1 Introduction
The Brownian motion is a stochastic process of interest for pure and applied mathematicians. The probability distribution of the Brownian motion is given by the Gaussian kernel
\[ G(x,t)=\frac{1}{{(4\pi t)^{d/2}}}\exp (-\| x{\| ^{2}}/4t),\hspace{1em}x\in {\mathbb{R}^{d}},t>0,\]
which is the source-type solution (that is $G(x,0)=\delta (x)$, where $\delta (x)$ represents Dirac’s delta function) to the parabolic heat equation
The main unwanted feature of this solution is that it is inevitably positive everywhere in its domain of definition; i.e., the Brownian motion scatters with unbounded velocity. A way to overcome this feature is to consider the porous medium equation which is a nonlinear diffusion equation
having the source-type solution given by
\[ U(x,t)=C{t^{-\alpha d}}{\left(1-\frac{\| x{\| ^{2}}}{{c^{2}}{t^{2\alpha }}}\right)_{+}^{\frac{1}{m-1}}},\hspace{1em}\alpha >0,c>0,\]
where ${(x)_{+}}:=\max (x,0)$ and C is a suitable constant such that $\textstyle\int U(x,t)\mathrm{d}x=1$. The solution $U(\cdot ,t)$ 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 (1.1) the reader can consult [33]. The connection between the porous medium equation and the theory of stochastic processes has been investigated, for instance, in [16–18, 2, 11, 19, 29] and [9].The aim of this paper is to study a class of functions generalizing the Barenblatt solution $U(x,t)$. For fixed $t>0$, we consider the map
where $\alpha >0,\beta >0$ and $\gamma >0$, and
(1.2)
\[ {\mathbb{R}^{d}}\ni x\mapsto u:=u(x,t):=C{t^{-\alpha d}}{\left(1-{\left(\frac{\| x\| }{c{t^{\alpha }}}\right)^{\beta }}\right)_{+}^{\gamma }},\]
\[ C:=C(\beta ,\gamma ,d):=\frac{\beta }{{c^{d}}\sigma ({\mathbb{S}^{d-1}})\mathrm{Beta}(\frac{d}{\beta },\gamma +1)}\]
is a positive constant determined by the condition $\| u(x,t){\| _{{L^{1}}({\mathbb{R}^{d}},\mathrm{d}x)}}=1$ (the property of the mass conservation), $\mathrm{d}x$ denoting the Lebesgue measure on $({\mathbb{R}^{d}},\mathcal{B}({\mathbb{R}^{d}}))$, $\sigma ({\mathbb{S}^{d-1}}):=2{\pi ^{d/2}}/\Gamma (d/2)$ represents the surface area of the $(d-1)$-dimensional sphere ${\mathbb{S}^{d-1}}$ with radius one and $\mathrm{Beta}(a,b):=\frac{\Gamma (a)\Gamma (b)}{\Gamma (a+b)},a,b>0$. We observe that u is a probability density function with an associated absolutely continuous probability measure given by
having the following features:-
• The density function u is compactly supported; i.e., for every $t>0$, the support of $u(\cdot ,t)$ is given by representing a closed ball with radius $r(t):=c{t^{\alpha }}$. This property implies the finite speed of propagation of u; i.e., the $(d-1)$-dimensional sphere with radius $r(t)$ denoted by ${\mathbb{S}_{r(t)}^{d-1}}$ provides the free boundary separating the regions $\{(x,t)\in {\mathbb{R}^{d}}\times (0,\infty ):u(x,t)>0\}$ and $\{(x,t)\in {\mathbb{R}^{d}}\times (0,\infty ):u(x,t)=0\}$.
-
• The probability measure ${\mu _{t}}$ is rotationally invariant; that is, ${\mu _{t}}(\mathrm{d}x)=u(\| x\| ,t)\mathrm{d}x$, or equivalently, $\mu (MA)=\mu (A)$ for $A\in \mathcal{B}({\mathbb{R}^{d}})$ and $M\in O(d)$, where $O(d)$ is the group of $d\times d$ orthogonal matrices acting in ${\mathbb{R}^{d}}$, where $d\ge 2$. As a direct consequence of this property we derivewhere $\mathrm{Beta}(x;a,b)={\textstyle\int _{0}^{x}}{y^{a-1}}{(1-y)^{b-1}}\mathrm{d}y,x\in \mathbb{R}$, is the incomplete Gamma function.
(1.4)
\[\begin{aligned}{}{\mu _{t}}({B_{a}})& =\sigma ({\mathbb{S}^{d-1}}){\int _{0}^{a}}{r^{d-1}}u(r,t)\mathrm{d}r\\ {} & =\frac{\mathrm{Beta}\left({(a/c{t^{\alpha }})^{\beta }};\frac{d}{\beta },\gamma +1\right)}{\mathrm{Beta}(\frac{d}{\beta },\gamma +1)},\hspace{1em}0<a<c{t^{\alpha }},\end{aligned}\]
We refer to (1.2) as the class of the Barenblatt-type solutions. The family of functions (1.2) contains as particular cases, for instance, weak solutions of several nonlinear and linear diffusion equations (see Section 2). Furthermore, in [13] it is proved that the mean exit time of a symmetric 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 3). 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 (1.2) emerge from different frameworks (linear hyperbolic, nonlinear parabolic and nonlocal). In this way, objects requiring different mathematical tools have common features.
2 Barenblatt-type solutions to diffusion equations
The aim of this section is to highlight that the class of density functions of the form (1.2) is very general. Solutions belonging to family (1.2) appear in different frameworks (linear hyperbolic, nonlinear parabolic and nonlocal). 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 $\mathcal{F}$ and the inverse transform ${\mathcal{F}^{-1}}$ of a function $v\in {L^{1}}({\mathbb{R}^{d}},\mathrm{d}x)$ are defined by
\[ \mathcal{F}v(\xi )={\int _{{\mathbb{R}^{d}}}}v(x){e^{ix\cdot \xi }}\mathrm{d}x,\hspace{2em}{\mathcal{F}^{-1}}v(x)=\frac{1}{{(2\pi )^{d}}}{\int _{{\mathbb{R}^{d}}}}v(\xi ){e^{-ix\cdot \xi }}\mathrm{d}\xi ,\]
with $\xi \in {\mathbb{R}^{d}}$.2.1 Nonlinear diffusions: p-Laplacian equation
We mean here the p-Laplacian equation (PLE for short) studied, for instance, in [20] and [23], which is the following nonlinear degenerate parabolic evolution equation
subject to the initial condition
where $u:=u(x,t)$, with $x\in {\mathbb{R}^{d}},d\ge 1$, is a scalar function defined on ${\mathbb{R}^{d}}\times {\mathbb{R}^{+}}$. The Cauchy problem (2.1)–(2.2) admits a unique nonnegative fundamental solution: it is a function $u\ge 0$ solving (2.1)–(2.5) in a weak sense (see [20] and [23] for the detailed definition of weak solution to PLE). This solution is given by the following probability density function
where
(2.1)
\[ \frac{\partial u}{\partial t}=\text{div}\left(|\nabla u{|^{p-2}}\nabla u\right),\hspace{1em}p>2,\hspace{1em}t>0,\](2.3)
\[ u(x,t)={t^{-k}}{\left(\mathfrak{c}-q{\left(\frac{\| x\| }{{t^{k/d}}}\right)^{\frac{p}{p-1}}}\right)_{+}^{\frac{p-1}{p-2}}},\]
\[ k:={\left(p-2+\frac{p}{d}\right)^{-1}},\hspace{2em}q:=\frac{p-2}{p}{\left(\frac{k}{d}\right)^{\frac{1}{p-1}}}\]
and $\mathfrak{c}:=\mathfrak{c}(p,d)$ is a constant determined by the condition $\textstyle\int u(x,t)\mathrm{d}x=1$. By setting $\beta =\frac{p}{p-1},\gamma =\frac{p-1}{p-2},\alpha =\frac{k}{d},C={\mathfrak{c}^{\frac{p-1}{p-2}}}$ and $c={(\mathfrak{c}/q)^{\frac{p-1}{p}}}$, the Barenblatt-type solution (1.2) coincides with (2.3).2.2 Nonlinear diffusions: nonlocal porous medium equation
The Nonlocal Porous Medium Equation (NPME), studied in [3] and [4], is the following degenerate nonlinear and nonlocal evolution equation
subject to the initial condition
The pseudo-differential operator ${\nabla ^{\nu -1}}$ is the fractional gradient denoting the nonlocal operator defined as ${\nabla ^{\nu -1}}u:={\mathcal{F}^{-1}}(i\xi \| \xi {\| ^{\nu -2}}\mathcal{F}u)$. This notation highlights that ${\nabla ^{\nu -1}}$ is a pseudo-differential (vector-valued) operator of order $\nu -1$. Equivalently, we can define ${\nabla ^{\nu -1}}$ as $\nabla {(-\Delta )^{\frac{\nu }{2}-1}}$, where ${(-\Delta )^{\frac{\nu }{2}}}u={\mathcal{F}^{-1}}(\| \xi {\| ^{\nu }}\mathcal{F}u)$ is the fractional Laplace operator, i.e., a Fourier multiplier with the symbol $\| \xi {\| ^{\nu }}$. For $\nu =2$, (2.4) becomes the classical nonlinear porous medium equation
If we restrict our attention to nonnegative solution $u(x,t)$, equation (2.6) becomes
which is usually adopted to model the flow of a gas through a porous medium.
(2.4)
\[ \frac{\partial u}{\partial t}=\text{div}\left(|u|{\nabla ^{\nu -1}}(|u{|^{m-2}}u)\right),\hspace{1em}m>1,\hspace{0.1667em}\nu \in (0,2],\hspace{0.1667em}t>0,\](2.6)
\[ \frac{\partial u}{\partial t}=\text{div}\left(|u|\nabla (|u{|^{m-2}}u)\right)=\text{div}\left((m-1)|u{|^{m-1}}\nabla u\right).\]Let $\nu \in (0,2]$ and $m>1$. A weak solution, in the sense of Definition 1 in [4], is given by
where $\alpha :=\frac{1}{d(m-1)+\nu }$, $k:=\frac{d\Gamma (d/2)}{(d(m-1)+\nu ){2^{\nu }}\Gamma (1+\frac{\nu }{2})\Gamma (\frac{d+\nu }{2})}$ and
(2.8)
\[ u(x,t)=C{t^{-d\alpha }}{\left(1-{k^{\frac{2}{\nu }}}\frac{\| x{\| ^{2}}}{{t^{2\alpha }}}\right)_{+}^{\frac{\nu }{2(m-1)}}},\]
\[ \hspace{1em}C:=\frac{\Gamma (\frac{d}{2}+\frac{\nu }{2(m-1)}+1){k^{\frac{d}{\nu }}}}{{\pi ^{\frac{d}{2}}}\Gamma (\frac{\nu }{2(m-1)}+1)}.\]
Furthermore, $u(x,t)$ is the pointwise solution of equation (2.4) for $\| x\| \ne {k^{-\frac{1}{\nu }}}{t^{\alpha }}$. The link between (2.8) and random flights has been investigated in [8]. For $\nu =2$, the solution (2.3) becomes the Barenblatt–Kompanets–Zel’dovich–Pattle solution of the porous medium equation (2.7) supplemented with the initial condition $u(x,0)=\delta (x)$ (see, for instance, [33]).2.3 Euler–Poisson–Darboux equation
It is well known that the fundamental solution of the Euler–Poisson–Darboux (EPD) equation
has the form
and therefore it belongs to the family of probability density functions with compact support (1.2) with $\beta =2$, $\alpha =1$, $k=d$ and $\gamma =\nu -1$. There is a wide literature about the EPD equation and its applications. We refer to Bresters [5] for the construction of weak solutions of the initial value problem for the EPD equation based on distributional methods. We recall that, in the one-dimensional case, the solution to the Cauchy problem
can be represented as the Erdélyi–Kober fractional integral (see definition (3.4) below) of the D’Alembert solution of the wave equation (see [10])
(2.9)
\[ \frac{{\partial ^{2}}u}{\partial {t^{2}}}+\frac{d+2\nu -1}{t}\frac{\partial u}{\partial t}={c^{2}}\Delta u,\hspace{1em}\nu >0,\hspace{2.5pt}t>0,\hspace{2.5pt}c>0,\](2.10)
\[ u(x,t)=\frac{\Gamma (\nu +\frac{d}{2})}{{\pi ^{d/2}}\Gamma (\nu )}\frac{1}{{(ct)^{d}}}{\left(1-\frac{\| x{\| ^{2}}}{{(ct)^{2}}}\right)_{+}^{\nu -1}}\](2.11)
\[ \left\{\begin{array}{l}\displaystyle \frac{{\partial ^{2}}u}{\partial {t^{2}}}+\displaystyle \frac{2\xi }{t}\displaystyle \frac{\partial u}{\partial t}={c^{2}}\displaystyle \frac{{\partial ^{2}}u}{\partial {x^{2}}},\hspace{1em}\\ {} u(x,0)=f(x),\hspace{1em}\\ {} \displaystyle \frac{\partial u}{\partial t}(x,t){\big|_{t=0}}=0,\hspace{1em}\end{array}\right.\]This means that
The first probabilistic interpretation of this analytic representation, discussed by Rosencrans [31] and more recently by Garra and Orsingher [12], is the following one: solution (2.13) can be written as
where
and ${(N(t))_{t\ge 0}}$ is the nonhomogeneous Poisson process with rate $\lambda (t)=\frac{\xi }{t}$, $U(0)$ is a uniformly distributed r.v. on $\{-c,c\}$ (furthermore ${(N(t))_{t\ge 0}}$ and $U(0)$ are supposed independent). By means of the general Proposition 1 (see the next section), we here obtain a new interesting probabilistic interpretation of the fundamental solution of the EPD equation.
(2.13)
\[\begin{aligned}{}u(x,t)& =\frac{2}{\mathrm{Beta}(\xi ,\frac{1}{2})}{\int _{0}^{1}}{(1-{y^{2}})^{\xi -1}}\left[\frac{f(x+yct)+f(x-yct)}{2}\right]\mathrm{d}y.\end{aligned}\](2.14)
\[ u(x,t)=\mathbf{E}\left[\frac{f(x+\hspace{0.1667em}\mathcal{U}(t))+f(x-\hspace{0.1667em}\mathcal{U}(t))}{2}\right],\]Moreover, we have the following interesting picture that underlines the role of the Barenblatt-type solution as a bridge between nonlinear and linear PDEs:
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.
2.4 Nonlinear time-fractional diffusive equations admitting Barenblatt-type solutions
There is a wide literature about the probabilistic interpretation of linear space and time-fractional diffusive equations (see, e.g., [24, 1, 6, 27] 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 equations has been investigated (see, e.g., [30]). 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 (1.2).
Let us consider the following nonlinear diffusive equation
where $\frac{{\partial ^{\nu }}}{\partial {t^{\nu }}}$ denotes the Riemann–Liouville derivative
(2.16)
\[ \frac{1}{{t^{2\nu }}}\frac{{\partial ^{\nu }}u}{\partial {t^{\nu }}}+\frac{1}{{t^{\nu }}}\frac{{\partial ^{2}}u}{\partial {x^{2}}}+{\left(\frac{\partial u}{\partial x}\right)^{2}}=0,\]
\[ \frac{{\partial ^{\nu }}f(x,t)}{\partial {t^{\nu }}}=\frac{1}{\Gamma (1-\nu )}\frac{\partial }{\partial t}{\int _{0}^{t}}{(t-s)^{-\nu }}f(x,s)\mathrm{d}s,\hspace{1em}\nu \in (0,1),\]
and $f(x,\cdot )$ is a suitable well-behaved function (see [22] for details about the functional setting).We start our analysis from a simple ansatz: equation (2.16) admits a solution in the form
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 $\nu >0$ and $\beta >-1$,
Then, by substituting (2.17) in (2.16) and by applying (2.18), we have
and balancing similar terms we have that
Therefore
(2.17)
\[ u(x,t)=\frac{{C_{1}}}{{t^{\nu }}}-{C_{2}}\frac{{x^{2}}}{{t^{3\nu }}},\hspace{1em}\nu \in \left(0,1/3\right)\setminus \{1/4\},\hspace{2.5pt}(x,t)\in \mathbb{R}\times {\mathbb{R}^{+}},\](2.18)
\[ \frac{{\partial ^{\nu }}}{\partial {t^{\nu }}}{t^{\beta }}=\frac{\Gamma (1+\beta )}{\Gamma (1+\beta -\nu )}{t^{\beta -\nu }}.\](2.19)
\[ \frac{\Gamma (1-\nu )}{\Gamma (1-2\nu )}\frac{{C_{1}}}{{t^{4\nu }}}-\frac{\Gamma (1-3\nu )}{\Gamma (1-4\nu )}\frac{{C_{2}}\hspace{2.5pt}{x^{2}}}{{t^{6\nu }}}-\frac{2{C_{2}}}{{t^{4\nu }}}+\frac{4{C_{2}^{2}}{x^{2}}}{{t^{6\nu }}}=0\](2.20)
\[ \left\{\begin{array}{l}\displaystyle \frac{\Gamma (1-\nu )}{\Gamma (1-2\nu )}{C_{1}}-2{C_{2}}=0,\hspace{1em}\\ {} 4{C_{2}^{2}}-\displaystyle \frac{\Gamma (1-3\nu )}{\Gamma (1-4\nu )}{C_{2}}=0.\hspace{1em}\end{array}\right.\](2.21)
\[ \left\{\begin{array}{l}{C_{1}}=\displaystyle \frac{1}{2}\displaystyle \frac{\Gamma (1-3\nu )}{\Gamma (1-4\nu )}\displaystyle \frac{\Gamma (1-2\nu )}{\Gamma (1-\nu )},\hspace{1em}\\ {} {C_{2}}=\displaystyle \frac{1}{4}\displaystyle \frac{\Gamma (1-3\nu )}{\Gamma (1-4\nu )}.\hspace{1em}\end{array}\right.\]Observe that the constraint on the real order of derivation $\nu \in \left(0,\frac{1}{3}\right)$ in (2.17) is due to the application of (2.18).
Moreover, $\nu \ne 1/4$ because of the coefficients appearing in the solution (depending on the Euler Gamma functions). We can conclude that equation (2.16) admits a solution of the form (2.17) if $\nu \in (0,1/3)\setminus \{1/4\}$. Therefore, in particular, under the same constraints on ν, we can say that the time-fractional equation (2.16) admits a solution of the form
where ${C_{1}}$ and ${C_{2}}$ are given by (2.21).
(2.22)
\[ u(x,t)=\frac{{C_{1}}}{{t^{\nu }}}{\left(1-\frac{{C_{2}}}{{C_{1}}}\frac{{x^{2}}}{{t^{2\nu }}}\right)_{+}},\]Obviously, the solution (2.22) is not normalized but it is a Barenblatt-type solution belonging to the general family considered in this paper. A systematic study of equation (2.16) 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 (2.16); nevertheless a full rigorous analysis should be developed, but this is beyond the aims of this paper. The fractional equation (2.16) can be viewed as a hybrid between a diffusive equation with singular time-dependent coefficients (in some way similar to the EPD equation) and a nonlinear time-fractional porous medium type equation.
3 Main results
Let us start with our first result concerning the case $d=1$.
Proposition 1.
For $d=1$, the density function (1.2) can be written as
where ${\mathbf{E}_{V}}[\cdot ]$ stands for the mean value w.r.t. $V\stackrel{\textit{(law)}}{=}c{Y^{1/\beta }}$, where $Y\sim \mathrm{Beta}(\frac{1}{\beta },\gamma +1)$.
(3.1)
\[ u(x,t)={\mathbf{E}_{V}}\left[\frac{\delta (x-V{t^{\alpha }})+\delta (x+V{t^{\alpha }})}{2}\right],\]Proof.
Let $d=1$. We have
\[\begin{aligned}{}\hat{u}(\xi ,t)& =\mathcal{F}u(\xi ,t)\\ {} & =C{t^{-\alpha }}{\int _{-c{t^{\alpha }}}^{c{t^{\alpha }}}}{e^{i\xi x}}{\left(1-{\left(\frac{|x|}{c{t^{\alpha }}}\right)^{\beta }}\right)^{\gamma }}\mathrm{d}x\\ {} & =2C{t^{-\alpha }}{\int _{0}^{c{t^{\alpha }}}}\cos (\xi x){\left(1-{\left(\frac{x}{c{t^{\alpha }}}\right)^{\beta }}\right)^{\gamma }}\mathrm{d}x\\ {} & =\frac{\beta /c}{\mathrm{Beta}(\frac{1}{\beta },\gamma +1)}{\int _{0}^{c}}\cos (\xi v{t^{\alpha }}){\left(1-{(v/c)^{\beta }}\right)^{\gamma }}\mathrm{d}v\\ {} & ={\mathbf{E}_{V}}\left[\cos (\xi V{t^{\alpha }})\right]\\ {} & ={\mathbf{E}_{V}}\left[\frac{{e^{i\xi V{t^{\alpha }}}}+{e^{-i\xi V{t^{\alpha }}}}}{2}\right].\end{aligned}\]
Hence, by Fubini’s theorem we immediately obtain
\[ u(x,t)={\mathcal{F}^{-1}}\hat{u}(\xi ,t)={\mathbf{E}_{V}}\left[\frac{\delta (x-V{t^{\alpha }})+\delta (x+V{t^{\alpha }})}{2}\right],\]
where in the last step we used the result ${\mathcal{F}^{-1}}{e^{\pm ia\xi }}=\delta (x\mp a),a\in \mathbb{R}$ (see, e.g., [14]). □Based on Proposition 1 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 $X:={(X(t))_{t\ge 0}}$, where
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 $V{t^{\alpha }}$. Therefore V represents the random velocity of the particle which is initially fixed with the probability law ${f_{V}}$.
Corollary 1.
Proof.
The following corollary highlights the link between $u(x,t)$ and a model of nonlinear wave propagation.
Corollary 2.
Let $v>0$. Then
is the fundamental solution to the hyperbolic EPD-type partial differential equation
subject to the initial conditions $u(x,0)=\delta (x),\frac{\partial u}{\partial t}(x,t){\big|_{t=0}}=0$.
(3.3)
\[ \frac{{\partial ^{2}}u}{\partial {t^{2}}}+\frac{(1-\alpha )}{t}\frac{\partial u}{\partial t}={v^{2}}{\alpha ^{2}}{t^{2\alpha -2}}\frac{{\partial ^{2}}u}{\partial {x^{2}}}\]Proof.
We observe that $\frac{\delta (x-vs)+\delta (x+vs)}{2},v>0,s>0$, is the fundamental solution to the wave equation
subject to the initial condition $u(x,0)=\delta (x),\frac{\partial u}{\partial s}(x,s){\big|_{s=0}}=0$. Therefore, the change of variable $s={t^{\alpha }}$ and direct calculations permit to prove (3.3). □
Remark 3.1.
The Erdélyi–Kober fractional integral is defined as (see, e.g., [28])
where $\mu >0,\eta >0$ and $\zeta \in \mathbb{R}$. Evidently, for $\zeta =0,\eta =1$, (3.4) reduces to the Riemann–Liouville integral with a power weight.
(3.4)
\[ {I_{\eta }^{\zeta ,\mu }}f(x)=\frac{\eta \hspace{2.5pt}{x^{-\eta (\mu +\zeta )}}}{\Gamma (\mu )}{\int _{0}^{x}}{\tau ^{\eta (\zeta +1)-1}}{({x^{\eta }}-{\tau ^{\eta }})^{\mu -1}}f(\tau )\mathrm{d}\tau ,\]From Proposition 1 it is easy to check that the Fourier transform $\hat{u}(\xi ,t)$ and $u(x,t)$ are Erdélyi–Kober integrals of the cosine function ${f_{\alpha }}(v;\xi ,t):=\cos \left(\xi v{t^{\alpha }}\right)$ and ${g_{\alpha }}(v;x,t):=\frac{\delta (x-v{t^{\alpha }})+\delta (x+v{t^{\alpha }})}{2}$, respectively; i.e.,
and
A recent interesting probabilistic interpretation of the Erdélyi–Kober integral is also discussed in [32].
(3.5)
\[\begin{aligned}{}\hat{u}(\xi ,t)& =\frac{\Gamma (\frac{1}{\beta }+\gamma +1)}{\Gamma (\frac{1}{\beta })}{I_{\beta }^{\frac{1}{\beta }-1,\gamma +1}}{f_{\alpha }}(c;\xi ,t)\end{aligned}\](3.6)
\[\begin{aligned}{}u(x,t)& =\frac{\Gamma (\frac{1}{\beta }+\gamma +1)}{\Gamma (\frac{1}{\beta })}{I_{\beta }^{\frac{1}{\beta }-1,\gamma +1}}{g_{\alpha }}(c;x,t).\end{aligned}\]Remark 3.2.
The (centred) Wigner law is defined by the probability distribution
A simple calculation proves that the even moments are given by (scaled) Catalan numbers, that is,
\[ {\int _{-2\sqrt{t}}^{2\sqrt{t}}}{x^{2m}}\mathfrak{m}(x,t)\mathrm{d}x={C_{m}}{t^{m}},\hspace{1em}m\in \mathbb{N},\]
with ${C_{m}}=\frac{1}{m+1}\left(\genfrac{}{}{0.0pt}{}{2m}{m}\right)$. The probability law (3.7) is the density function of the free Brownian motion $S:={({S_{t}})_{t\ge 0}}$, i.e. for $0\le {t_{1}}<{t_{2}}<\infty $, the law of ${S_{{t_{2}}}}-{S_{{t_{1}}}}$ is given by $\mathfrak{m}(x,{t_{2}}-{t_{1}})$ and $\mathbf{E}({S_{{t_{2}}}}-{S_{{t_{1}}}})=0$, $\mathbf{E}{({S_{{t_{2}}}}-{S_{{t_{1}}}})^{2}}={t_{2}}-{t_{1}}$. For a, detailed introduction to the free probability and free Brownian motion the reader can consult, for instance, [34, 35] and [26].By setting $d=1,\alpha =1/2,\beta =2,\gamma =1/2$ and $c=2$, the function (1.2) coincides with $\mathfrak{m}(x,t)$. Furthermore, we observe that $\mathfrak{m}:=\mathfrak{m}(x,t)$ is equal to the time-rescaled, with $t={s^{2}}$, solution to the EPD equation (2.10), given by
Therefore some simple calculations allow to deduce
The study of the Barenblatt-type solutions for $d\ge 2$ leads to the following alternative representations of (1.2).
Proposition 2.
For $d\ge 2$, the probability density functions (1.2) have the representation
where ${\mathbf{E}_{Z}}[\cdot ]$ stands for the mean value w.r.t. $Z\stackrel{\textit{(law)}}{=}c{Y_{1}^{1/\beta }}$ where ${Y_{1}}\sim \mathrm{Beta}(\frac{1}{\beta }(\frac{d}{2}+1),\gamma +1)$.
(3.9)
\[ u(x,t)=\frac{\mathrm{Beta}(\frac{1}{\beta }(\frac{d}{2}+1),\gamma +1)}{\sigma ({\mathbb{S}^{d-1}}){(c{t^{\alpha }}\| x\| )^{\frac{d}{2}-1}}\mathrm{Beta}(\frac{d}{\beta },\gamma +1)}{\mathbf{E}_{Z}}\left[\delta (\| x\| -Z{t^{\alpha }})\right]\]Proof.
Let $d\ge 2$. Let σ be the measure on ${\mathbb{S}^{d-1}}$. We recall that (see (2.12), p. 690, [7]),
One has that
where $Z\stackrel{\text{(law)}}{=}c{Y^{1/\beta }}$ is the random variable with the density function given by
(3.10)
\[ {\int _{{\mathbb{S}^{d-1}}}}{e^{i\rho \xi \cdot \theta }}\mathrm{d}\sigma (\theta )={(2\pi )^{d/2}}\frac{{J_{\frac{d}{2}-1}}(\rho \| \xi \| )}{{(\rho \| \xi \| )^{\frac{d}{2}-1}}}.\](3.11)
\[\begin{aligned}{}\hat{u}(\xi ,t)& =\mathcal{F}u(\xi ,t)\\ {} & =C{\int _{0}^{c{t^{\alpha }}}}{\rho ^{d-1}}{t^{-\alpha d}}{\left(1-{\left(\frac{\rho }{c{t^{\alpha }}}\right)^{\beta }}\right)^{\gamma }}\mathrm{d}\rho {\int _{{\mathbb{S}^{d-1}}}}{e^{i\rho \xi \cdot \theta }}\mathrm{d}\sigma (\theta )\\ {} & =\frac{{t^{-\alpha d}}C{(2\pi )^{d/2}}}{\| \xi {\| ^{\frac{d}{2}-1}}}{\int _{0}^{c{t^{\alpha }}}}{\rho ^{\frac{d}{2}}}{\left(1-{\left(\frac{\rho }{c{t^{\alpha }}}\right)^{\beta }}\right)^{\gamma }}{J_{\frac{d}{2}-1}}(\rho \| \xi \| )\mathrm{d}\rho \\ {} & ={\left(\frac{2}{c{t^{\alpha }}\| \xi \| }\right)^{\frac{d}{2}-1}}\frac{\Gamma (d/2)\beta /c}{\mathrm{Beta}(\frac{d}{\beta },\gamma +1)}{\int _{0}^{c}}{(z/c)^{\frac{d}{2}}}{(1-{(z/c)^{\beta }})^{\gamma }}{J_{\frac{d}{2}-1}}\left({t^{\alpha }}z\| \xi \| \right)\mathrm{d}z\\ {} & ={\left(\frac{2}{c{t^{\alpha }}\| \xi \| }\right)^{\frac{d}{2}-1}}\frac{\Gamma (d/2)\mathrm{Beta}(\frac{1}{\beta }(\frac{d}{2}+1),\gamma +1)}{\mathrm{Beta}(\frac{d}{\beta },\gamma +1)}{\mathbf{E}_{Z}}\left[{J_{\frac{d}{2}-1}}\left(Z{t^{\alpha }}\| \xi \| \right)\right],\end{aligned}\]
\[ {f_{Z}}(z)=\frac{\beta /c}{\mathrm{Beta}(\frac{1}{\beta }(\frac{d}{2}+1),\gamma +1)}{(z/c)^{\frac{d}{2}}}{(1-{(z/c)^{\beta }})^{\gamma }}{1_{0<z<c}}.\]
Hence, by Fubini’s theorem we obtain
\[\begin{aligned}{}u(x,t)& ={\mathcal{F}^{-1}}u(\xi ,t)\\ {} & =\frac{1}{{(2\pi )^{d}}}{\int _{{\mathbb{R}^{d}}}}{e^{-ix\cdot \xi }}u(\xi ,t)\mathrm{d}\xi \\ {} & =(\text{by passing to spherical coordinates})\\ {} & =\frac{1}{{(2\pi )^{d}}}{\left(\frac{2}{c{t^{\alpha }}}\right)^{\frac{d}{2}-1}}\frac{\Gamma (d/2)\beta /c}{\mathrm{Beta}(\frac{d}{\beta },\gamma +1)}\\ {} & \hspace{1em}\times {\int _{0}^{c}}{(z/c)^{\frac{d}{2}}}{(1-{(z/c)^{\beta }})^{\gamma }}\mathrm{d}z\\ {} & \hspace{1em}\times {\int _{0}^{\infty }}{\rho ^{d/2}}{J_{\frac{d}{2}-1}}\left(z{t^{\alpha }}\rho \right)\left({\int _{{\mathbb{S}^{d-1}}}}{e^{-i\rho x\cdot \theta }}\mathrm{d}\sigma (\theta )\right)\mathrm{d}\rho \\ {} & =(\text{by explotinig the result}\hspace{0.1667em}\text{(3.10)})\\ {} & =\frac{1}{2{\pi ^{d/2}}}\frac{\Gamma (d/2)}{{(c{t^{\alpha }}\| x\| )^{\frac{d}{2}-1}}}\frac{\beta /c}{\mathrm{Beta}(\frac{d}{\beta },\gamma +1)}\\ {} & \hspace{1em}\times {\int _{0}^{c}}{(z/c)^{\frac{d}{2}}}{(1-{(z/c)^{\beta }})^{\gamma }}\mathrm{d}z{\int _{0}^{\infty }}\rho {J_{\frac{d}{2}-1}}\left(\rho {t^{\alpha }}u\right){J_{\frac{d}{2}-1}}(\rho \| x\| )\mathrm{d}\rho \\ {} & =\frac{1}{2{\pi ^{d/2}}}\frac{\Gamma (d/2)}{{(c{t^{\alpha }}\| x\| )^{\frac{d}{2}-1}}}\frac{\beta /c}{\mathrm{Beta}(\frac{d}{\beta },\gamma +1)}\\ {} & \hspace{1em}\times {\int _{0}^{c}}{(z/c)^{\frac{d}{2}}}{(1-{(z/c)^{\beta }})^{\gamma }}\delta (\| x\| -z{t^{\alpha }})\mathrm{d}z\\ {} & =(\text{by Proposition 2, in [21]})\\ {} & =\frac{1}{2{\pi ^{d/2}}}\frac{\Gamma (d/2)}{{(c{t^{\alpha }}\| x\| )^{\frac{d}{2}-1}}}\frac{\mathrm{Beta}(\frac{1}{\beta }(\frac{d}{2}+1),\gamma +1)}{\mathrm{Beta}(\frac{d}{\beta },\gamma +1)}{\mathbf{E}_{Z}}\left[\delta (\| x\| -Z{t^{\alpha }})\right]\end{aligned}\]
which coincides with (3.9). □Let
\[ {g_{uw}}(x,t):={\mathcal{F}^{-1}}{e^{i\| \xi \| uwt}}=\frac{1}{{(2\pi )^{d}}}{\int _{{\mathbb{R}^{d}}}}{e^{-ix\cdot \xi }}{e^{i\| \xi \| uwt}}\mathrm{d}{\xi _{d}}.\]
We are able to provide an alternative representation of $u(x,t)$ with respect to (3.9).Proposition 3.
For $d\ge 2$, the law (1.2) can be rewritten as
where $\mathfrak{U}\stackrel{\textit{(law)}}{=}c{Y_{2}^{1/\beta }}$, with ${Y_{2}}\sim \mathrm{Beta}(\frac{d}{\beta },\gamma +1)$, and $\mathfrak{W}$ admits the density function given by ${f_{\mathfrak{W}}}(w)=\frac{2}{\mathrm{Beta}(\frac{1}{2},\frac{d-1}{2})}{(1-{w^{2}})_{+}^{\frac{d-1}{2}-1}}$. Moreover $\mathfrak{U}$ and $\mathfrak{W}$ are independent.
(3.12)
\[ u(x,t)={\mathbf{E}_{\mathfrak{U}\hspace{0.1667em}\mathfrak{W}}}\left[\frac{{g_{\mathfrak{U}\hspace{0.1667em}\mathfrak{W}}}(x,{t^{\alpha }})+{g_{\mathfrak{U}\hspace{0.1667em}\mathfrak{W}}}(x,-{t^{\alpha }})}{2}\right]\]Proof.
From (3.11), one has that
where we have used
valid for $\mu >-\frac{1}{2},z\in \mathbb{R}$. Therefore, from (3.13) we get (3.12). Indeed,
(3.13)
\[\begin{aligned}{}\hat{u}(\xi ,t)& ={\left(\frac{2}{c{t^{\alpha }}\| \xi \| }\right)^{\frac{d}{2}-1}}\frac{\Gamma (d/2)\beta /c}{\mathrm{Beta}(\frac{d}{\beta },\gamma +1)}{\int _{0}^{c}}{(z/c)^{\frac{d}{2}}}{(1-{(z/c)^{\beta }})^{\gamma }}{J_{\frac{d}{2}-1}}\left({t^{\alpha }}z\| \xi \| \right)\mathrm{d}z\\ {} & =\frac{2\beta /c}{\mathrm{Beta}(\frac{d}{\beta },\gamma +1)\mathrm{Beta}(\frac{1}{2},\frac{d-1}{2})}{\int _{0}^{c}}{(z/c)^{d-1}}{(1-{(z/c)^{\beta }})^{\gamma }}\mathrm{d}z\\ {} & \hspace{1em}\times {\int _{0}^{1}}{(1-{w^{2}})^{\frac{d-1}{2}-1}}\cos (\| \xi \| zw{t^{\alpha }})\mathrm{d}w\\ {} & ={\mathbf{E}_{\mathfrak{U}\hspace{0.1667em}\mathfrak{W}}}\left[\cos (\| \xi \| \mathfrak{U}\hspace{0.1667em}\mathfrak{W}\hspace{0.1667em}{t^{\alpha }})\right]\end{aligned}\](3.14)
\[ {J_{\mu }}(z)=\frac{{(z/2)^{\mu }}}{\sqrt{\pi }\Gamma (\mu +\frac{1}{2})}{\int _{-1}^{+1}}{(1-{w^{2}})^{\mu -\frac{1}{2}}}\cos (zw)\mathrm{d}w\]
\[\begin{aligned}{}u(x,t)& ={\mathcal{F}^{-1}}\hat{u}(\xi ,t)\\ {} & =\frac{1}{{(2\pi )^{d}}}\frac{2\beta /c}{\mathrm{Beta}(\frac{d}{\beta },\gamma +1)\mathrm{Beta}(\frac{1}{2},\frac{d-1}{2})}{\int _{0}^{c}}{(z/c)^{d-1}}{(1-{(z/c)^{\beta }})^{\gamma }}\mathrm{d}z\\ {} & \hspace{1em}\times {\int _{0}^{1}}{(1-{w^{2}})^{\frac{d-1}{2}-1}}\mathrm{d}w{\int _{{\mathbb{R}^{d}}}}{e^{-ix\cdot \xi }}\left[\frac{{e^{i\| \xi \| zw{t^{\alpha }}}}+{e^{-i\| \xi \| zw{t^{\alpha }}}}}{2}\right]\mathrm{d}\xi \end{aligned}\]
which concludes the proof. □Remark 3.4.
We observe that:
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• For $d=2$, the density function ${f_{\mathfrak{W}}}$ becomes the probability law of the square root of T, where and ${(B(t))_{t\ge 0}}$ represents the standard one-dimensional Brownian motion. T leads to the well-known arcsin law of the Wiener process which is given by It is easy to prove that $\sqrt{T}\stackrel{\text{(law)}}{=}\mathfrak{W}$.
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• For $d=3$, the random variable $\mathfrak{W},t>0$, is uniformly distributed in $(0,1)$.
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• For $d\ge 4$, the density function ${f_{\mathfrak{W}}}$ represents a Wigner $(d-2)$-sphere law.
