Nonlocal Diffusion Operators for Normal and Anomalous Dynamics ^{†}^{†}thanks: This work was partially supported by NSFC 11421101, 11421110001 and 11671182.
Abstract
The Laplacian is the infinitesimal generator of isotropic Brownian motion, being the limit process of normal diffusion, while the fractional Laplacian serves as the infinitesimal generator of the limit process of isotropic Lévy process. Taking limit, in some sense, means that the operators can approximate the physical process well after sufficient long time. We introduce the nonlocal operators (being effective from the starting time), which describe the general processes undergoing normal diffusion. For anomalous diffusion, we extend to the anisotropic fractional Laplacian and the tempered one in . Their definitions are proved to be equivalent to an alternative one in Fourier space. Based on these new nonlocal diffusion operators, we further derive the deterministic governing equations of some interesting statistical observables of the very general jump processes with multiple internal states. Finally, we consider the associated initial and boundary value problems and prove their wellposedness of the Galerkin weak formulation in . To obtain the coercivity, we claim that the probability density function should be nondegenerate.
ump processes; Nonlocal normal diffusion; Anisotropic anomalous diffusion; Tempered Lévy flight; Multiple internal states; Wellposedness
1 Introduction
Diffusion phenomena are ubiquitous in the natural world, which describe the net movements of the microscopical molecules or atoms from a region of high concentration to a region of low concentration. The speed of diffusion can be characterized by the second moment of the particle trajectories . It is called normal diffusion if and anomalous diffusion [20, 32] if . The scaling limits of all the processes undergoing normal diffusion are Brownian motion. But without the scaling limits, most of the time, they are pure jump processes. For anomalous diffusion, the processes are always characterized by longrange correlation or broad distribution. The former includes fractional Brownian motion [22] and tempered fractional Brownian motion [7, 23], while the latter contains the processes with long tailed waiting time or jump length. In the framework of continuous time random walks (CTRWs) [19, 25], any one of the first moment of waiting time and the second moment of jump length diverging leads to the anomalous dynamics. If we extend to the processes with multiple internal states [35], then the diffusion phenomena will depend on the distribution of each internal states, transition matrix and initial distribution, involving more complex dynamics.
There are many microscopic/stochastic models to describe normal and anomalous diffusions and many different ways of deriving the macroscopic/deterministic equations governing the probability distribution functions of some particular statistical observables of the stochastic processes. For normal diffusion, in mathematical community, most people are more familiar with the deriving procedure based on the law of mass conservation and the assumption of Fick’s law. The commonly used stochastic models include CTRWs, Langevin type equations, and Lévy processes. The CTRWs consist of two important random variables, i.e., the waiting time and jump length . If both the first moment of waiting time and the second moment of jump length are finite, after taking the scaling limit, the CTRWs converge to Brownian motion. On the contrary, if diverges and is finite, the CTRW describes subdiffusion, while it characterizes superdiffusion if is bounded and infinite; if both and are unbounded, the type of diffusions is possible to be subdiffusion, superdiffusion, or even normal diffusion, depending on the dominant role played by or or that and are balanced each other. Two of the most important CTRW models undergoing anomalous diffusion are Lévy flights and Lévy walks. For Lévy flights, the with finite first moment and with infinite second moment are independent, and the divergence of the second moment of makes the processes propagate with infinite speed. Therefore, the physical realizations of such processes are quite hard and then rare. Lévy walks [37] can remedy the divergence of the second moment of jump length by coupling the distribution of and . This gives rise to a class of spacetime coupled processes. Different from Lévy walks, another idea to bound the second moment is to truncate the long tailed probability distribution of jump length [22, 24], i.e., modify as with being a small positive constant, leading to the tempered Lévy flights, which have the advantage of still being an infinitely divisible Lévy process. The Langevin type equations are built based on Newton’s second law with noise as random forces, and the CTRW models also have their corresponding Langevin pictures [14]. Sometimes, it is convenient to use this type of models if the external potentials are considered.
Another way to describe anomalous diffusion is the Lévy process (subordinated Lévy process, and inverse subordinated Lévy process). It is defined by its characteristic function and more convenient to deal with the stochastic process in high dimensional space. According to the LévyKhintchine formula [1], the characteristic function of Lévy process has a specific form
(1) 
where
(2) 
with , and is a positive definite symmetric matrix, is the indicator function of the set , is a finite Lévy measure on , implying that . If we take and to be zero and to be a rotationally symmetric (tempered) stable Lévy measure
(3) 
then its probability density function (PDF) of the position of the particles solves
(4) 
where the operators and are defined in [9, Eq. 34] by Fourier transform with
(5)  
here , and denotes the smallest integer that is bigger than or equal to . A similar operator appears in [30, Eq. 3], where the only difference is the term . However, their physical background is completely different. The term in (5) is strictly derived in [9, Eq. 34], where we consider the compound Poisson processes with tempered power law jump lengths, i.e., take the Lévy measure to be . But for the formula in [30, Eq. 3], it is inspired by the Schrödinger operator with the free Hamiltonian of the form in [4], and naturally extended to the form with fractional order .
The two equations in (5) describe the isotropic movements of microscopic particles with (tempered) Lévy distribution. At the same time, in the natural world, anisotropic motions are very popular. So we need to develop models for characterizing the corresponding physical reality. Compte [3] generalized the scheme of CTRWs and showed the diffusionadvection equation and the mean square displacement in three kinds of shear flows. Meerschaert et al [21] made an extension to higher dimensions and provided an operator being mixture of directional derivatives taken in each radial direction, where the operator was directly given in Fourier space and the associated fractional advectiondispersion equation was derived. Ervin and Roop [13] discussed directional integral and directional differential operators in two dimensions, and defined the appropriate fractional directional derivative spaces. For more details we refer the interested readers to these literatures and the references cited therein. In this paper, we start from the compound Poisson process to discuss more general nonlocal normal diffusion and anomalous diffusion. It is well known that, most of the time, anomalous diffusions are described by nonlocal differential equations. But for normal diffusion, a compound Poisson process with Gaussian jumps indeed leads to a nonlocal differential equations. For the isotropic movement and the movement just allowed in axis directions, their associated diffusion equations are different, though the scaling limit makes them become the same classical diffusion equation. For the nonlocal normal diffusion, we still can discuss the problem of escape probability [5, 11, 17] and the way of specifying the boundary conditions of their corresponding macroscopic equations is the same as the models for anomalous diffusion. We also discuss the anomalous diffusion undergoing anisotropic movements in , and derive the associated diffusion equations with anisotropic tempered fractional Laplacian ( corresponds to the one without tempering and the subscript means that this new operator depends on the probability density function or first appeared in (15)). Similar to the operator in [21, Eq. 2], we also give the tempered one in Fourier space and show its equivalence with the just derived one . Then we discuss the space fractional partial differential equations (PDEs) with the anisotropic tempered fractional Laplacian in , endowed with generalized Dirichlet and Neumann boundary conditions, and prove their wellposedness. One of the key requests is to have the coercivity of the variational formulation of the PDEs in , being proved by the technique in presented in [39] under some assumptions on the probability density function .
All the models mentioned above are for the diffusion with single internal state, implying that the processes have the same distributions of waiting time and jump length throughout the time. Intrigued by applications, e.g., the particles moving in multiphase viscous liquid composed of materials with different chemical properties, we further generalize the processes with multiple internal states. In fact, the case of two internal states is considered in [15, 28] with applications, including trapping in amorphous semiconductors, electronic burst noise, movement in systems with fractal boundaries, the digital generation of noise, and ionic currents in cell membranes; Niemann et al [26] detailedly investigate a stochastic signal with multiple states, in which each state has an associated joint distribution for the signal’s intensity and its holding time. Xu and Deng [35] derived the FokkerPlanck and FeymannKac [33, 34] equations for the particles undergoing the anomalous diffusion with multiple temporal internal states. Here, we further present the fractional FokkerPlanck and FeymannKac equations with multiple internal states, both temporally and spatially.
The rest of this paper is organized as follows. In Section 2, we show two kinds of processes with Gaussian jumps, leading to different nonlocal macroscopic equations describing normal diffusions. More general anisotropic processes undergoing anomalous diffusions are discussed in Section 3, and we also give two kinds of definitions of anisotropic (tempered) fractional Laplacian for two different motivations and prove their equivalences. In Section 4, the fractional FokkerPlanck and FeymannKac equations of anisotropic (tempered) fractional Laplacian with multiple internal states are derived. The initial and boundary value problems with generalized Dirichlet and Neumann boundary conditions are given in Section 5, and their wellposednesses are proved in Section 6. We conclude the paper with some discussions in the last section.
2 Nonlocal normal diffusion
As all we know, except Brownian motion with drift, the paths of all other proper Lévy processes are discontinuous. From the viewpoint of [8, 9], the PDEs governing the PDFs of these processes should be endowed with the generalized boundary conditions, since the boundary itself can not be hit by the majority of discontinuous sample trajectories. For nonlocal normal diffusion, it is a pure jump process with Gaussian jumps. Therefore, the boundary conditions of their corresponding PDEs should be specified on the domain . By the central limit theorem, the scaling limits of all these processes are Brownian motion. But without scaling limit, these processes are different and should be distinguished.
Now we consider the compound Poisson process with Gaussian jump length, in which Poisson process is taken as the renewal process. We denote Poisson process by satisfying , where the rate denotes the mean number of jumps per unit time. Then the compound Poisson process is defined as , where are i.i.d. random variables and their length obeys Gaussian distribution. The characteristic function of has a specific form as [9, Eq. 9]
(6) 
where ,N(t). Denoting the probability measure of the jump length by , we have
(7) 
which is the same as the LévyKhintchine formula (2) by taking and ( contains and the third term in the integral of (2)). Although the length of obeys Gaussian distribution, the distribution of the direction of the movement has many different choices. Here, we consider two specific cases in two dimensional space, and derive their corresponding deterministic equations governing the PDF of position of the particles undergoing normal diffusion. The first case is that the particles spread uniformly in all directions while the second one is that the particles move only in horizontal and vertical direction. Considering the definition of Fourier transform and (6), we have
(8) 
which implies that the equation in space is
(9) 
Next, we give the specific expressions of (or ) for these two cases.
Case 1: Since the particles spread uniformly in all directions, is taken as
where is the variance. Then we obtain
(10) 
which implies
(11) 
by taking the inverse Fourier transform
Case 2: Since the particles spread in either horizontal or vertical direction, we take to be
Similar to Case 1, we have
(12) 
and thus derive the equation
(13) 
From (11) and (13), it can be noted that different ways of movement of microscopic particles lead to different macroscopic equations. Furthermore, these macroscopic equations are both nonlocal, and should be endowed with the generalized boundary conditions. But the scaling limits of the Gaussian jump processes of the above two cases are both Brownian motion. In fact, let and , while the product
is kept finite, where is the diffusion coefficient with unit [2]. Then, both (10) and (12) become, up to a multiplier,
resulting in the classical heat equation
(14) 
where is the usual Laplacian in .
To illustrate the relationship between Case 1 and Case 2, we simulate the trajectories of the particles undergoing Gaussian jumps. Two pictures in the top row are for the 400 jumps performed uniformly (a) and just in horizontalvertical direction (b), while another two pictures in the bottom row display 40000 jumps, respectively. The differences between Case 1 and Case 2 are apparent for a relatively small number of jumps. But after many thousands of jumps, they gradually disappear, as both processes are converging to the same Brownian motion.
Besides the two cases above, more generally, the particles can move in a variety of different ways, depending on the environment. There may be more particles spreading in one direction or some particles spreading faster in another direction. This phenomenon is named as anisotropic diffusion, and can be expressed clearly by the Lévy measure . More precisely, still in two dimensional space, by polar coordinates transformation, take to be
(15) 
where is the normalized parameter, , denotes the different directions, denotes the probability distribution of particles spreading in direction, satisfying , , and denotes the possibly different variance or speed of particles spreading in direction. Different from (3), this contains a new probability density function which only depends on direction. Turning back to the Cartesian coordinate system and following (7), we have
where the probability density function is abused by and is in the Cartesian coordinate system, while it really means , only depending on the radial direction of . The notation will be used in the subsequent sections. Then similarly to (11) and (13), we can derive the equation
(16) 
If we take , being a constant or in (15), then Eq. (16) reduces to (11) and (13), respectively.
All the above discussions, including Case 1 and Case 2, and even the case of (15), are for pure jump processes (without the scaling limit). The models are different and their associated macroscopic equations should be endowed with generalized boundary conditions. But after the scaling limit, Case 1 and Case 2 are equivalent, and only local boundary conditions for their macroscopic equations (14) are needed.
3 Anisotropic anomalous diffusion
Here, we discuss the anomalous diffusion with the property of anisotropy. Still based on the compound Poisson processes in the previous section, but with the diffusion processes being anisotropic (tempered) stable, we try to derive their corresponding deterministic equations undergoing anomalous diffusion. Taking in (9) leads to
(17) 
where
(18) 
Here, different from (7), we add a term to overcome the possible divergence of the integral of (18) because of the possible strong singularity of at zero for the case of anomalous diffusion. For an isotropic stable anomalous diffusion process in dimensional space, its distribution of jump length is , which means that
(19) 
When , the term can be omitted due to weak singularity (the integral in (18) is convergent at origin). If , though the singularity is strong, this term can also be omitted due to the possible symmetry of the Lévy measure , i.e., (the integral in (18) at origin can be understood in the sense of Cauchy principal value). Therefore, if meets with the asymmetry of , this term is required. Based on the analyses above, we will keep the term formally for in the following, though it vanishes in some appropriate situations.
Two special cases have been considered in [9], i.e., the isotropic one (19) and the horizontalvertical one
(20) 
where and is the component of , i.e., . Their corresponding macroscopic equations are
(21) 
and
(22) 
where is the fractional Laplacian in w.r.t. . Besides the two cases, there are also a large number of irregular motions the microscopic particles perform. In general, we call it anisotropy. With the aid of LévyKhintchine formula (2), we will give the concrete form of in two and three dimensions.
Following (17) and (18), with inverse Fourier transform, we have
(23) 
where . Taking
(24) 
then (23) becomes
(25) 
We can make the meaning of clear by transforming this equation into polar coordinate system. In the two and three dimensional cases, (25) becomes, respectively,
and
where the probability density function or specifies the distribution of particles spreading in the radial direction of ; among them, is defined on , satisfying , while is defined on a rectangular domain, satisfying .
For the tempered Lévy flight, we can describe the movement of microscopic particles and derive the macroscopic equations by defining
(26) 
and (23) becomes
(27) 
We write Eqs. (25) and (27), respectively, as
(28) 
and
(29) 
where the notation () denotes the anisotropic (tempered) fractional Laplacian in ; and their definitions are the right hand sides of (25) and (27).
We simulate the trajectories of the particles with the anisotropic movements. Figure 2 shows five random trajectories of 2000 steps of Lévy flight with (Gaussian), and tempered Lévy flight with and in two dimensions. All trajectories start from the origin . Three pictures on the top row correspond to the isotropic case, i.e., for , while another three on bottom row correspond to the anisotropic case, where we choose for and for . Note that (a) and (d) depict the isotropic and anisotropic Gaussian jump processes introduced in Section 2. By horizontal comparison, the lengths of Gaussian jumps in (a) have almost the same sizes, while Lévy flight in (b) preforms rare but large jumps. And an exponential truncation in (c) with even little excludes large jumps compared with (b). By vertical comparison, in the bottom row, particles are more inclined to move upward and thus finally farther than the isotropic case with the same steps.
Different from (25) and (27), an alternative definition of the anisotropic (tempered) fractional Laplacians is given by Fourier transform [21, Eq. 2], with an analogous tempered one presented here:
(30) 
and
(31) 
It seems that these definitions are natural for the study of the governing equations, since the symbol for denotes order fractional directional derivative. Now we consider the question of when the two ways of defining the operators are equivalent. To establish the relationship between them, we focus on two cases:
In Case II, since the high singularity makes the integral divergent, we use the notation p.f. to denote its finite part in the Hadamard sense.
Then we have the following theorem; see Appendix for the proof, which further implies the equality (35). {theorem} Let be any probability density function on the unit sphere and . The definitions of the anisotropic (tempered) fractional Laplacians in both Case I and Case II are, respectively, equivalent to in (30) and (31) in .
We have just defined the anisotropic (tempered) fractional Laplacian by extending the Lévy measure with different probability distribution in different directions. More generally, another two variables jump length exponent and truncation exponent can also be generalized to be anisotropic, i.e., and , sometimes abused by and similar to . Let and . When , it goes back to anisotropic fractional Laplacian. Following (30), (31), (33) and (35), the definitions of new anisotropic (tempered) fractional Laplacian are, respectively,

Case I: or is symmetric,
(36) 
Case II: and is asymmetric,
(37) where .
In Fourier space, the new operator has the form
(38) 
We also simulate the trajectories of particles with the new anisotropic Lévy measure defined in (36). As Figure 3 shows, we take the isotropic and the anisotropic to be for and for in ; the particles move farther downward than upward. In , only difference with the parameter in is the anisotropic being for and for . This choice of aims to balance the anisotropic ; as the second graph shows, the movements of particles become almost isotropic. In , we take the isotropic and , but the anisotropic to be for and for ; the particles move farther downward than upward.
Remark 3.1
In the practical problem, the directional measure may depend on concentration gradient. To emphasize the effects caused by the directional gradient, the definition of the anisotropic (tempered) fractional Laplacian in (36) can be extended to
(39) 
where should be an increasing function of directional gradient .
As a complement to the definition of the anisotropic (tempered) fractional Laplacian (30) and (31), we also present the definition of the operator in the case that , i.e., let , which still is a nonlocal operator. For the sake of simplicity, we assume that is symmetric, then the term in (23) can be omitted. For the one dimensional asymmetric operators with , see [18] for the details.
Let and . If the probability density function is symmetric, then the Fourier symbols of the anisotropic fractional Laplacian and the corresponding tempered one, respectively, are
(40) 
and
(41) 
We firstly prove the tempered case. Taking the Fourier transform of the right hand side of (27), we have
where the term vanishes due to the symmetry of . By polar coordinate transformation and integration by parts, we have
from which Eq. (41) can be directly obtained by using [16, Eq. 4.441(12)].
For the proof of (40), taking in (41) leads to
Furthermore, if is isotropic, then
where is the measure of the dimensional unit sphere, if and when ; the rotation invariance [27, Proposition 3.3] of the integrand is used in the second equality, and denotes one of the components of vector ,
Following (41), the Fourier symbol of the new anisotropic tempered fractional Laplacian when is
All the discussions above are based on compound Poisson processes with different probability distribution of jump length for (tempered) Lévy flights, which render the deterministic governing equations with classical first derivative temporally. If instead, the fractional Poisson processes are taken as the renewal process, in which the time interval between each pair of events follows the power law distribution. Then the deterministic governing equations with Caputo fractional derivative temporally can be derived. More precisely, let be a nondecreasing subordinator [6] with Laplace exponent ,