R) and it is the infinitesimal generator of, in general, a nonsymmetric and noncontraction semigroup.This operator generalizes the differentiation of high order (so (1.4) generalizes (1.1), when α is even [13,16]) and it generalizes also the fractional differential operators in [8,11,15,20] for 0 < α ≤ 2. It is selfadjoint only when δ = 0, in this case, it coincides with the fractional power of the Laplacian.Evidently, when α = 2, it is the Laplacian itself.Furthermore, it is proven in [5] with the Riemann-Liouville differential operator.From [15], D α δ can be represented for 1 < α < 2, by and for 0 < α < 1, by where M δ − and M δ + are two nonnegative constants satisfying M δ − + M δ + > 0 and 1 (−∞,0) and 1 (0,+∞) are the indicator functions of the intervals (−∞,0), (0,+∞), respectively, and ϕ is a smooth function for which the integrals exist, and ϕ is its derivative.For more details about this operator see [5].
It is proven that for Definitions 2.1 and 2.2 and for smooth functions, we have the following relationship between the fractional differential operator and the Fourier transform (see [28] for Riemann-Liouville operator and see [2,24] for Nishimoto operator): For 0 < α < 2 and using the fractional operators above, we can represent the solution of (1.4) by an α-stable process.Definition 2.4 [30].A real stochastic process {X t , t ≥ 0} defined on a probability space (Ω,Ᏺ,P) is a Lévy process if the following conditions are satisfied: (i) it has independent increments, that is, for any choice of n ≥ 1 and 0 ≤ t 0 ≤ t 1 ≤ ••• ≤ t n , the random variables X t0 ,X t1 − X t0 ,...,X tn − X tn−1 are independent, (ii) it has stationary increments, that is, for all s,t ≥ 0 the distribution of X t+s − X t does not depend on t, (iii) X 0 = 0 a.s., (iv) it is stochastically continuous; that is, ∀t > 0 and ε > 0, (2.10) The parameters α, σ, β, μ are called, respectively, stable, scale, skewness, shift parameter.When 0 < α < 1, β = 1, and μ ≥ 0, the stable Lévy process is called subordinator.
Definition 2.5.Let (Ω,Ᏺ − ,P − ) be a signed probability space; that means P − is a signed measure which could be only additive and Ᏺ − could be only an algebra but It is clear that a pseudoprocess is a process in the classical sense when P − is a probability measure.We use the same notations as in probability: E denotes the "expectation" E( f ) = f dP − α and the characteristic function is given by the formula E[exp(i f λ)].Definition 2.6.A pseudoprocess {X − (t), t ≥ 0} defined on the signed probability space (Ω,Ᏺ − ,P − ) is called stable Lévy pseudoprocess if it satisfies conditions (i)-(v) in Definition 2.4 with respect to the signed probability P − and its characteristic function is given by the formula (2.11) where The notation X − α (1) ∼ S α (σ,β,μ) means that the characteristic function of X − α (1) is given by (2.10) where α ∈ R/2N + 1 and −1 < β < 1.The notation ∂ α /∂x α is used to designate any of the fractional differential operators introduced in Definitions 2.1, 2.2, and 2.3.We take κ equal to −(cos(απ/2)) −1 for Definitions 2.1 and 2.2 and equal to (cos(δπ/2)) −1 for Definition 2.3.

Resolution of the fractional equation, some properties, and quasi-probabilistic approach
First, we assume that the fractional differential operator in (1.4) is the Riemann-Liouville operator (∂ α /∂x α = D α ) and applying the Fourier transform, we find the differential equation where (−iλ) α = |λ| α exp(−isgn(λ)(απ/2)).The solution of this differential equation taking into account the initial condition f is By an identical way and across the complex analysis, we obtain the same result with the Nishimoto fractional differential operator.However, for (1.4) with the fractional operator D α δ given in Definition 2.3, we find ) is absolutely integrable, let p α (t,x) be its inverse Fourier transform.From the relationship u α (t,λ) = f (λ) p α (t,λ), we get The function p α (t,x) is the fundamental solution of (1.4).In the sequel of this section, we take the fractional differential operator in (1.4) equal to D α δ with general δ and we take κ > 0. Let us denote the fundamental solution, in this case by δ p α,c,κ (t,x), then (3.5) In the following lemma, we give some properties of the function δ p α,c,κ (t,x).
(x) It is sufficient to prove this property for the function δ p α,0,1 (1,x) when x > 0. In fact, using properties (iii), (iv), and (v), we get (x) for δ p α,c,κ (1,x), and using the representation to which the same calculus also applies, we obtain the result for the derivatives.We are interested in the case α > 2, for the case 0 < α ≤ 2 this result can be deduced from [19,32].
The function δ p α,0,1 (1,x) can be written as Let 0 < r, R < ∞, and let the curve , where [r,R] design the segment in the real axis between r and R, the symbol ∨ means followed by and * means that the curve is taken in the opposite direction.By the Cauchy theorem, the integral of the function exp[−izx − z α e −i(δπ/2) ] over C δ vanishes, further the integrals over the two arcs tend to zero when R tends to infinity and r tends to zero, so +∞ 0 exp − iλx − λ α e −i(δπ/2) dλ = e i(πδ/2α) +∞ 0 exp − iλxe i(πδ/2α) − λ α dλ.
(xi) Let c = 0, using the properties (vi) and (x), we prove that δ p α,0,κ (t,x) tends to zero when x = 0 and tends to infinity when x = 0.The case c = 0 is easily obtained thanks to the property (iv).Proposition 3.2.For 0 < α < 2 and α = 1, there exists a probability space (Ω,Ᏺ,P α ) such that the solution of (1.4) is represented by Latifa Debbi 9 where X α = {X α (t), t ≥ 0} is the canonical process on this space and which is α-stable Lévy process totally skewed to the right (i.e., β = 1).For α ∈ (2,+∞) | N, there exists a signed probability space (Ω,Ᏺ − ,P − α ) such that the solution of (1.4) is represented by where X − α = {X − α (t), t ≥ 0} is the canonical pseudo stable Lévy process on this space.Proof.We will use the same notation in the two cases.
The signed probability P α and the pseudo stable Lévy process X α have also the following properties.

(iii) X α is an infinitely divisible pseudoprocess and its spectral representation (generalized Lévy Khinchine canonical formula) when δ
where Q is a real nonnegative number, (iv) let f be a deterministic function, a formal integral of f with respect to the pseudoprocess , where x α = sgn(x)|x| α .This definition is a formal extension of that given in [29] for stable processes.

Representation of the solution using probabilistic tools
In this section we take First, we introduce certain functions that generalize Airy function.These functions are more general than those given in [6], and simpler and more general than those given in [31].The functions in [31] are given for integral exponents, k ∈ N, and they are represented by integrals over k + 1 curves.
Lemma 4.1.Let β > 1 and n ∈ N be fixed.The function ψ β,n defined on R by is well defined and it is infinitely differentiable.Further when β = 2m + 1 and n = m + 1, the real part of this function, {ψ 2m+1,m+1 (x)}, is solution of the equation where v (2m) is the derivative of order 2m of the function v.
Proof.For fixed x, consider the analytic function with z β is a branch of the multiform exponent function.Let the curve C β,n given by By Cauchy theorem, we have Latifa Debbi 11 On the other hand, r β e i(−1) n+1 βθ re i(−1) n+1 θ dθ.
Proof.First we consider the case c = 0. From (3.5), we have then we can write where  where It is clear that for fixed y and x, the sequence of functions Φ α,M (x, y) tends to the function 2)/α ) when M tends to infinity, where the function ψ α/(α−[α]2),1 is given in Lemma 4.1.Further, we can apply Lebesgue theorem.In fact, using (4.8), we obtain y]dλ, and by Fubini's theorem, we get But the second integral on the right-hand side of the above inequality is the Laplace transform of the subordinator X α−[α]2 (t) in time t, so it is equal to exp(−(cos((α by the same technique as above, we can see that the integral on the right-hand side in this last inequality is finite.Therefore
Proof.Thanks to property (iv) in Lemma 3.1, we can take without restriction c = 0.
When n = 0, we have

.25)
By Proposition 3.2, there exist a probability space (Ω,Ᏺ,P α ) and a Lévy motion It is known that the density of X 1/2 (t) is given by Latifa Debbi 15 For n > 0 and similarly in the proof of Theorem 4.4, we regard h n+1/2 (t,λ) as   The function p 3/2 (1,x) is the density of the 3/2-stable law, (ii) α = 5/2 and c = 0, we have The function p 5/2 (t,x) can represent the probability density of escaping the island vicinity after being in its neighborhood for a time t, in vortex medium [18].

Some other PDEs connected with (1.4).
In this subsection, we consider two equations connected with (1.4).The first one corresponds to the vectorial case (x ∈ R m ).In the second one the variable x is real and we use a sum of fractional differential operators. (

Remark 4 . 3 .
The particular case β = 2m + 1 and n = m + 1 has special interest in this work as we will see below.