Alternative probabilistic representations of Barenblatt-type solutions
Alessandro De Gregorio Roberto Garra
math.PR



Research Article





[type=corresp,id=cor1]Corresponding author.
Department of Statistical Sciences, “Sapienza” University of Rome,P. le Aldo Moro, 5 – 00185, Rome, Italy






A. De Gregorio, R. GarraAlternative probabilistic representations of Barenblatt-type solutions



A general class of probability density functions


    u(x,t)=C t^-α d (1- (x/ct^α )^β )_+^γ,   x∈ℝ^d,t>0,

is considered, containing as particular case the Barenblatt solutions arising, for instance,
in the study of nonlinear heat equations.

Alternative probabilistic
representations of the Barenblatt-type solutions u(x,t) are proposed. In the one-dimensional
case, by means of this approach,

u(x,t) can be connected with the
wave propagation.



Anomalous diffusion
Beta random variable
Euler–Poisson–Darboux equation
Fourier transform
nonlinear diffusion equation
random velocity


[MSC2010]
60G07


29 11 2019
4 3 2020
13 3 2020
23 3 2020





§ INTRODUCTION



The Brownian motionBrownian motion is a stochastic process of interest for pure
and applied mathematicians. The probability distribution of the Brownian
motionBrownian motion is given by the Gaussian kernel


    G(x,t)=1/(4π t)^d/2exp (-x^2/4t),    x∈ℝ^d, t>0,

which is the source-type solution (that is G(x,0)=δ (x), where
δ (x) represents Dirac's delta function) to the parabolic heat
equationparabolic heat equation


    ∂ u/∂ t=Δ u.

The main 
unwanted feature of this solution is that

it is inevitably
positive everywhere in its domain of definition; i.e., the Brownian motionBrownian motion
scatters with unbounded velocity. A way to overcome this 
feature is to
consider the porous medium equationPorous Medium Equation (PME) which is a nonlinear diffusion equation


    ∂ u/∂ t=Δ (u^m),   m>1,

having the source-type solution given by


    U(x,t)=Ct^-α d (1-x^2/c^2t^2α )_+^1/m-1,  α >0,c>0,

where (x)_+:=max (x,0) and C is a suitable constant such that
∫ U(x,t) dx=1. The solution U(· ,t) is a compactly
supported function and it is called the  Barenblatt solution. For a complete
description of the mathematical analysis 
related to the partial
differential equation (<ref>) the reader can consult
<cit.>. The connection between the porous medium equationPorous Medium Equation (PME) and the
theory of stochastic processes has been investigated, for instance, in
<cit.> and <cit.>.

The aim of this paper is to study a class of functions generalizing the
Barenblatt solution U(x,t).Barenblatt solution For fixed t>0, we 
consider the map


    ℝ^d∋ x↦ u:=u(x,t):=C t^-α d (1- (
    x/ct^α )^β )_+^γ,

where α >0,β >0 and γ >0, and


    C:=C(β ,γ ,d):=
    β/c^dσ (𝕊^d-1)Beta(d/β,γ +1)

is a positive constant determined by the condition
u(x,t)_L^1(ℝ^d,dx)=1 (the property of the mass conservation), dx denoting the Lebesgue measure on
(ℝ^d,ℬ(ℝ^d)),
σ (𝕊^d-1):=2π ^d/2/Γ (d/2) represents the surface
area of the (d-1)-dimensional sphere 𝕊^d-1 with radius one
and Beta(a,b):=Γ (a)Γ (b)/Γ (a+b), a,b>0. We observe
that u is a probability density functionprobability density function with an associated absolutely continuous
probability measure given by


    μ _t(dx)=u(x,t)dx,

having the following features:



  * The density function udensity function is compactly supported; i.e., for every
t>0, the support of u(· ,t) is given by


    supp  u(· ,t)= B_r(t):={x∈ℝ^d:x≤ r(t)
    },

representing a closed ball with radius r(t):=ct^α. This property
implies the finite speed of propagation of u; i.e., the (d-1)-dimensional
sphere with radius r(t) denoted by 𝕊^d-1_r(t) provides
the free boundary separating the regions
{(x,t)∈ℝ^d× (0,∞ ):u(x,t)>0} and
{(x,t)∈ℝ^d× (0,∞ ):u(x,t)=0}.

  * The probability measure μ _t is rotationally invariant; that
is, μ _t(dx)= u(x,t)dx, or equivalently,
μ (M A)=μ (A) for A∈ℬ(ℝ^d) and
M∈ O(d), where O(d) is the group of d× d orthogonal matrices
acting in ℝ^d, where d≥ 2. As a direct consequence of this
property we derive


    μ _t(B_a)   =σ (𝕊^d-1)∫ _0^a r^d-1u(r,t)
    dr
       =Beta ((a/ct^α)^β;d/β,γ +1 )/Beta(d/β,γ +1),
       0<a<ct^α,

where Beta(x;a,b)=∫ _0^x y^a-1(1-y)^b-1dy, x
∈ℝ, is the incomplete Gamma function.

  * The density u is a self-similar function. Indeed


    u(x,t)=L^dαu(L^αx, Lt),    L>0.



We refer to (<ref>) as the class of the Barenblatt-type solutions.
The family of functions (<ref>) contains as particular cases, for
instance, weak solutions of several nonlinear and linear diffusion equations
(see Section <ref>). Furthermore, in <cit.> it is proved that
the mean exit time of a symmetric Lévy stable processsymmetric Lévy stable process from a ball admits
a representation belonging to the Barenblatt-type solution class. In this
paper we provide alternative probabilistic representations of Barenblatt-type
density functions in terms of mean value of delta functions containing
random terms (see Section <ref>). At least in the case d=1, our
approach permits to shed light on the connection of the nonlinear diffusion
with the propagation of waves and spherical waves (which are described
by means of linear partial differential equations). The main novelty of
this interpretation is that a wave performs random displacements nonlinearly
with respect to time. It is worth to mention that solutions belonging
to family (<ref>) emerge from different frameworks (linear hyperbolic,linear hyperbolic
nonlinear parabolicnonlinear parabolic and nonlocalnonlocal). 
In this way, objects requiring different
mathematical tools have 
common features.

It is worth to 
note that the connections between stochastic processes
and nonlinear Fokker–Planck equations have also been analyzed in
<cit.> and in references therein.




§ BARENBLATT-TYPE SOLUTIONS TO DIFFUSION EQUATIONS



The aim of this section is to highlight that the class of density functionsdensity function
of the form (<ref>) is very general. Solutions belonging to family
(<ref>) appear in different frameworks (linear hyperbolic,linear hyperbolic nonlinear
parabolicnonlinear parabolic and nonlocalnonlocal). Therefore, in what follows, we list some diffusion
equations studied by means of different approaches. Nevertheless,

their solutions share the same analytic form.

The Fourier transform ℱ and the inverse transform
ℱ^-1 of a function
v∈ L^1(ℝ^d, dx) are defined by


    ℱ v(ξ )=∫ _ℝ^d v(x) e^i x·ξdx,    ℱ^-1 v(x)=1/(2π )^d∫ _ℝ^d v(ξ ) e^-i x·ξdξ ,

with ξ∈ℝ^d.




 §.§ Nonlinear diffusions:nonlinear diffusions p-Laplacian equation



We 
mean here the p-Laplacian equation (PLE for short) studied, for instance,
in <cit.> and <cit.>, 
which is the following nonlinear degenerate
parabolic evolution equation


    ∂ u/∂ t=div (|∇ u|^p-2∇ u
     ),   p>2,    t>0,

subject to the initial condition


    u(x,0)=δ (x),

where u:=u(x,t), with x∈ℝ^d,d≥ 1, is a scalar function
defined on ℝ^d×ℝ^+. The Cauchy problem
(<ref>)–(<ref>) admits a unique nonnegative fundamental
solution:fundamental solution it is  a function u≥ 0 solving (<ref>)–(<ref>)
in a weak sense (see <cit.> and <cit.> for the detailed definition
of weak solution to PLE). This solution is given by the following probability
density functionprobability density function


    u(x,t)= t^-k (𝔠-q (x/t^k/d )^p/p-1 )_+^p-1/p-2,

where


    k:= (p-2+p/d )^-1,     q:=p-2/p (
    k/d )^1/p-1

and 𝔠:=𝔠(p,d) is a constant determined by the condition
∫ u(x,t)dx=1. By setting
β =p/p-1, γ =p-1/p-2, α =k/d, C=
𝔠^p-1/p-2 and
c=(𝔠/q)^p-1/p, the Barenblatt-type solution
(<ref>) coincides with (<ref>).




 §.§ Nonlinear diffusions:nonlinear diffusions nonlocal porous medium equation



The Nonlocal Porous Medium EquationNonlocal Porous Medium Equation (NPME) (NPME), studied in <cit.> and
<cit.>, is 
the following degenerate nonlinear and nonlocalnonlocal
evolution equation


    ∂ u/∂ t=div (|u|∇ ^ν -1(|u|^m-2u)
     ),   m>1, ν∈ (0,2], t>0,

subject to the initial condition


    u(x,0)=u_0(x).

The pseudo-differential operator ∇ ^ν -1 is the fractional gradient
denoting the nonlocal operatornonlocal operator defined as
∇ ^ν -1u:=ℱ^-1(iξξ^ν -2ℱu).
This notation highlights that ∇ ^ν -1 is a pseudo-differential
(vector-valued) operator of order ν -1. Equivalently, we can define
∇ ^ν -1 as ∇ (-Δ )^ν/2-1, where
(-Δ )^ν/2u=ℱ^-1(ξ^νℱu) is
the fractional Laplace operator,fractional Laplace operator i.e., a Fourier multiplier

with the symbol ξ^ν. For ν =2, (<ref>) becomes the
classical nonlinear porous medium equationPorous Medium Equation (PME)


    ∂ u/∂ t=div (|u|∇ (|u|^m-2u)
     )=div ((m-1)|u|^m-1∇ u ).

If we restrict our attention to nonnegative solution u(x,t), equation
(<ref>) becomes


    ∂ u/∂ t=m-1/mΔ (u^m),

which is usually adopted to model the flow of a gas through a porous medium.

Let ν∈ (0,2] and m>1. A weak solution, in the sense of Definition
1 in <cit.>, is given by


    u(x,t)= Ct^-dα (1-k^2/νx^2/t^2α )_+^ν/2(m-1),

where α :=1/d(m-1)+ν,
k:=
dΓ (d/2)/(d(m-1)+ν )2^νΓ (1+ν/2)Γ (d+ν/2)
and


    C:=
    Γ (d/2+ν/2(m-1)+1)k^ d/ν/π ^d/2Γ (ν/2(m-1)+1).

Furthermore, u(x,t) is the pointwise solution of equation
(<ref>) for x≠ k^-1/νt^α. The link
between (<ref>) and random flights has been investigated in
<cit.>. For ν =2, the solution (<ref>) becomes the
Barenblatt–Kompanets–Zel'dovich–Pattle solution of the porous medium equationPorous Medium Equation (PME)
(<ref>) supplemented with the initial condition
u(x,0)=δ (x) (see, for instance, <cit.>).

The function (<ref>) reduces to (<ref>) for
γ =ν/2(m-1), β =2 and c=1/k^2/ν.




 §.§ Euler–Poisson–Darboux equation



It is well known that the fundamental solutionfundamental solution of the Euler–Poisson–Darboux
(EPD) equation


    ∂ ^2 u/∂ t^2+d+2ν -1/t∂ u/∂ t = c^2Δ u,   ν >0,  t>0,
     c>0,

has the form


    u(x,t) = Γ (ν +d/2)/π ^d/2Γ (ν )1/(ct)^d (1-x^2/(ct)^2 )_+^ν -1

and therefore it belongs to the family of probability density functions
with compact support (<ref>) with β = 2, α = 1,
k = d and γ = ν -1. There is a wide literature about the EPD
equationEPD equation and its applications. We refer to Bresters <cit.> for the construction
of weak solutions of the initial value problem for the EPD equationEPD equation based
on distributional methods. We recall that, in the one-dimensional case,
the solution to the Cauchy problem


    ∂ ^2 u/∂ t^2+2ξ/t∂ u/∂ t= c^2∂ ^2 u/∂ x^2,
    
     u(x,0)=f(x),
    
    ∂ u/∂ t(x,t) |_t=0=0,

can be represented as the Erdélyi–Kober fractional integral (see
definition (<ref>) below) of the D'Alembert solution of the wave
equation (see <cit.>)


    ∂ ^2 w/∂ t^2= c^2∂ ^2 w/∂ x^2,
    
     w(x,0)= f(x),
    
    ∂ w/∂ t(x,t) |_t=0=0.


This means that


    u(x,t)   =2/Beta(ξ , 1/2)∫ _0^1 (1-y^2)^ξ -1 [f(x+yct)+f(x-yct)/2 ]dy.

The first probabilistic interpretationprobabilistic interpretation of this analytic representation, discussed
by Rosencrans <cit.> and more recently by Garra and Orsingher
<cit.>, is the following one: solution (<ref>) can be written
as


    u(x,t) = 𝐄 [
    f(x+ 𝒰(t))+f(x- 𝒰(t))/2 ],

where


    𝒰(t) = U(0)∫ _0^t(-1)^N(s)ds,

and (N(t))_t≥ 0 is the nonhomogeneous Poisson process with rate
λ (t)= ξ/t,  U(0) is a uniformly distributed r.v.
on {-c,c} (furthermore (N(t))_t≥ 0 and U(0) are supposed independent).
By means of the general Proposition <ref> (see the next section), we here obtain
a new interesting probabilistic interpretationprobabilistic interpretation of the fundamental solutionfundamental solution
of the EPD equation.EPD equation

Moreover, we have the following interesting picture that underlines the
role of the Barenblatt-type solution as a bridge between nonlinear and linear
PDEs:



  * The Erdélyi–Kober fractional integral of the solution of the D'Alembert
equation leads to the solution of the Euler–Poisson–Darboux equation under
the same initial conditions.

  * As recently pointed out in <cit.> and <cit.>, the time-rescaled
Kompanets–Zel'do­vichZelddovich–Barenblatt solution of the Porous Medium EquationPorous Medium Equation (PME)
(PME) coincides with the fundamental solutionfundamental solution of the Euler–Poisson–Darboux
equation.

Therefore, we have found, by means of new analytical representations, a
direct connection between nonlinear parabolic equations and linear hyperbolic
equations.From the probabilistic point of view this connection could be expected
because in both the cases we have generalizations of the diffusion equation
leading to a finite speed propagation.




 §.§ Nonlinear time-fractional diffusive equations admitting Barenblatt-type solutions



There is a wide literature about the probabilistic interpretationprobabilistic interpretation of linear
space and time-fractional diffusive equations (see, e.g.,
<cit.> and the references therein). On the other hand,
a probabilistic approach to time-fractional nonlinear diffusive-type equations
is still completly missing. 
Recently, the existence and uniqueness
of compactly supported solutions for time-fractional porous medium equationsPorous Medium Equation (PME)
has been investigated (see, e.g., <cit.>). However, up to our knowledge,
it is not possible to find an explicit form of the Barenblatt-type solution.
On the other hand, time-fractional diffusive equations are actracting an
increasing interest in the literature. Explicit Barenblatt-type solutions
for nonlinear time-fractional equations can play a relevant role for
future studies in this context. We here consider a new family of nonlinear
time-fractional diffusive equations admitting a Barenblatt-type solution
of the form (<ref>).

Let us consider the following nonlinear diffusive equation


    1/t^2ν∂ ^νu/∂ t^ν+
    1/t^ν∂ ^2 u/∂ x^2+ (
    ∂ u/∂ x )^2 = 0,

where ∂ ^ν/∂ t^ν denotes the Riemann–Liouville
derivative


    ∂ ^νf(x,t)/∂ t^ν =
    1/Γ (1-ν )∂/∂ t∫ _0^t (t-s)^-
    νf(x,s) ds,   ν∈ (0,1),

and f(x,· ) is a suitable well-behaved function (see
<cit.> for details about the functional setting).

We start our analysis from a simple ansatz:  equation (<ref>) admits
a solution in the form


    u(x,t) = C_1/t^ν-C_2x^2/t^3ν,   ν∈ (0,1/3 )∖{1/4} ,  (x,t)∈ℝ×ℝ^+,

where C_1 and C_2 are real constants that we are going to find.
We now directly check the correctness of this conjecture. We first recall
that, for ν >0 and β >-1,


    ∂ ^ν/∂ t^νt^β=
    Γ (1+β )/Γ (1+β -ν )t^β -ν.

Then, by substituting (<ref>) in (<ref>) and by applying
(<ref>), we have


    Γ (1-ν )/Γ (1-2ν )C_1/t^4ν-
    Γ (1-3ν )/Γ (1-4ν )C_2 x^2/t^6ν- 2C_2/t^4ν+
    4C_2^2 x^2/t^6ν=0

and balancing similar terms we have that


    Γ (1-ν )/Γ (1-2ν )C_1 - 2 C_2 =0,
    
     4C_2^2-Γ (1-3ν )/Γ (1-4ν )C_2=0.

Therefore


    C_1 = 1/2Γ (1-3ν )/Γ (1-4ν )Γ (1-2ν )/Γ (1-ν ),
    
     C_2 = 1/4Γ (1-3ν )/Γ (1-4ν ).


Observe that the constraint on the real order of derivation
ν∈ (0,1/3 ) in (<ref>) is due to the application
of (<ref>).

Moreover, ν≠ 1/4 because of the coefficients appearing in the solution
(depending on the Euler Gamma functions). We can conclude that  equation
(<ref>) admits a solution of the form (<ref>) if
ν∈ (0,1/3)∖{1/4}. Therefore, in particular, under the
same constraints on ν, we can say that the time-fractional equation
(<ref>) admits a solution of the form


    u(x,t) = C_1/t^ν (1-C_2/C_1x^2/t^2ν )_+,

where C_1 and C_2 are given by (<ref>).

Obviously, the solution (<ref>) is not normalized but it is a Barenblatt-type
solution belonging to the general family considered in this paper. A systematic
study of equation (<ref>) should be an object of further investigation,
both from the physical and mathematical points of view. We conjecture that
this is a source-type solution of the nonlinear time-fractional equation
(<ref>); nevertheless a full rigorous analysis should be developed,
but this is beyond the aims of this paper. The fractional equation
(<ref>) can be viewed as a hybrid between a diffusive equation with
singular time-dependent coefficients (in some way similar to the EPD equationEPD equation)
and a nonlinear time-fractional porous medium type equation.Porous Medium Equation (PME)




§ MAIN RESULTS



Let us start with our first result concerning the case d=1.



For d=1, the density functiondensity function (<ref>) can be written  as


    u(x,t)=𝐄_V [
    δ (x-Vt^α)+δ (x+Vt^α)/2 ],

where 𝐄_V[· ] stands for the mean value w.r.t.
V(law)=cY^1/β, where Y∼Beta (
1/β,γ +1).


Let d=1. We have


    û(ξ ,t)   =ℱu(ξ ,t)
    
       =Ct^-α∫ _-ct^α^ct^αe^iξ x (1-
     (|x|/ct^α )^β )^γdx
    
       =2 Ct^-α∫ _0^ct^αcos (ξ x) (1- (
    x/ct^α )^β )^γdx
    
       =β /c/Beta(1/β,γ +1)∫ _0^ccos (ξ vt^α) (1- (v/c)^β )^γdv
    
       =𝐄_V [ cos (ξ Vt^α) ]
    
       =𝐄_V [
    e^iξ Vt^α+e^-iξ Vt^α/2 ].

Hence, by Fubini's theorem we immediately obtain


    u(x,t)=ℱ^-1û(ξ ,t)=𝐄_V [
    δ (x-Vt^α)+δ (x+Vt^α)/2 ],

where in the last step we used the  result
ℱ^-1 e^± iaξ=δ (x∓ a), a∈ℝ (see,
e.g., <cit.>).


Based on Proposition <ref> we argue the following random model. Let
D be a random variable uniformly distributed on {-1,1}, which is
independent from V. We deal with the  stochastic process
X:=(X(t))_t≥ 0, where


    X(t):=D  V  t^α

represents the position, at time t>0, of a particle starting from the
origin of the real line, which initially chooses with the same probability
to move leftward or rightward and performs a random displacement of length
equal to V t^α. Therefore V represents the random velocity of
the particle which is initially fixed with the probability lawprobability law f_V.


At time t>0


    P(X(t)∈dx)=𝐄_V [
    δ (x-Vt^α)+δ (x+Vt^α)/2 ]
    dx

and the cumulative distribution function of X(t) is given by


    𝐅(x)=1/2 [1+sgn(x)
    Beta(|x|/ct^α;1/β,γ +1)/Beta(1/β,γ +1) ],   x∈ [-ct^α,ct^α].



Given V=v, one has that


    P(X(t)∈ A|V=v)   =∫ _Aν _v(x,t)dx,   A∈ℬ(ℝ),

where


    ν _v(x,t):=
    δ (x-vt^α)+δ (x+vt^α)/2

represents the singular probability measure of X_1 (w.r.t. the Lebesgue
measure dx), and then


    μ _t(dx)   =P(X(t)∈dx)=𝐄_V [
    ν _V(x,t) ]dx.


Some simple calculations and the result (<ref>) lead to
(<ref>).


The following corollary highlights the link between u(x,t) and a model
of nonlinear wave propagation.


Let v>0. Then


    ν _v(x,t):=
    δ (x-vt^α)+δ (x+vt^α)/2

is the fundamental solutionfundamental solution to the  hyperbolic EPD-type partial
differential equation


    ∂ ^2 u/∂ t^2+(1-α )/t∂ u/∂ t=v^2α ^2 t^2α -2∂ ^2 u/∂ x^2

subject to the initial conditions
u(x,0)=δ (x), ∂ u/∂ t(x,t) |_t=0=0.


We observe that δ (x-vs)+δ (x+vs)/2, v>0,s>0, is the
fundamental solutionfundamental solution to the wave equation


    ∂ ^2 u/∂ s^2=v^2∂ ^2 u/∂ x^2,

subject to the initial condition
u(x,0)=δ (x), ∂ u/∂ s(x,s) |_s=0=0. Therefore,
the change of variable s=t^α and direct calculations permit to
prove (<ref>).




The Erdélyi–Kober fractional integral is defined as (see, e.g.,
<cit.>)


    I^ζ , μ_ηf(x) =
    η x^-η (μ +ζ )/Γ (μ )∫ _0^xτ ^η (ζ +1)-1(x^η-τ ^η)^μ -1f(τ )
    dτ ,

where μ >0,η >0 and ζ∈ℝ. Evidently, for
ζ =0,η =1, (<ref>) reduces to the Riemann–Liouville integral
with a power weight.

From Proposition <ref> it is easy to check that the Fourier transform
û(ξ ,t) and u(x,t) are Erdélyi–Kober integrals of the cosine
function
f_α(v;ξ ,t):=cos (ξ v t^α ) and
g_α(v;x,t):=
δ (x-vt^α )+δ (x+vt^α )/2, respectively;
i.e.,


    û(ξ ,t)    =
    Γ (1/β+γ +1)/Γ (1/β)I_β^1/β-1,γ +1f_α(c;ξ ,t)

and


    u(x,t)    =
    Γ (1/β+γ +1)/Γ (1/β)I_β^1/β-1,γ +1g_α(c;x,t).

A recent interesting probabilistic interpretationprobabilistic interpretation of the Erdélyi–Kober integral
is also discussed in <cit.>.




The (centred) Wigner law is defined by the  probability distribution


    𝔪 (x,t)=1/2π t√(4t-x^2),   |x|≤ 2
    √(t).

A simple calculation proves that the even moments are given by (scaled) Catalan
numbers, that is,


    ∫ _-2√(t)^2√(t)x^2m𝔪 ( x,t)dx=C_m
    t^m,    m∈ℕ,

with C_m=1/m+12mm. The probability lawprobability law
(<ref>) is the density functiondensity function of the free Brownian motionBrownian motion
S:=(S_t)_t≥ 0, i.e. for 0≤ t_1<t_2<∞, the law of
S_t_2-S_t_1 is given by 𝔪 (x,t_2-t_1) and
𝐄(S_t_2-S_t_1)=0,
𝐄(S_t_2-S_t_1)^2=t_2-t_1. For a, detailed introduction
to the free probability and free Brownian motionBrownian motion the reader can consult,
for instance, <cit.> and <cit.>.

By setting d=1,α =1/2,β =2, γ =1/2 and c=2, the function
(<ref>) coincides with 𝔪 (x,t). Furthermore, we observe
that 𝔪:=𝔪 ( x,t) is equal to the time-rescaled, with
t=s^2, solution to the EPD equation (<ref>), given by


    ∂ ^2 u/∂ s^2+1/s∂ u/∂ s = 4
    ∂ ^2 u/∂ s^2,      s>0.

Therefore some simple calculations allow to deduce


    ∂ ^2𝔪/∂ t^2+1/2t∂𝔪/∂ t=1/t∂ ^2𝔪/∂ x^2.



The study of the Barenblatt-type solutions for d≥ 2 leads to the following
alternative representations of (<ref>).



For d≥ 2, the probability density functionsprobability density function (<ref>) have the
 representation


    u(x,t)=
    Beta(1/β(d/2+1),γ +1)/σ (𝕊^d-1)(ct^αx)^d/2-1Beta(d/β,γ +1)𝐄_ Z [δ (x- Z t^α) ]

where 𝐄_Z[· ] stands for the mean value w.r.t.
Z(law)=cY_1^1/β where Y_1∼Beta(
1/β(d/2+1),γ +1).


Let d≥ 2. Let σ be the measure on 𝕊^d-1. We recall
that (see (2.12), p. 690pag.690, <cit.>),


    ∫ _𝕊^d-1e^iρξ·θdσ (θ)=(2π )^d/2J_d/2-1(ρξ)/(ρξ)^d/2-1.

One has that


    û(ξ ,t)   =ℱu(ξ ,t)
    
       =C∫ _0^ct^αρ ^d-1 t^-α d (1- (
    ρ/ct^α )^β )^γdρ∫ _𝕊^d-1e^iρξ·θdσ (θ)
    
       =t^-α dC(2π )^d/2/ξ^d/2-1∫ _0^ct^αρ ^d/2 (1- (ρ/ct^α )^β )^γJ_d/2-1(ρξ)
    dρ
       = (2/ct^αξ )^d/2-1Γ (d/2)β /c/Beta(d/β,γ +1)∫ _0^c (z/c)^d/2(1-(z/c)^β)^γJ_d/2-1 (t^αzξ )dz
    
       = (2/ct^αξ )^d/2-1Γ (d/2)Beta(1/β(d/2+1),γ +1)/Beta(d/β,γ +1)𝐄_ Z [J_d/2-1 (Zt^αξ )
     ],

where Z(law)=cY^1/β is the random variable
with the density functiondensity function given by


    f_Z(z)=
    β /c/Beta(1/β(d/2+1),γ +1)(z/c)^d/2(1-(z/c)^β)^γ1_0<z<c.

Hence, by Fubini's theorem we obtain


    u(x,t)   =ℱ^-1u(ξ , t)
    
       =1/(2π )^d∫ _ℝ^de^- i x·ξu(ξ ,
    t) dξ
       =(by passing to spherical coordinates)
    
       =1/(2π )^d (2/ct^α )^d/2-1Γ (d/2)β /c/Beta(d/β,γ +1)
         ×∫ _0^c (z/c)^d/2(1-(z/c)^β)^γdz
    
         ×∫ _0^∞ρ ^d/2 J_d/2-1 (zt^αρ ) (∫ _𝕊^d-1e^-i ρ x·θdσ (θ )  )dρ
       =(by explotinig the result (<ref>))
    
       =1/2π ^d/2Γ (d/2)/(ct^αx)^d/2-1β /c/Beta(d/β,γ +1)
         ×∫ _0^c (z/c)^d/2(1-(z/c)^β)^γdz∫ _0^∞ρ J_d/2-1 (ρ t^α u )J_d/2-1 (ρx)dρ
       =1/2π ^d/2Γ (d/2)/(ct^αx)^d/2-1β /c/Beta(d/β,γ +1)
         ×∫ _0^c (z/c)^d/2(1-(z/c)^β)^γδ (x- zt^α)
    dz
    
       =(by Proposition 2, in <cit.>)
    
       =1/2π ^d/2Γ (d/2)/(ct^αx)^d/2-1Beta(1/β(d/2+1),γ +1)/Beta(d/β,γ +1)𝐄_ Z [δ (x- Z t^α) ]

which coincides with (<ref>).





For d=2, the representation (<ref>) is particularly simple and
reads as


    u(x,t)=1/2π𝐄_ Z [δ (x- Z t^α)
     ].



Let


    g_uw(x,t):=ℱ^-1 e^i ξ u w t=
    1/(2π )^d∫ _ℝ^d e^-i x·ξ e^i ξ u w tdξ _d .

We are able to provide an alternative representation of u(x,t) with respect
to (<ref>).


For d≥ 2, the law (<ref>) can be rewritten as


    u(x,t)=𝐄_𝔘 𝔚 [
    g_𝔘 𝔚(x,t^α)+g_𝔘 𝔚(x,-t^α)/2 ]

where 𝔘(law)=cY_2^1/β, with
Y_2∼Beta(d/β,γ +1), and 𝔚 admits the density
functiondensity function given by
f_𝔚(w)=2/Beta(1/2,d-1/2)(1-w^2)_+^d-1/2-1. Moreover 𝔘 and 𝔚 are independent.



From (<ref>), one has that


    û(ξ ,t)   = (2/ct^αξ )^d/2-1Γ (d/2)β /c/Beta(d/β,γ +1)∫ _0^c (z/c)^d/2(1-(z/c)^β)^γJ_d/2-1 (t^αzξ )dz
    
       =
    2β /c/Beta(d/β,γ +1)Beta(1/2,d-1/2)∫ _0^c(z/c)^d-1(1-(z/c)^β)^γdz
    
         ×∫ _0^1 (1-w^2)^d-1/2-1cos (ξ zw
    t^α)dw
    
       =𝐄_𝔘 𝔚 [cos ( ξ𝔘 𝔚 t^α) ]

where we have used


    J_μ(z)=(z/2)^μ/√(π)Γ (μ +1/2)∫ _-1^+1(1-w^2)^μ -1/2cos (z w)dw

valid for μ >-1/2,z∈ℝ. Therefore, from
(<ref>) we get (<ref>). Indeed,


    u(x,t)   =ℱ^-1û(ξ , t)
    
       =1/(2π )^d2β /c/Beta(d/β,γ +1)Beta(1/2,d-1/2)∫ _0^c(z/c)^d-1(1-(z/c)^β)^γdz
    
         ×∫ _0^1 (1-w^2)^d-1/2-1dw
    ∫ _ℝ^d e^-i x·ξ [
    e^i ξ zw t^α+e^- iξ z w t^α/2 ]dξ

which concludes the proof.





We observe that:



  * For d=2, the density functiondensity function f_𝔚 becomes the probability
lawprobability law of the square root of T, where


    T:=meas{t∈ [0,1]: B(t)>0},

and (B(t))_t≥ 0 represents the standard one-dimensional Brownian
motion. T leads to the well-known arcsin law of the Wiener process which
is given by


    1/π√(w(1-w))1_0<w<1.

It is easy to prove that
√(T)(law)=𝔚.

  * For d=3, the random variable 𝔚, t>0, is uniformly distributed
in (0,1).

  * For d≥ 4, the density functiondensity function f_𝔚 represents a
Wigner (d-2)-sphere law.






It is simple to prove by direct calculations that the mean squared displacement
goes like t^2α, recovering the entire range of behaviours from
sub-diffusion (α <1/2) to super-diffusion (α >1/2); i.e.,


    ∫ _ℝ^dx^2 u(x,t)dx=
    Γ (1/β(d+2))Γ (d/β+γ +1)/Γ (d/β)Γ (1/β(d+2)+γ +1)c^2t^2
    α.







The authors wish to thank the referees for their comments which allowed
the improvement of the previous version of the manuscript.








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