Modern Stochastics: Theory and Applications

We deal with a Nonlocal Porous Medium Equation (NPME) studied in [3, 4], given by the following degenerate nonlinear and nonlocal evolution equation subject to the initial condition where $u:=u(x,t)$, with $x:=({x_{1}},\dots ,{x_{d}})\in {\mathbb{R}^{d}},d\ge 1$, is a scalar function defined on ${\mathbb{R}^{d}}\times {\mathbb{R}^{+}}$ and ${\partial _{t}}:=\partial /\partial t$. The pseudo-differential operator ${\nabla ^{\alpha -1}}$ is the fractional gradient denoting the nonlocal operator defined as ${\nabla ^{\alpha -1}}u:={\mathcal{F}^{-1}}(i\xi ||\xi |{|^{\alpha -2}}\mathcal{F}u)$, where the Fourier transform $\mathcal{F}$ and the inverse transform ${\mathcal{F}^{-1}}$ of a function $v\in {L^{1}}({\mathbb{R}^{d}})$ are defined by with $\xi \in {\mathbb{R}^{d}}$. This notation highlights that ${\nabla ^{\alpha -1}}$ is a pseudo-differential (vector-valued) operator of order $\alpha -1$. Equivalently, we can define ${\nabla ^{\alpha -1}}$ as $\nabla {(-\Delta )^{\frac{\alpha }{2}-1}}$, where ${(-\Delta )^{\frac{\alpha }{2}}}u={\mathcal{F}^{-1}}(||\xi |{|^{\alpha }}\mathcal{F}u)$ is the fractional Laplace operator; i.e. a Fourier multiplier with the symbol $||\xi |{|^{\alpha }}$. We observe that ${\nabla ^{1}}=\nabla $ is the classical gradient and that div$({\nabla ^{\alpha -1}})={\nabla ^{\frac{\alpha }{2}}}\cdot {\nabla ^{\frac{\alpha }{2}}}=-{(-\Delta )^{\frac{\alpha }{2}}}$. Another equivalent definition of the fractional gradient ${\nabla ^{\alpha -1}}$ involves the Riesz potential; that is ${\nabla ^{\alpha -1}}=\nabla {I_{2-\alpha }}$ where ${I_{\beta }}={(-\Delta )^{-\frac{\beta }{2}}}$ is a Fourier multiplier with symbol $||\xi |{|^{-\beta }},\beta \in (0,2)$ (for more details on this point see [3, 4]).

In [3, 4], explicit and compactly supported nonnegative self-similar solutions of (1) are constructed. These explicit solutions generalize the well-known Barenblatt–Kompanets–Zel’dovich–Pattle solutions of the porous medium equation (4) below. Furthermore, the authors proved the existence of sign-changing weak solution to the Cauchy problem (1)–(2) for ${u_{0}}(x)\in {L^{1}}({\mathbb{R}^{d}})$, and the hypercontractivity ${L^{1}}\mapsto {L^{p}}$ estimates.

By exploiting Darcy’s law, it is possible to interpret the equation (1) as a transport equation ${\partial _{t}}u=\text{div}(|u|\mathbf{v})$, where $\mathbf{v}:=\nabla \mathbf{p}:=\nabla {I_{2-\alpha }}(|u{|^{m-2}}u)$ is a vector velocity field with nonlocal and nonlinear pressure p in the case of nonnegative initial data. We observe that the fractional operator $\nabla {I_{2-\alpha }}$ represents the long-range diffusion effects. The one-dimensional version of the pseudo-differential equation (1) describes the dynamics of dislocations in crystals (see [2]).

For $\alpha =2$, (1) becomes the classical nonlinear porous medium equation If we restrict our attention to nonnegative solution $u(x,t)$, the equation (3) becomes which is usually adopted to model the flow of a gas through a porous medium. The reader interested in the theory of porous medium equation can consult, for instance, [33].

Other types of nonlocal porous medium equations have been proposed in literature. For instance, [5, 6] introduced the porous medium equation with fractional diffusion effects with nonlocal pressure $p:={(-\Delta )^{-s}}u,0<s<1$, and $u\ge 0$. For $\alpha =2-2s\in (0,2)$ we obtain the equation (1) with $m=2$; i.e. ${\partial _{t}}u=\text{div}(u{\nabla ^{\alpha -1}}u)$. In [32] the nonlinear diffusion equation (5) is generalized as follows with $m>1$ and initial condition $u(x,0)={u_{0}}(x)$ which is nonnegative bounded with compact support or fast decaying at infinity. The main contribution in [32] concerns the study of the property of finite/infinite speed of propagation of the solutions to (6) with varying m.

The following equation is studied in [34], where it is also proved that the self-similar solutions of (7) enjoy the ${L^{1}}$-contraction property and then they are unique. Nevertheless, these solutions are not compactly supported. Explicit self-similar solutions to (6) and (7) have been obtained by [20] for particular values of m.

The main goal of this paper is to investigate the relationship between (1) and some random models. In particular, we focus our attention to the probabilistic interpretations of the weak solution to NPME. The idea to study stochastic processes associated to the classical porous medium equation (4) was developed by different authors; see, for instance, [21–23, 12, 13, 24, 29]. In the listed papers the authors introduced different types of Markov chains on lattice and interacting particle systems having a dynamic which macroscopically converges to the solution of (4). By [17], the Barenblatt solution of (4) can be viewed as the mean of the first passage time of a symmetric stable process to exterior of a ball. In [1], the authors provided a probabilistic interpretation of (4) in terms of stochastic differential equations. Recently, [11] highlighted the connection between (4) and the Euler–Poisson–Darboux equations by taking into account time-rescaled random flights.

Up to our knowledge, this paper is the first attempt concerning the probabilistic interpretation of the fractional porous medium equation (1). Similarly to [11], we can exploit stochastic models defined by continuous-time random walks in ${\mathbb{R}^{d}},d\ge 1$, arising in the description of the displacements of a particle choosing uniformly its directions; i.e. the so-called isotropic transport processes or random flights. In a suitable time-rescaled frame, the probability law of the above processes is given by the solution (8) below. Therefore, this paper represents a generalizations of some results contained in [11]. We point out that the proposed random processes recover some features of the Barenblatt weak solution (8) to nonlinear evolution equations like finite speed of propagation and the anomalous diffusivity. For this reason the random flights seem to represent a natural way to describe the real phenomena studied by means of (1).

In Section 2, we recall the definition of weak solution to (1) as well as its basic properties. In Section 3 the isotropic transport processes are introduced. Furthermore, Section 3 contains our main results; i.e. Propositions 3.1, 3.2. From these propositions we are able to give a reasonable interpretation of the solutions to (1). In the last section we sum up the main contribution of the paper.

Let us recall the definition of weak solution to the nonlocal operator equation (1) and its main properties (see [4]).

Let ${(x)_{+}}:=\max (x,0)$. The following theorem, proved in [4], represents our starting point.

It is worth to mention that the family of functions (8) represents a class of nonnegative compactly supported solutions of (1). Moreover, (8) is a self-similar solution under a suitable space-time rescaling; i.e. It is crucial to observe that the constant C appearing in (8) guarantees the mass conservation or equivalently and then $u(x,t)$ (as well as ${u_{0}}(x)$) is a probability density function with compact support ${\overline{\mathbb{B}}_{\mathrm{c}{t^{\beta }}}}:=\{x\in {\mathbb{R}^{d}}:||x||\le \mathrm{c}{t^{\beta }}\}$. By setting ${R^{2}}=\frac{1}{{k^{2/\alpha }}}$, the solution (8) coincides with (2.4) in [4].

We point out that NPME has the property of finite speed of propagation. We are able to explain this property as follows. The solution to NPME is a continuous function $u(x,t)$ such that for any $t>0$ the profile $u(\cdot ,t)$ is nonnegative, bounded and compactly supported. Hence, the support expands eventually to penetrate the whole space, but it is bounded at any fixed time. Therefore, for fixed $t>0$, the support of (8) is given by the closed ball ${\overline{\mathbb{B}}_{\mathrm{c}{t^{\beta }}}}$, while the free boundary (that is the set separating the region where the solution is positive) is given by the sphere ${\mathbb{S}_{\mathrm{c}{t^{\beta }}}^{d-1}}:=\{x\in {\mathbb{R}^{d}}:||x||=\mathrm{c}{t^{\beta }}\}$.

The finite speed of propagation of NPME is in contrast with the infinite speed of propagation of the classical heat equation; that is, a nonnegative solution of the heat equation is positive everywhere in ${\mathbb{R}^{d}}$.

The next proposition contains the explicit Fourier transform of (8). A similar result has been already proved, for instance, in [4], Lemma 4.1.

In this section, we analyze the link between the weak solution of the nonlocal equation (1) and the transport processes. We follow the approach developed in [11]. Let us start with introducing isotropic transport processes and recalling their main features.

An isotropic transport process, also called random flight, is a continuous-time random walk in ${\mathbb{R}^{d}}$ described by a particle starting at the origin with a randomly chosen direction and with finite speed $c>0$. The direction of the particle changes whenever a collision with some scattered obstacles in the environment happens and then a new direction of motion is taken. For $d\ge 2$, all the directions are independent and identically distributed. The directions are chosen uniformly on the sphere ${\mathbb{S}_{1}^{d-1}}=\{x\in {\mathbb{R}^{d}}:||x||=1\}$. For $d=1$, we have two possible directions alternatively taken by the moving particle. The random flights have been studied, for instance, in [30, 31, 9, 26, 27, 10, 7, 18, 28]. Recently, in [14, 15] the relationship between the isotropic transport processes and some fractional Klein–Gordon equations has been analyzed. Furthermore, stochastic models like random flights are associated to the Euler–Poisson–Darboux partial differential equations as argued in [16].

Rigorously speaking, we introduce the isotropic transport processes as follows. Let $({T_{k}},k\in {\mathbb{N}_{0}})$ be a sequence of random arrival epochs with ${T_{0}}:=0$. Furthermore, let $({V_{k}},k\in {\mathbb{N}_{0}})$ be a sequence of random variables defined for $d=1$, by ${V_{k}}:=V(0){(-1)^{k}}$, where V(0) is a uniform r.v. on $\{-1,+1\}$, while for $d\ge 2$ they are independent $({\mathbb{S}_{1}^{d-1}},\mathcal{B}({\mathbb{S}_{1}^{d-1}}))$-valued random variables where $\mathcal{B}({\mathbb{S}_{1}^{d-1}})$ denotes the Borel class on ${\mathbb{S}_{1}^{d-1}}$. We assume that during the interval $[0,t]$ the particle takes a new direction, ${V_{0}},{V_{1}},\dots ,{V_{n}}$, $n+1$ times at random moments ${T_{0}},{T_{1}},\dots ,{T_{n}}$, respectively. Therefore, we can define an isotropic random flight on $(\varOmega ,({\mathcal{F}_{t}^{n}},t\ge 0))$ as follows where ${X^{n}}(t)$ stands for the position, at time $t\ge 0$, reached by the moving particle according to the mechanism described above and $({V^{n}}(t),t\ge 0)$ is the jump process with $0\le k\le n$; i.e. ${X^{n}}$ is adapted to the filtration $({\mathcal{F}_{t}^{n}},t\ge 0)$ where where $\mathfrak{B}:=\{-1,+1\}$, if $d=1$, or $\mathfrak{B}:=\mathcal{B}({\mathbb{S}_{1}^{d-1}})$, if $d>1$. Therefore ${X^{n}}(t)$ represents a random motion with finite velocity c and ${X^{n}}(t)\in {\mathbb{B}_{ct}}$ a.s. for a fixed $t>0$. The components of ${X^{n}}(t)$ can be written explicitly as in formula (1.6) of [10]. Important assumptions in our paper are: the random vector of the renewal times $({\tau _{1}},\dots ,{\tau _{n}})$, where ${\tau _{k+1}}:={T_{k+1}}-{T_{k}}$, has the joint density equal to and for $d\ge 2$, or for $d\ge 3$, where The distributions (14) and (15) are rescaled Dirichlet distributions, with parameters $(d-1,\dots ,d-1),\hspace{0.1667em}d\ge 2$, and $(\frac{d}{2}-1,\dots ,\frac{d}{2}-1),\hspace{0.1667em}d\ge 3$, respectively. Generalized versions of the Dirichlet density functions (13) and (14) have been used in [8] to generalize the family of random walks defined above.

In the one-dimensional case the process (12) is the well-known telegraph process and admits the density given by (see [9]) We observe that for n odd, we have that Under the assumptions (14) and (15), [10] provides (Theorem 2 in [10]) the explicit density functions of the random flights ${X^{n}}(t)$; that is,

Hereafter, we discuss the main results of the paper; i.e. Propositions 3.1, 3.2 below. Therefore, we provide a reasonable probabilistic interpretation of the weak solution (8) in terms of a time-rescaled random flights. From the features of ${X^{n}}$ it emerges that the random flights share with (8) the crucial property of finite speed of propagation in the space. For this reason the transport process (12) seems to represent a fine choice to model phenomena described by nonlinear diffusion equation with nonlocal pressure (1). Our first result is the following theorem and it represents a generalization of Theorem 1 in [11].

To enhance the features of the random models ${Y^{n}},n\ge 1$, it is useful to introduce the Euclidean distance process ${\mathsf{R}^{n}}:=({\mathsf{R}^{n}}(t),t\ge 0)$; that is ${\mathsf{R}^{n}}(t):=||{Y^{n}}(t)||$. For a fixed $t\ge 0$, ${\mathsf{R}^{n}}(t)\in [0,\mathrm{c}{t^{\beta }}]$ a.s. The next result will be useful for arguing on the anomalous diffusivity of ${Y^{n}}$.

The Barenblatt–Kompanets–Zel’dovich–Pattle solution to the classical PME does not spread in the space linearly over the time and then we can argue that the phenomena described by the equation (4) represent anomalous diffusion (see, for instance, [33]). Similar considerations hold for (8). By means of Theorem 3.2, we infer that the stochastic models ${Y^{n}},n\ge 1,$ behave similarly to an anomalous diffusion. From (23) and (24), we observe that For $d=1$, one has $2\beta =\frac{2}{m-1+\alpha }=\frac{4k}{\alpha (2k+1)},k\ge 1$ (condition (i) in Proposition 3.1). Therefore, for a fixed $k\in \mathbb{N}$, we can find the values $\alpha \in (0,2]$, such that the process ${Y^{n}}$ spreads over the real line like a sub-diffusion or a super-diffusion. Therefore ${Y^{n}}$ has the following properties: Analogous remarks hold in higher dimensions. For a fixed $n\in \mathbb{N}$ and $d\ge 2$ (resp. $d\ge 3$), under the condition (ii) (resp. (iii)) in Proposition 3.1, the random process ${Y^{n}}$ has the following properties:

We are able to provide a probabilistic interpretation of the weak solution (8) to NPME. In particular, we deal with random flight models (12) with a suitable rescaling of the time coordinate. These random processes enjoy the main features of (8), at least for particular values of m:

In conclusion, the isotropic transport processes seem to describe well the real phenomena studied by means of the degenerate nonlinear diffusion equation with fractional pressure (1).