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 α-stable subordinator. We compute the mean square displacement of the proposed process and show that the process exhibits subdiffusion when 0 < α < 1/3, normal diffusion when α = 1/3, and superdiffusion when 1/3 < α < 1. 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.


Introduction
Anomalous diffusion is found in a wide diversity of systems (see review articles [1][2][3][4] and references therein).In one dimension, it is characterized by a mean square displacement (MSD) of the form with  ̸ = 1, which deviates from the linear dependence on time found in normal diffusion.The coefficient   is generalized diffusion constant.It is called subdiffusion for 0 <  < 1 and superdiffusion for  > 1 [2].
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 [5].In a continuum limit, the process has been considered by Fogedby [6] via coupled Langevin equations where () is a white Gaussian noise with ⟨()⟩ = 0, ⟨()(  )⟩ = ( −   ), 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  () = inf { :  () > } . (3) Mathematically, the fundamental approach to describe the combined process () = (()) is based on subordination technique, which was first introduced by Bochner [7].Using the notation of subordination, the process (), (), and () are named parent process, subordinator, and inverse subordinator, respectively.
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 [32] and active biological movements [33] or in financial market dynamics [34].Recently, three correlated CTRW models are introduced to model the random walks with some forms of memory [35][36][37].Some advances in the field of CTRWs with correlated temporal or/and spatial structure can be also found in [38][39][40][41][42][43][44][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 by and 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 [46].It also appears in the description of statistical properties of the Burgers equation with Brownian initial velocity [47].Some further results of the integrated Brownian motion can be found in the paper [48] 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 timeaveraged 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.

Model
We begin by recalling the general framework for CTRW theory.Let {  } ≥1 be the sequence of nonnegative independent identically distributed (IID) random variable representing waiting times between jumps of a particle.We set (0) = 0 and () = ∑  =1   , that is, the time of the th jump.Let {  } ≥1 be the sequence of IID jump lengths of the particle, which are assumed to be independent of waiting times.We set (0) = 0 and () = ∑  =1   , that is, the position of the particle after the th jump.Then, the position of the particle at time  is given by where () = max{ ≥ 0 : () ≤ } 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 to where   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 φ() is given by In the continuous limit, we get the following set of coupled Langevin equations for the position  and time  of the CTRW where () and () are the same as those in (2) and () is the standard Brownian motion with ⟨()⟩ = 0, ⟨()()⟩ = min(, ).
An equivalent representation of (9) in the form of subordination is Here the parent process () has the form and the inverse subordinator   () is defined by where () = ∫  0 (  )  is an -stable totally skewed Lévy motion with characteristic function
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 moment we obtain Thus, the MSD of the subordinated process () is Let us turn to the inverse subordinator   ().Observing the equivalence from ( 12) we obtain the relation which gives the formula for the PDF (, ) in terms of the PDF ℎ(, ): Taking the Laplace transform for (22) about variable , we get Thus, the MSD of the subordinated process () in Laplace space is which implies that the MSD of () is It is easy to observe from ( 25) that the process is subdiffusive when 0 <  < 1/3, normally diffusive when  = 1/3, and superdiffusive when 1/3 <  ≤ 1.
It is well-known that the MSD of the process given by ( 2) is of the form Comparing ( 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 equivalence Here,  2 (Δ) is the time-averaged MSD of the process (), defined as where Δ is the lag time and  is the overall measure time.
At last, we consider the propagator (, ) associated with the subordinated process ().By the total probability formula, we obtain an integral representation of (, ): For fixed  > 0, the random variable () = ∫  0 (  )  is normally distributed.From ( 15) and ( 16), we have It follows from and the Laplace transform   →  for g(, ) that we obtain After taking the inverse Laplace transform   →  for g(, ), we get where is the Mittag-Leffler function with parameter  [56].
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.

Conclusions
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 0 <  < 1/3, normal diffusive when  = 1/3, and superdiffusive when 1/3 <  ≤ 1.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.