Research Article | Open Access
Modeling Anomalous Diffusion by a Subordinated Integrated Brownian Motion
We consider a particular type of continuous time random walk where the jump lengths between subsequent waiting times are correlated. In a continuum limit, the process can be defined by an integrated Brownian motion subordinated by an inverse -stable subordinator. We compute the mean square displacement of the proposed process and show that the process exhibits subdiffusion when , normal diffusion when , and superdiffusion when . The time-averaged mean square displacement is also employed to show weak ergodicity breaking occurring in the proposed process. An extension to the fractional case is also considered.
Anomalous diffusion is found in a wide diversity of systems (see review articles [1–4] and references therein). In one dimension, it is characterized by a mean square displacement (MSD) of the formwith , which deviates from the linear dependence on time found in normal diffusion. The coefficient is generalized diffusion constant. It is called subdiffusion for and superdiffusion for .
A fundamental account to anomalous diffusion is provided by a stochastic process called continuous time random walk (CTRW), which was originally introduced by Montroll and Weiss in 1965 . In a continuum limit, the process has been considered by Fogedby  via coupled Langevin equationswhere is a white Gaussian noise with , , and is a white -stable Lévy noise, taking positive values only and independent of .
In (2), the random walk is parametrized in terms of a continuous variable , which is subjected to a random time change. This random time change to the physical time is described by the second equation. The combined process in the physical time is then given by , where is the inverse process to defined as
Mathematically, the fundamental approach to describe the combined process is based on subordination technique, which was first introduced by Bochner . Using the notation of subordination, the process , , and are named parent process, subordinator, and inverse subordinator, respectively.
In recent years, (2) consisting of Brownian motions with or without external field and inverse -stable subordinator are becoming a hot topic [8–17]. There are also several other processes considered as parent processes within the subordination framework, for example, Lévy-stable process [18, 19], arithmetic Brownian motion , geometric Brownian motion [21, 22], Ornstein-Uhlenbeck process [23, 24], tempered stable process , fractional Brownian motion [26, 27], and fractional Lévy-stable process . Here, we note that, apart from inverse -stable subordinator, inverse tempered -stable subordinator and infinitely divisible subordinators are also considered in the literatures [16, 20, 25–31].
In the simplest CTRW process, after each jumps, a new pair of waiting time and jump length is drawn from the associated distributions, independent of the previous values. This independence giving rise to a renewal process is not always justified, for instance, by observations of human motion patterns  and active biological movements  or in financial market dynamics . Recently, three correlated CTRW models are introduced to model the random walks with some forms of memory [35–37]. Some advances in the field of CTRWs with correlated temporal or/and spatial structure can be also found in [38–45].
In this work, we consider a jump-correlated CTRW model which has the subordination form . Here, the parent process is an integrated Brownian motion, defined byand inverse subordinator is the inverse of one-side -stable Lévy process , defined by
The integrated Brownian motion is called the random acceleration process in the physical literature and has been studied by many authors. For instance, it appears in the continuum description of the equilibrium Boltzmann weight of a semiflexible polymer chain . It also appears in the description of statistical properties of the Burgers equation with Brownian initial velocity . Some further results of the integrated Brownian motion can be found in the paper  reviewing this subject.
The structure of the paper is as follows. In Section 2, we introduce the jump-correlated CTRW model. In Section 3, we compute MSD of the proposed process and observe the corresponding anomalous diffusive behaviors. The time-averaged MSD is also employed to show weak ergodicity breaking occurring in the proposed process. In Section 4, we generalize the integrated Brownian motion to the fractional integral of Brownian motion and compute the corresponding MSD. The conclusions are given in Section 5.
We begin by recalling the general framework for CTRW theory. Let be the sequence of nonnegative independent identically distributed (IID) random variable representing waiting times between jumps of a particle. We set and , that is, the time of the th jump. Let be the sequence of IID jump lengths of the particle, which are assumed to be independent of waiting times. We set and , that is, the position of the particle after the th jump. Then, the position of the particle at time is given bywhere is the number of jumps up to time . The process is called CTRW.
In what follows, we analyze a particular type of CTRW where the jump lengths are correlated. Assume that each jump is equal towhere are IID random variables with finite second moment (for simplicity we assume that their second moment is equal to 1). Moreover, we assume that each waiting time is nonnegative IID random variable, whose characteristic function is given by
In the continuous limit, we get the following set of coupled Langevin equations for the position and time of the CTRWwhere and are the same as those in (2) and is the standard Brownian motion with , .
An equivalent representation of (9) in the form of subordination isHere the parent process has the formand the inverse subordinator is defined bywhere is an -stable totally skewed Lévy motion with characteristic function
At first, let us compute the MSD of subordinated process .
Assume that , , and are PDFs of subordinated process , parent process , and inverse subordinator , respectively. In terms of subordination, we have
Since the first moment of parent process and the second momentwe obtain
Thus, the MSD of the subordinated process is
Let us turn to the inverse subordinator . Observing the equivalence from (12) we obtain the relationwhich gives the formula for the PDF in terms of the PDF :Taking the Laplace transform for (22) about variable , we get
Thus, the MSD of the subordinated process in Laplace space iswhich implies that the MSD of is
It is easy to observe from (25) that the process is subdiffusive when , normally diffusive when , and superdiffusive when .
It is well-known that the MSD of the process given by (2) is of the formComparing (25) with (26), we see that Fogedby’s model can only represent anomalous subdiffusion, but our model can represent subdiffusion, normal diffusion, and superdiffusion.
Next, we study weak ergodicity breaking of the subordinated process .
In an ergodic system, one can find the equivalenceHere, is the time-averaged MSD of the process , defined aswhere is the lag time and is the overall measure time.
For anomalous diffusion, the behavior of the ensemble MSD and the time-averaged MSD (28) may be fundamentally different. The disparity is usually called weak ergodicity breaking (or weak nonergodicity) . In recent years, weak nonergodicity of anomalous diffusion process attracts more and more attentions [49–55].
Since, for any , parent process satisfieswhere means an equality in distribution, we have
In the limit ,
At last, we consider the propagator associated with the subordinated process . By the total probability formula, we obtain an integral representation of :
It follows from and the Laplace transform for that we obtainAfter taking the inverse Laplace transform for , we getwhere is the Mittag-Leffler function with parameter .
4. An Extension to the Fractional Case
In this section, we introduce the dependent sequence of jump lengths in the following manner:where is a memory function. The continuous limit is of the form
Integrating (42) we get
Here, we are interested in the competition between the memory parameter and stability index . In what follows, we will not discuss any properties of motion other than the MSD.
In terms of (44), we getBy denoting and exchanging the order of quadratic integral , we obtainThus,where and is Beta function.
We observe from (48) that, in the limiting case , memory function , the parent process defined by (43) reduces to the form defined by (11), and the second moment of computed by (48) reduces to , where , the same form as (16). We are also interested in the limiting case . At the moment, the memory function is a Dirac -function; defined by (43) reduces to the standard Brownian motion.
Let us turn to the MSD of the subordinated process . In terms of (18) and (48), we obtainIn the Laplace space, the MSD is of the formTaking the inverse Laplace transform for , we havewhere . In the limiting case , the parameter reduces to Thus, (51) reduces to (25).
It is easy to observe from (51) that there exists a competition between the memory parameter and stability index . For the case , the subordinated process exhibits subdiffusive behaviors. For the case , the process is subdiffusive when , normal diffusive when , and superdiffusive when .
We introduce an integrated Brownian motion subordinated by inverse -stable one-sided Lévy motion, which is a continuous limit of a jump-correlated CTRW. In terms of the ensemble MSD of the proposed process, we conclude that the process is subdiffusive when , normal diffusive when , and superdiffusive when . The time-averaged MSD is also employed to show weak ergodicity breaking occurring in the proposed process.
We also generalize the process to the case, where the parent process is fractional integral of Brownian motion. In terms of the MSD, we observe a competition between the memory parameter and stability index . Other types of inverse subordinators may be also candidates.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work is supported by the National Natural Science Foundation of China (Grant no. 11671343) and the Scientific Research Project of Hunan Provincial Education Department (no. 17B258).
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