# Generalized Diffusion Equation Associated with a Power-Law Correlated Continuous Time Random Walk

Authors: Long Shi
Journal: Advances in Mathematical Physics
 (2019)

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## Abstract

In this work, a generalization of continuous time random walk is considered, where the waiting times among the subsequent jumps are power-law correlated with kernel functionM(t)=tρ(ρ>-1). In a continuum limit, the correlated continuous time random walk converges in distribution a subordinated process. The mean square displacement of the proposed process is computed, which is of the form 〈x2(t)〉∝tH=t1/(1+ρ+1/α). The anomy exponent H varies from α to α/(1+α) when -1<ρ<0 and from α/(1+α) to 0 when ρ>0. The generalized diffusion equation of the process is also derived, which has a unified form for the above two cases.


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## Body

## 1. Introduction

Continuous time random walk (CTRW), which was originally introduced to physics by Montroll and Weiss [1], has been applied successfully in many fields (e.g., the reviews [2–4] and references therein).In a continuum one-dimensional space, a CTRW is a process where the motion of a random walker is described by a sequence of independent identically distributed (IID) positive waiting timesT1,T2,T3,… and a sequence of IID random jumps X1,X2,X3,…. Set t0=0 and tn=∑i=1nTi as the time of n-th jump. Then, the process N(t)=max⁡{n≥0:tn≤t} counts the number of jumps of the walker up to time t. Consequently, the CTRW process, defined as(1)xt=xNt=∑i=1NtXi,describes the position of the walker at time t.The independence among the waiting times given rise to a renewal process is not always justified. As soon as the random walk has some form of memory, the variables become nonindependent. Examples are found in financial market dynamics [5], human motion patterns [6], and so on. Recently, these facts impel one to introduce the correlated CTRWs [7–18].There exists two simple approaches to CTRW with correlated waiting times. One was introduced by Chechkin et al. in Ref. [7]. Authors assumed the corresponding waiting times to be weighted sums of independent random variables in the following form:(2)Ti=∑j=1iMi-j+1ξj,where M(t)=t-μ/Γ(1-μ) with 0<μ<1 and ξj is the sequence of IID α-stable random variables with the one-sided totally skewed probability density function (PDF). The characteristic function of the random variables {ξj} is given by(3)eikξj=exp⁡-kαexp⁡-iπα2sgn⁡k),0<α<1.Another correlated temporal structure was introduced by Tejedor et al. in Ref. [9]. The authors assumed that the waiting times Ti equaled(4)Ti=∑j=1iξj,where the distribution of the IID random variables {ξj} is symmetric and α-stable with Fourier transform:(5)eikξj=exp⁡-12kα,0<α≤2.Magdziarz et al. generalized these two models by combining the underlying correlation mechanisms for waiting timesTi in the following manner [12]:(6)Ti=∑j=1iMi-j+1ξj,where M(t)=tρ(ρ∈R) is the memory function and {ξj} is the sequence of IID α-stable random variables with the Fourier transform:(7)eikξj=exp⁡-kα1-iβtan⁡πα2sgn⁡k,0<α<1,β≤1.Note that, for β=1, random variables ξj are positive. By choosing β=1 and ρ=-μ, one obtains the correlated CTRW introduced in the study by Chechkin et al. [7]. On the other hand, choosing β=0 and M(t)=1, one obtains the case 0<α<1 of the correlated CTRW derived in the study by Tejedor et al. [8].In the scaling limit, correlated CTRWs converge to the subordinated processx(t)=x(s(t)). Here, s(t) is a continuum analog of the count process N(t), defined by(8)st=inf⁡s≥0:ts>t.The process t(s) is a continuum analog of hitting time tn.One way to explore the statistical characteristics of the subordinated processx(t)=x(s(t)) is to consider its probability distribution. Here, we are interested in generalized diffusion equation associated with the subordinated process x(t)=x(s(t)). Note that reflecting boundary condition in waiting times is the disadvantage of deriving generalized diffusion equation. Therefore in this work, we limit β=1 in Eq. (7) for our purpose. That is, we assume(9)Ti=∑j=1ii-j+1ρξj,ρ>-1,where {ξj} is IID random variables with a one-sided Lévy α-stable PDF Lα(t), whose Laplace transform is(10)L~αu=∫0∞Lαte-utdt=exp⁡-uα,0<α<1.The structure of this work is as follows. In Section2, we introduce the Langevin description of power-law correlated CTRW model. In Section 3, we compute the MSD of the subordinated process x(t) and derive its generalized diffusion equation. The conclusions are given in Section 4.


## 2. Model

The Langevin equations for the positionx and the time t corresponding to the continuous-time limit process of the above-defined correlated CTRW have the form:(11)ddsxs=ddsBs,ddsts=∫0ss-s′ρdlαs′,0<α<1,ρ>-1.Here, B(s) is a Brownian motion with variance 2s and lα(s) is an α-stable totally skewed Lévy motion with Laplace transform:(12)e-ulαs=exp⁡-suα.Note that when -1<ρ<0 one obtains the correlated CTRW introduced in Ref. [7].To solve Eqs. (11), one first solves the first equation to produce the driving process B(s). Next, one solves the second equation to obtain the process t(s), thus yielding the process s(t), which is inverse to t(s). Finally, one assembles both processes B(s) and s(t) to obtain the solution as the subordination(13)xt=Bst.Here the subordinator s(t), defined by Eq. (8), is the inverse of(14)ts=∫0s∫0s′s′-s′′ρdlαs′′ds′.By exchanging integral order for Eq. (14), the process t(s) can be rewritten as(15)ts=1ρ+1∫0ss-s′ρ+1dlαs′.


## 3. Discussions

Proposition I. The second moment (MSD) of the process x(t)=B(s(t)) is finite and proportional to tH with H=1/(1+ρ+1/α).Proof.
Assume thatp(x,t), f(x,s), and g(s,t) are the PDFs of processes x(t), B(s), and s(t), respectively. Using the total probability formula, we obtain(16)px,t=∫0∞fx,sgs,tds.Thus,(17)x2t=∫-∞∞x2px,tdx=∫-∞∞dx∫0∞x2fx,sgs,tds=2∫0∞sgs,tds=2st.
Since(18)ts=1ρ+1∫0ss-s′ρ+1dlαs′=dlα1αρ+1∫0ss-s′ρ+1s′1/α-1ds′=lα1αρ+1s1+ρ+1/α∫011-τρ+1τ1/α-1dτ=As1+ρ+1/αlα1=As1/Hlα1,where the sign “=d” means the same finite-dimensional distributions, A=B(1/α,2+ρ)/α(ρ+1) and B(·,·) is beta function. We obtain the PDF h(t,s) of the process t(s) in Laplace space(19)h~u,s=e-uts=e-uAs1/Hlα1=exp⁡-sα/HAαuα.
Considering the relationg(s,t)=d/ds∫t∞ht′,sdt′, we have(20)g~s,u=-dds1uh~u,s=αHAαsα/H-1uα-1exp⁡-sα/HAαuα.Thus,(21)su=∫0∞sg~s,uds=αHAαuα-1∫0∞sα/Hexp⁡-sα/HAαuαds=Aαuα-1∫0∞τH/αexp⁡-Aαuατdττ=sα/H=ΓH/α+1AHu-H-1.Taking inverse Laplace transform on u↦t, we get(22)st=ΓH/ααAHΓHtH.
Substituting Eq. (22) into Eq. (17), we obtain(23)x2t∝tH.
Note that anomy exponentH=1/(1+ρ+1/α) decreases from α to α/1+α with the parameter ρ varying from -1 to 0 and decreases from α/1+α to 0 with the parameter ρ varying from 0 to ∞.Proposition II. The PDF p(x,t) of x(t) satisfies the generalized diffusion equation:(24)∂∂tpx,t=∂2∂x2Gpx,t,t>0,where the operator G acts on variable t and is defined by(25)Gpx,t=Lu→t-112πi∫c-i∞c+i∞ΓH/α+1Ψu-vH/αΨ-1vvp~x,Ψ-1vdv.Here Ψ-1(u) denotes the inverse in point u of Ψ(u)=(Au)α and Lu→t-1 is the inverse Laplace transform u→t.Proof.
From Eqs. (16) and (20), we obtain in Laplace space(26)p~x,u=∫0∞px,te-utdt=∫0∞fx,sg~s,uds=αHAαuα-1∫0∞fx,ssα/H-1exp⁡-sα/HAαuαds=Aαuα-1∫0∞fx,τH/αexp⁡-Auατdττ=sα/H=Ψuuf~1x,Ψu,where f1(x,τ)=f(x,τH/α) and Ψ(u)=(Au)α.
SinceΨ(u) is monotone, we notice that after transformation u→Ψ-1(u) one obtains(27)f~1x,u=Ψ-1uup~x,Ψ-1u.
Since(28)∂∂τf1x,τ=∂∂τfx,τH/α=∂∂sfx,s·dsdτs=τH/α=∂2∂x2fx,s·HατH/α-1=∂2∂x2f1x,τ·HατH/α-1,Eq. (28) in Laplace space takes the form(29)uf~1x,u-f1x,0=∂2∂x212πi∫c-i∞c+i∞f~1x,vΓH/α+1u-vH/αdv.
After changing the variableu→Ψ(u) in Eq. (29) and using the initial condition p(x,0)=f(x,0)=f1(x,0)=δ(x), we obtain that in the Laplace domain the PDF p(x,t) of x(t) obeys(30)up~x,u-px,0=Ψuf~1x,Ψu-f1x,0=∂2∂x212πi∫c-i∞c+i∞f~1x,vΓH/α+1Ψu-vH/αdv=∂2∂x212πi∫c-i∞c+i∞ΓH/α+1Ψu-vH/αΨ-1vvp~x,Ψ-1vdv.Thus, after taking inverse Laplace transform on u↦t for Eq. (30), we obtain the generalized diffusion equation:(31)∂∂tpx,t=∂2∂x2Lu→t-112πi∫c-i∞c+i∞ΓH/α+1Ψu-vH/αΨ-1vvp~x,Ψ-1vdv=∂2∂x2Gpx,t.


## 4. Conclusions

In this work, a CTRW model with power-law correlated waiting times is considered. The kernel functionM(t)=tρ contains two cases: -1<ρ<0 and ρ>0. The MSD of the subordinated process x(t)=x(s(t)) based on the proposed correlated CTRW is computed, which is proportional to tH=t1/(1+ρ+1/α). One can observe that in the case -1<ρ<0 the exponent H decreases from α to α/1+α and in the case ρ≥0 the exponent H decreases from α/1+α to 0. The proposed model supplements the discussions in Ref. [7] on CTRW with correlated waiting times. A generalized diffusion equation associated with the subordinated process x(t)=x(s(t)) is also derived by using subordination and Laplace transform technique.



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*Source: 3479715-2019-06-02.xml*