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Asymptotic behaviour of random walks with correlated temporal structure Marcin Magdziarz, Wladyslaw Szczotka and Piotr Zebrowski Proc. R. Soc. A 2013 469, 20130419, published 28 August 2013
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Asymptotic behaviour of random walks with correlated temporal structure rspa.royalsocietypublishing.org
Marcin Magdziarz1 , Władysław Szczotka2 and Piotr Żebrowski3 1 Hugo Steinhaus Center, Institute of Mathematics and Computer
Research Cite this article: Magdziarz M, Szczotka W, Żebrowski P. 2013 Asymptotic behaviour of random walks with correlated temporal structure. Proc R Soc A 469: 20130419. http://dx.doi.org/10.1098/rspa.2013.0419 Received: 26 June 2013 Accepted: 30 July 2013
Subject Areas: statistical physics, applied mathematics Keywords: continuous-time random walk, Langevin equation, subdiffusion, convergence in distribution, stable distribution, ergodicity breaking Author for correspondence: Marcin Magdziarz e-mail: [email protected]
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2013.0419 or via http://rspa.royalsocietypublishing.org.
Science, Wrocław University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wrocław, Poland 2 Institute of Mathematics, University of Wrocław, Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland 3 Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956 Warsaw, Poland We introduce a continuous-time random walk process with correlated temporal structure. The dependence between consecutive waiting times is generated by weighted sums of independent random variables combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. We derive the corresponding diffusion limit and prove its subdiffusive character. Analysing the set of corresponding coupled Langevin equations, we verify the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking.
1. Introduction The standard mathematical description of diffusion processes is in terms of partial differential equations (Fokker–Planck, Kolmogorov equations) and, equivalently, in terms of Itô or Stratonovich stochastic differential equations. Studies of the diffusion processes have a glorious and exciting history. They began in 1827. In this year, Scottish botanist Robert Brown observed and investigated, for the first time, the random motion of pollen grains suspended in a liquid. It took almost 80 years to describe this chaotic (Brownian) motion of particles using mathematical equations. Einstein [1], in one of his four ‘annus mirabilis’ articles, derived an
2013 The Author(s) Published by the Royal Society. All rights reserved.
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(a) The correlated model and its scaling limit Let us begin with recalling the general definition of the CTRW process. We denote by Ti , i = 1, 2, . . ., the sequence of positive random variables, which represent the waiting times between consecutive jumps of the walker. Then, the process N(t) = max{k ≥ 0 : ki=1 Ti ≤ t} counts the number of jumps of the walker up to time t. Next, let Ji , i = 1, 2, . . ., be the sequence of random
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2. Limiting behaviour
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equation for Brownian motion from microscopic principles. Today, it is called the diffusion equation and has the well-known form ∂p/∂t = K(∂ 2 p/∂x2 ). This equation was first described by Fick [2]. Einstein’s explanation of diffusion was based on two fundamental assumptions: (i) the consecutive steps of the particle are independent with finite second moment and (ii) the mean time taken to perform a step is finite. However, in the 1970s researchers started to investigate situations in which the assumptions made by Einstein do not hold. Surprisingly, the way that photocopier machines work was the trigger for these developments [3]. At that time, the theory of ‘anomalous diffusion’ was born, but it has only been in the last decade that one has been able to observe its vivid development [4]. Today, an increasing number of experimentally observed processes can be described as anomalous. Starting from the signalling of biological cells to the foraging behaviour of animals, it seems that in many cases the overall motion of a particle is better described by steps and waiting times that are not independent and that can broaden power-law distributions [4]. The most popular models of anomalous dynamics are continuous-time random walks (CTRWs), which were introduced to physics by Montroll & Weiss [5]. A CTRW is a process in which the motion of a random walker is described by consecutive jumps and waiting times between them. Each pair of jumps and waiting times is drawn from some associated probability distributions. If both Einstein’s assumptions hold, the CTRW converges in distribution to the Brownian motion (Wiener process). However, if we relax one of these assumptions, then we arrive at the class of fractional dynamics and anomalous diffusions [6–8]. In this paper, we analyse CTRWs for which the second of Einstein’s assumptions does not hold. More precisely, we assume that waiting times have infinite mean. Moreover, we make an additional assumption that they are dependent. The dependence between consecutive waiting times is generated by weighted sums of independent random variables with diverging mean, combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. Such correlated CTRWs are good candidates for modelling human mobility [9], financial market dynamics [10], or chaotic and turbulent flows [11]. The introduced model generalizes those studied in Chechkin et al. [12] and Tejedor & Metzler [13]. Recent progress in the field of CTRWs with correlated temporal and spatial structure can be found in Meerschaert et al. [14] and Magdziarz et al. [15,16]. The Langevin description of some classes of anomalous diffusions has been recently studied in Magdziarz [17], Magdziarz et al. [18] and Teuerle et al. [19]. We would like to emphasize that the word ‘correlated’ is used here in the broad sense and should be understood as ‘dependent’. As the waiting times considered in this paper are heavy tailed, the usual correlation coefficient between them is not defined. In §2, we define the correlated model and derive its diffusion limit. We verify that it is subdiffusive and analyse its speed of relaxation. We also check the asymptotic behaviour of a correlated CTRW for the case of an exponential kernel. In §3, we introduce the Langevin description of the limit process and extend it to the non-zero external forces. Taking advantage of the Langevin picture we analyse the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking. The last section is the summary of the obtained results.
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describes the position of the walker at time t. In this paper, we assume that the jumps Ji are independent, identically distributed (iid) α-stable random variables with the Fourier transform given in the well-known stretched exponential form [20] α
E[eikJi ] = e−|k| ,
0 < α ≤ 2.
(2.2)
Moreover, we introduce the dependent sequence of waiting times Ti in the following manner: i (2.3) M(i − j + 1)ξj , Ti = j=1 where M is the memory function and ξj , j = 1, 2, . . . , is the sequence of iid γ -stable random variables with the Fourier transform [20] πγ sgn(k) , 0 < γ < 1, |β| ≤ 1. (2.4) E[eikξj ] = exp −|k|γ 1 − iβ tan 2 Note that, for β = 1, random variables ξi are positive. However, for other values of the skewness parameter β, ξi are supported also on the negative half-line. Therefore, the absolute value (reflecting boundary condition) must appear in (2.3). We emphasize that taking the absolute value of sum (2.3) is one of a few different ways of reflecting the process at the origin. Another, common in the literature, way of reflecting the jump process is by subtracting its running infimum (see Bertoin [21, ch. 6]). Observe that, in definition (2.3), we use only values of function M at points that are positive integers. Hence, the sequence of waiting times Ti is in fact uniquely determined by the sequence of random variables ξj and the values of M at integer points. Thus, with no restrictions of generality, we will further assume for technical reasons that M is continuous and bounded on interval (0, 1). In order to investigate the limit behaviour of the CTRW (2.1), we need some assumptions on the scaling properties of function M, which are outlined in appendix C. In this paper, however, we will concentrate mainly on the wide class of regularly varying functions [22]. Recall that function M(t) mapping (0, ∞) into (0, ∞) is regularly varying, if for any λ ∈ R lim
t→∞
M(λt) = λρ . M(t)
The number ρ ∈ R \ {0} is the index of regular variation of M(t). A function L(t) satisfying limt→∞ L(λt)/L(t) = 1 for any λ ∈ R is called slowly varying. Regularly varying functions play a key role in calculus, differential equations, probability and number theory [22]; in addition, they have found widespread applications in other scientific fields, such as statistical physics, insurance mathematics or renewal theory (see [23] and references therein). The prominent examples of regularly varying functions are: power-law functions M(t) = tρ , logarithms M(t) = log(t) or constants M(t) = const. We emphasize that, by choosing β = 1 in (2.3) and M(t) = tρ in (2.4), we obtain the correlated model introduced in Chechkin et al. [12]. On the other hand, choosing β = 0 and M(t) = const., we arrive at the correlated CTRW derived in Tejedor & Metzler [13]. Thus, the CTRW (2.1) analysed in our paper generalizes these two models, by combining the underlying correlation mechanisms for waiting times Ti (weighted sum of ξi ’s combined with a reflecting boundary condition). Moreover, as compared with Chechkin et al. [12], we extend the choice of the kernel M(t) to the wide class of regularly varying functions. In the last part of this section, we separately analyse the case of the exponential kernel function, which does not belong to the family of regularly varying functions.
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variables representing the consecutive jumps of the walker. Consequently, the CTRW process defined as N(t) Ji (2.1) R(t) =
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In the next theorem, which is the main result of this section, we derive the asymptotic diffusion limit of the correlated CTRW (2.1).
(2.5)
1+1/γ
M(Bn )/n → 1, as n → ∞. Lα (t) is the Here, the deterministic scaling sequence {Bn } is such that Bn α α-stable Lévy motion with Fourier transform E[eikLα (t) ] = e−t|k| . Moreover, τ s (2.6) S(τ ) = (s − y)ρ dTγ (y) ds, 0
0
where Tγ (y) is the γ -stable Levy motion with Fourier transform E[eikTγ (y) ] = exp{−y|k|γ (1 − iβ tan(π γ /2) sgn(k))} independent of Lα (t), whereas S−1 (t) is the inverse of S(τ ), i.e. S−1 (t) = inf{τ > 0 : S(τ ) > t}.
Proof. See appendix E.
The scaling sequence Bn can be written in the form Bn = L(n)n1/(1+ρ+1/γ ) , where L(n) is some slowly varying function. A more general result extending theorem 2.1 is formulated as theorem D.1 in appendix D. d
The notation ‘−→’ in the above theorem means convergence in distribution. However, in the appendices, we prove even stronger results—functional convergence in the Skorohod J1 topology (see Billingsley [24]). In particular, this convergence implies convergence of all finite-dimensional distributions. Now, let us explain the structure of the limit process X(t). The α-stable Lévy motion Lα (t) appears here as the limit of rescaled cumulated jumps ni=1 Ji . Moreover, the cumulated waiting n i−1 n times i=1 Ti = i=1 | j=0 M(i − j)ξj | converge (after proper rescaling) to the process S(t). It follows that the inverse process S−1 (t) is the scaling limit of the counting process N(t). Finally, the limit of the correlated CTRW has the form of subordination X(t) = Lα (S−1 (t)). It appears that the limit process X(t) has nice scaling properties. Note first that for any a > 0 τ ua aτ s (s − y)ρ dTγ (y) ds u=s/a (ua − y)ρ dTγ (y) du = a S(aτ ) = 0
0
z=y/a 1+ρ
= a
0
0
τ u d 1+ρ+1/γ (u − z)ρ dTγ (az) du = a S(τ ). 0
0
Thus, S(τ ) is self-similar with index 1 + ρ + 1/γ . It follows from the property P[S−1 (t) ≤ τ ] = P[t ≤ S(τ )] that S−1 (t) is self-similar with index 1/(1 + ρ + 1/γ ). Finally, applying the wellknown fact that Lα (t) is 1/α-self-similar, we get that X(t) = Lα (S−1 (t)) is self-similar with index H = 1/α(1 + ρ + 1/γ ). Note that the parameter H can take any value from the infinite interval (0, ∞). Thus, depending on the choice of the parameters, the process X(t) can display scaling, which is characteristic for sub-, super- or normal diffusion [25]. However, the second moment is finite only for α = 2. Then, the mean square displacement yields E[X2 (t)] = K · t1/(1+ρ+1/γ ) , where K = E[X2 (1)] is the generalized diffusion constant. As the parameter 1/(1 + ρ + 1/γ ) takes values in the interval (0, 1/2], the mean square displacement of X(t) displays subdiffusive behaviour.
(b) Exponential kernel Now, we investigate in detail the diffusion limit of the correlated CTRW with exponential kernel M(t). This model was introduced in Chechkin et al. [12]. We show that, in the limit, the exponential
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R(nt) d −→ X(t) = Lα (S−1 (t)) as n → ∞. (Bn )1/α
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Theorem 2.1. Let Ji , i ∈ N, be the sequence of iid α-stable random variables with Fourier transform (2.2). Let Ti , i ∈ N, be the correlated sequence of waiting times defined by (2.3) with ξi given by (2.4) and M(t) regularly varying with index ρ > 0. Assume that the jumps Ji , i ∈ N, and the waiting times Ti , i ∈ N, are independent. Then, the corresponding CTRW process R(t) (2.1) satisfies
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kernel kills the dependence between waiting times. As a result, the scaling limit is the same as for the uncorrelated temporal structure.
exp{−c(i − j + 1)}ξj ,
c > 0,
(2.7)
j=0
where ξi satisfy (2.4) with β = 1. Assume that the jumps Ji , i ∈ N, and the waiting times Ti , i ∈ N, are independent. Then, the corresponding CTRW process R(t) (2.1) satisfies R(nt) d −→ Y(t) = Lα (Sˆ −1 (t)) as n → ∞. nγ /α
(2.8)
ˆ ) is the Here, Lα (t) is the α-stable Lévy motion with Fourier transform E[eikLα (t) ] = e−t|k| . Moreover, S(τ γ -stable subordinator with Fourier transform πγ ˆ sgn(k) , (2.9) E[eikS(τ ) ] = exp −τ (ec − 1)−γ |k|γ 1 − i tan 2 α
ˆ ), i.e. Sˆ −1 (t) = inf{τ > 0 : S(τ ˆ ) > t}. which is independent of Lα (t), whereas Sˆ −1 (t) is the inverse of S(τ
Proof. See appendix F.
The above result confirms that the limit process Y(t) of the correlated CTRW with an exponential kernel is equal (up to a constant) to the limit of the CTRW with uncorrelated waiting times Ti [4,8,26]. This shows that the exponential kernel kills the dependence between waiting times, and the correlation between successive rests of the walker can be observed only in the primary stage of the motion. It should be noted that the probability density function of the scaling limit Y(t) satisfies the following fractional diffusion equation [4]: ∂w(x, t) 1−γ = K0 Dt ∇ α w(x, t). ∂t 1−γ
stands for the fractional derivative of the Riemann–Liouville type, and Here, the operator 0 Dt ∇ α is the Riesz fractional derivative with the Fourier transform F {∇ μ f (x)} = −|k|α f˜(k). K > 0 is the generalized diffusion coefficient. For α = 2, the second moment of Y(t) is finite and given by E[Y2 (t)] = c · tγ , where c = E[Y2 (1)]. This result agrees with the asymptotic subdiffusive behaviour of the mean square displacement derived in Chechkin et al. [12] for large times.
3. Langevin framework (a) Coupled Langevin equations The great advantage of the obtained diffusion limit (2.5) is that it can be used to describe the corresponding correlated dynamics in the framework of coupled Langevin equations. Applying the method of Fogedby [27], we get the following set of coupled Langevin equations for position x and time t of the considered process: s ρ ˙ (3.1) x˙ (s) = Γα (s), t(s) = (s − y) dz(y) and z˙ (s) = Γγ (s). 0
Here, Γα (s) = dLα (s)/ds and Γγ (s) = dTγ (s)/ds are two independent noises. The first equation in (3.1) is the usual Langevin equation in the operational time s. The next two equations describe the relationship between the operational time s and the physical time t. Solving the above system of equations is rather straightforward. First, one solves the first equation to get the driving process
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i
Ti =
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Theorem 2.2. Let Ji , i ∈ N, be the sequence of iid α-stable random variables with Fourier transform (2.2). Let Ti , i ∈ N, be the correlated sequence of waiting times given by
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0
0
0
Finally, we obtain the solution of (3.1) in the subordination form X(t) = x(s(t)) = Lα (s(t)), where s(t) is the inverse of t(s), i.e. s(t) = inf{s > 0 : t(s) > t}. So, we arrive at exactly the same process as in (2.5). To include external force F(x) in the description of correlated dynamics, one needs to add the drift term in the first equation in (3.1). Then, we obtain the following set of equations: s F(x(s)) (3.3) + Γα (s), ˙t(s) = (s − y)ρ dz(y) and z˙ (s) = Γγ (s), x˙ (s) = mη 0 which describes the correlated dynamics in the presence of external force F(x). Here, m is the particle mass and η denotes friction. The solution to the above set of equations is obtained in the same way as (3.1). We get that the solution is equal to X(t) = x(s(t)), where s(t) is the inverse of t(s) given in (3.2) and x(s) satisfies the following stochastic differential equation: dx(s) =
F(x(s)) ds + dLα (s). mη
Based on (3.3), one can efficiently approximate numerically trajectories of X(t). It is enough to simulate the noises Γα (s) and Γγ (s), which can be done by the well-known method of simulating stable processes [20] and plugging the obtained results into (3.3).
(b) Asymptotic properties Now, let us apply previously obtained representations (3.1) and (3.3) to analyse the asymptotic properties of the correlated model. We start with the speed of single-mode relaxation. Assume that F ≡ 0 and denote by w(k, t) = E[exp(ikX(t))] the characteristic function (Fourier transform) of X(t). We have w(k, t) = E[exp(−|k|α s(t))]. Recall that s(t) is the inverse of t(s) given in (3.2). Denote s u a(s) = (u − y)ρ dTγ (y) du and b(s) = sρ+1 sup |Tγ (u)|. 0 0
u∈[0,s]
Then we have a(s) ≤ t(s) ≤ b(s) or, equivalently, exp(−|k|α a−1 (t)) ≤ exp(−|k|α s(t)) ≤ exp(−|k|α b−1 (t)). Here, a−1 (t) and b−1 (t) are the inverses of a(s) and b(s), respectively. Now, as both processes a(s) and b(s) are heavy-tailed with index γ and self-similar with index 1 + ρ + 1/γ , applying the above inequalities and Tauberian theorems, we get that w(k, t) 1/tγ for large t. The notation w(k, t) 1/tγ means that there exist positive constants c1 and c2 , such that c1 t−γ < w(k, t) < c2 t−γ for large t. Thus, the correlated model displays power-law single-mode relaxation. In what follows, we will analyse properties which require the existence of moments of the process. Therefore, from now until the end of the paper, we assume that α = 2. This means that the process Lα=2 (t) is just the standard Brownian motion. We will denote it by B(t). Thus, its moments of any order exist. In particular, E[B2 (t)] = 2Dt, where D > 0 is the diffusion constant. Consequently, the second moment of the correlated model yields E[X2 (t)] = E[B2 (s(t))] = 2DE[s(t)] = 2DDγ t1/(1+ρ+1/γ ) ,
(3.4)
which is typical for subdiffusion. Here, Dγ = E[s(1)] depends on the distribution of the correlated waiting times. It is natural to ask about the stationary solution of (3.3). As the subordinator s(t) tends to infinity as t → ∞, we obtain that the stationary solution of (3.3) has the well-known form wst (x) ∝ exp(−V(x)/Dmη). Here, V(x) = − F(x) dx denotes the external potential. Comparing
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0
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x(s) = Lα (s) acting in the operational time s. In the next step, one solves the second and third equations to get z(s) = Tγ (s) and s u s u (3.2) t(s) = (u − y)ρ dz(y) du = (u − y)ρ dTγ (y) du.
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the above expression with the classical Gibbs–Boltzmann equilibrium distribution weq ∝ exp(−V(x)/[kB T]), we obtain that the correlated model satisfies the generalized Einstein relation (3.5)
which connects the diffusion parameter D with the dissipative parameter η and the energy kB T. Further on, if we assume that the force is constant F ≡ F0 , we obtain the following expression for the first moment of the correlated process X(t): EF0 [X(t)] =
F0 F0 F0 E[s(t)] + E[B(s(t))] = E[s(t)] = Dγ t1/(1+ρ+1/γ ) . mη mη mη
Comparing the above expression with the second moment in the force-free case (3.4), we arrive at the second Einstein relation EF0 [X(t)] = (F0 /2)(E[X2 (t)]/kB T). Similar to other models of subdiffusion based on CTRWs, the studied process displays weak ergodicity breaking [28–33]. To see this, let us analyse the time-averaged mean square displacement t− 1 δ 2 (, t) = [X(τ + ) − X(τ )]2 dτ . (3.6) t− 0 Here, is the lag time and t represents the total length of the observation. For standard diffusion processes, δ 2 (, t) scales as 2D for large t. However, after some standard calculations, we get the following expression for the correlated process: E[δ 2 (, t)] ∼
2DDγ t1−1/(1+ρ+1/γ )
for t. Thus, the correlated process displays weak ergodicity breaking. Another feature of anomalous dynamics displayed by CTRW models is ageing [34,35], manifested by the temporal decay of the response of the process to a sinusoidal, time-dependent force. Applying the subordination method to the correlated process under the influence of timedependent force F(t) = F0 sin(ωt) (see [17,34] for details), we get that the corresponding first moment of X(t) has the form F0 t E[X(t)] = sin(ωu) dE[s(u)] mη 0
t F0 Dγ 1 u1/(1+ρ+1/γ )−1 sin(ωu) du. = mη 1 + ρ + 1/γ 0 Because of the factor u1/(1+ρ+1/γ )−1 inside the integral, the first moment decays as time proceeds and we observe the so-called ‘death of linear response’ [34].
4. Summary In this paper, we have introduced the CTRW process with correlated temporal structure. Consecutive waiting times were defined as weighted sums of independent random variables combined with a reflecting boundary condition. The weights, in turn, were determined by the regularly varying memory function. Our model generalizes two different correlated models introduced in Chechkin et al. [12] and Tejedor & Metzler [13]. We have analysed various asymptotic properties of the studied process. We have derived the corresponding diffusion limit and proved its subdiffusive character. It should be emphasized that the functional limit theorems proved in the appendices apply to the class of correlated processes, which is much wider than the CTRWs studied in the main part of the paper. Applying the subordination scheme we have extended the correlated model to include external forces. The derived coupled Langevin equations have been further used to analyse Einstein relations, speed of relaxation, equilibrium distribution, ergodicity breaking and ageing properties. We have separately analysed the case of exponential memory function. We have shown that in such a setting the exponential kernel kills the dependence between waiting times and the
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kB T , mη
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D=
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Appendix The structure of this appendix is as follows. Necessary definitions and notations are gathered in appendix A. In appendix B, we define two useful mappings and show when they are continuous. Next, in appendix C, we formulate two general assumptions on the sequence of jumps Jk and the sequence of innovations ξk and the function M defining the correlated time structure of the CTRW process. We also discuss in what situations these general assumptions are satisfied. Results from appendices B and C will be used in appendix D, in which we prove the general result dealing with the weak convergence of the sequence of scaled CTRW processes. In appendix E, we give the proof of theorem 2.1, which is the conclusion from appendix D. In appendix F, we prove theorem 2.2.
Appendix A. Definitions and notations Let D denote the space of functions mapping [0, ∞) into R that are right-continuous having limits from the left (cádlág functions). We equip this space with the Skorohod J1 topology. We define also two subspaces of D; namely, D↑ being the space of all cádlág functions starting from 0 that are non-negative, non-decreasing and unbounded from above, and the space Dv containing all cádlág functions of bounded variation on any interval [0, r], r > 0. Let M and Ms denote the space of Radon measures on [0, ∞) and the space of signed Radon measures on [0, ∞), respectively. For any function y ∈ D↑ there exists measure μy ∈ M such that μy ((a, b]) = y(b) − y(a) for any interval (a, b], 0 ≤ a < b. Any function z ∈ Dv may be written in the form z = z+ − z− , where z+ , z− ∈ D↑ . Hence, to any z ∈ Dv corresponds the measure μz ∈ Ms defined as μz = μz+ − μz− . r In the following appendices, we write 0 f (x)μy (dx) to denote the Lebesgue integral of f r with respect to the Radon measure μy , y ∈ D↑ , while 0 f (x) dy(x) is to be understood in the Riemann–Stieltjes sense with f and y being the integrand and the integrator, respectively. For r r r r z ∈ Dv , we write 0 f (x)μz (dx) = 0 f (x)μz+ (dx) − 0 f (x)μz− (dx). Recall that 0 f (x) dz(x) exists if f is continuous on [0, r], and then r r r + f (x) dz(x) = f (x) dz (x) − f (x) dz− (x) 0
0
=
r 0
0
f (x)μz+ (dx) −
r 0
f (x)μz− (dx) =
r 0
f (x)μz (dx).
Given functions yn , y ∈ D↑ , we say that sequence {μyn } of measures from M converges vaguely ∞ ∞ v to some measure μy ∈ M, which we denote as μyn → μy , if 0 g(u)μyn (du) → 0 g(u)μy (du) for any function g : [0, ∞) → [0, ∞), which is continuous and has compact support. It is equivalent to convergence yn (t) → y(t) for all t being continuity points of y. Similarly, given functions zn , v z ∈ Dv , we say that measures μzn ∈ Ms converge vaguely to the measure μz ∈ Ms if μz+n → μz+ v
and μz−n → μz− .
Appendix B. Continuity of two useful mappings For any x ∈ D and y ∈ D↑ , we define mapping as t def h(x, y)(t) = x(t − u)μy (du), 0
t ≥ 0.
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Funding statement. The research of M.M. was partially supported by an NCN Maestro grant.
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correlation between successive rests of the walker can be observed only in the primary stage of the motion. However, the diffusion limit is the same as for the renewal CTRW with independent waiting times.
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as n → ∞.
Proof. See the electronic supplementary material. For any x ∈ D, z ∈ Dv and κ ∈ M we define mapping t s def f (x, y, κ)(t) = x(s − u)μz (du) κ(ds), 0
0
t ≥ 0.
− Lemma B.2. Let xn , x be as in lemma B.1 and assume that sequence {(zn , z+ n , zn )} ⊂ Dv × D↑ × D↑ + + − + converges to (z, z , z ) ∈ Dv × D↑ × D↑ in the Skorohod J1 topology, where zn = zn − z− n and z = z − v − z . Moreover, let κn , κ ∈ M, such that κn → κ. Then for any r > 0
sup |f (xn , zn , κn )(t) − f (x, z, κ)(t)| → 0 as n → ∞. 0≤t≤r
Proof. See the electronic supplementary material.
Appendix C. General assumptions on a continuous-time random walk with correlated temporal structure The construction of a CTRW with correlated temporal structure is based on the sequence of random vectors {(Ji , ξi )}i≥1 defined on the common probability space (Ω, F , P) and the function M. Consecutive jumps of the walker are given by the sequence {Ji }, while waiting times between the jumps are of the form Ti = | ij=1 M(i − j + 1)ξj |. We define the counting process N(t) = max{k ≥ N(t) 0 : T1 + · · · + Tk ≤ t} and the CTRW R(t) = i=1 Ji . Below we state assumptions (A1) and (A2) on the sequence {(Ji , ξi )} and the function M, respectively. Under these assumptions, we will be able to obtain asymptotic behaviour of the sequence of scaled CTRWs with correlated temporal structure. (A1) There exist sequences {an } and {bn } such that the sequence of processes of scaled partial sums (Ln (t), Ξn (t), Ξn+ (t), Ξn− (t)) ⎛ ⎞ [nt] [nt] [nt] [nt] def ⎝ −1 = an Ji , b−1 ξi , b−1 ξi 1(ξi > 0), −b−1 ξi 1(ξi < 0)⎠ n n n i=1
i=1
i=1
i=1
converges weakly in the Skorohod J1 topology to the process (L(t), Ξ (t), Ξ + (t), Ξ − (t)) such that Ξ = Ξ + − Ξ − and the trajectories of processes Ξ + and Ξ − are the elements of D↑ , almost surely. (A2) There exists a sequence {cn } and a continuous function m mapping [0, ∞) into R, such that ∀r > 0 sup |Mn (t) − m(t)| → 0 as n → ∞, 0≤t≤r
where
def Mn (t) = c−1 n M(nt + 1),
t ≥ 0.
Before proceeding to investigate the asymptotic of the CTRW with correlated temporal structure (see appendix D), we discuss the situations in which the above assumptions hold true. The following lemma, which is an extension of [36, theorem 7.1], is a useful tool for checking whether (A1) holds.
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sup |h(xn , yn )(t) − h(x, y)(t)| → 0 0≤t≤r
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Lemma B.1. Assume that the sequence of functions xn ∈ D, n ≥ 1, converges to a continuous function x uniformly on any interval [0, r], r > 0, and let the sequence yn ∈ D↑ , n ≥ 1, converge to y ∈ D↑ in the Skorohod J1 topology. Then for any r > 0
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Lemma C.1. Let {Xn,j ≡ (X1,n,j , X2,n,j )}n,j≥1 be the array of random vectors such that {Xn,j }j≥1 is the iid sequence for any n ≥ 1. Assume that
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(i)
v
nP(Xn,1 ∈ ·) → ν0 (·) as n → ∞, where ν0 is a Lévy measure on R2 \ {0}; (ii) P(X2,n,1 > 0) → p and P(X2,n,1 < 0) → q = 1 − p
as n → ∞;
(iii) 2 lim lim sup nE(Xi,n,1 1(|Xi,n,1 | ≤ ε)) = 0,
ε0 n→∞
i = 1, 2.
Let {Yn,j ≡ (Y1,n,j , Y2,n,j , Y3,n,j , Y4,n,j )}n,j≥1 be the array of random vectors, where def
(Y1,n,j , Y2,n,j , Y3,n,j , Y4,n,j ) = (X1,n,j , X2,n,j , X2,n,j 1(X2,n,k > 0), −X2,n,j 1(X2,n,k < 0)). Define the sequence of partial sum processes Yn ≡ (Y1,n , Y2,n , Y3,n , Y4,n ) [nt] def
Yn (t) =
(Yn,k − E(Yn,k 1(Yn,k ∞ ≤ 1))),
k=1
where x∞ = maxi=1,2,3,4 |xi |, x ∈ R4 . Then (a) Yn ⇒ Y in the Skorohod J1 topology, where Y ≡ (Y1 , Y2 , Y3 , Y4 ) is the Lévy process with the Fourier transform k,Y(t) ik,x Ee = exp t (e − 1 − ik, x1(0 < x∞ ≤ 1))ν(dx) , k ∈ R4 , x=0
where Lévy measure ν is defined as def
ν(dx) = pν0 (dx1 × dx2 )δx2 (dx3 )δ0 (dx4 ) + qν0 (dx1 × dx2 )δ0 (dx3 )δ(−x2 ) (dx4 ); (b) processes Y3 and Y4 are independent and Y2 = Y3 − Y4 .
Proof. See the electronic supplementary material.
Remark C.2. One can prove results similar to [36, theorem 7.1] and lemma C.1 in situations where {Xn,k } is the array of dependent random vectors and the first coordinate of the limit process Y has a gaussian component. For more details, see [37,38]. Assumption (A2) is also quite weak. In particular, it is satisfied for functions from a broad family of regularly varying functions with index ρ > 0 (see appendix E). Assumption (A2) also holds when Mn (t) → m(t) for every t ≥ 0 and functions Mn are non-decreasing.
Appendix D. Asymptotic of a continuous-time random walk with correlated temporal structure Theorem D.1. Let the assumptions (A1) and (A2) be satisfied with sequences {bn } and {cn } such that −1 in the nbn cn → ∞ as n → ∞. Define the sequence of processes Rn (t) = a−1 n R(nbn cn t). Then Rn ⇒ L ◦ S t s Skorohod J1 topology, where S(t) = 0 | 0 m(s − y) dΞ (y)| ds. def
Proof. See the electronic supplementary material.
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=
t
s
0
0
m(s − y) dΞ (y) ds =
t t−y 0
t
t
0
y
m(s − y) ds dΞ (y)
m(u) du dΞ (y).
0
Furthermore, one can easily show that ∀t ≥ 0
t [nt] 1 def M(i) → m(u) du = g(t) as n → ∞ ncn 0 i=1
and then S(t) =
t
0 g(t − y) dΞ (y).
Appendix E. Proof of theorem 2.1 Let sequences {Jk } and {ξk } and function M be as in §2a. We will obtain theorem 2.1 as a conclusion from theorem D.1. Proof of theorem 2.1. In order to use theorem D.1, we need to check if its assumptions are satisfied, i.e. if sequence {(Jk , ξk )} fulfils condition (A1) and the regularly varying function M satisfies condition (A2). We begin with checking condition (A1). The main tool we use to verify that this condition holds will be lemma C.1; thus, we need to check its assumptions first. Before doing so, recall that {Jk } is the iid sequence with Fourier transform given by (2.2). As shown in [39, §1.2.6] we D
have that Jk ∼ Sα (1, 0, 0), n−1/α (J1 + · · · + Jn ) = J, where J ∼ Sα (1, 0, 0) is independent of {Jk } and the distribution of Jk is infinitely divisible with Lévy measure 1 −1−α x dx, x > 0, def να (dx) = 21 −1−α dx, x < 0. |x| 2 v
Then, we also have that nP(n−1/α J1 ∈ ·) → να (·) as n → ∞ (e.g. [40, theorem 3.2.2]). Similarly, the assumption that ξk have Fourier transform (2.4) implies that ξk ∼ Sγ (1, β, 0), D
n−1/α (ξ1 + · · · + ξn ) = ξ , where ξ ∼ Sγ (1, β, 0) is independent of {ξk }, the distribution of ξk is infinitely divisible with Lévy measure ⎧ 1 + β −1−γ ⎪ ⎨ dx, x > 0, x def 2 νγ (dx) = ⎪ ⎩ 1 − β |x|−1−γ dx, x < 0 2 v
and that nP(n−1/γ ξ1 ∈ ·) → νγ (·) as n → ∞. Now, we define the array of random vectors {Xn,k } ≡ {(n−1/α Jk , n−1/γ ξk )} and check that it fulfils the assumptions of lemma C.1. Take arbitrary A, B ∈ B(R) such that {(0, 0)} ∈ / A × B. Then, as sequences {Jk } and {ξk } are assumed to be independent, we have that nP(Xn,1 ∈ A × B) = nP(n−1/α J1 ∈ A)P(n−1/γ ξ1 ∈ B) να (A)1B (0), if 0 ∈ B, → νγ (B)1A (0), if 0 ∈ A. Hence condition (i) holds with ν0 (dx × dy) ≡ να (dx)δ0 (dy) + δ0 (dx)νγ (dy). Assumption (ii) is obviously satisfied with p = P(ξ1 > 0) and q = P(ξ1 < 0).
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S(t) =
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Remark D.2. Let the assumptions of theorem D.1 be satisfied. If we additionally assume that ξi > 0 almost surely for all i ≥ 1 and that function M is non-negative, then
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Next observe that for arbitrary small ε > 0
nE((n−1/γ ξ1 )2 1(|n−1/γ ξ1 | ≤ ε)) →
0 0 : ⎩ ⎭ ⎫ ⎬
⎧ ⎨
i=1
⎞ k 1⎝ 1 1 = Ti ≤ n1/γ t + 1⎠ = N(n1/γ t) + . max k ≥ 0 : ⎭ ⎩ n n n ⎛
i=1
We define the auxiliary sequence of CTRW processes as
1/γ t) N(n1/γ t) 1 def R(n −1 = L (t) − = L , Rn (t) = S n n n n n n−1/α
t ≥ 0.
By [36, corollary 7.1], it follows that Ln ⇒ Lα and that sequences of processes Ln , Sn are ˆ in the Skorohod independent. This, together with (F 3), implies joint convergence (Ln , Sn ) ⇒ (Lα , S) J1 topology. Then from [41, theorem 3.6], it follows that Rn ⇒ Lα ◦ Sˆ−1 in the Skorohod J1 topology.
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= (ec − 1)−1 Ξn (t) − (ec − 1)−1 Zn (t),
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j=1
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We conclude the proof of theorem 2.2 with the observation that R(nt)/nγ /α = Rnγ (t). As nγ → ∞ with n → ∞, n−γ /α R(nt) has the same limit as sequence Rn and convergence (2.8) follows.
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Asymptotic behaviour of random walks with correlated temporal structure Marcin Magdziarz, Wladyslaw Szczotka and Piotr Zebrowski Proc. R. Soc. A 2013 469, 20130419, published 28 August 2013
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Asymptotic behaviour of random walks with correlated temporal structure rspa.royalsocietypublishing.org
Marcin Magdziarz1 , Władysław Szczotka2 and Piotr Żebrowski3 1 Hugo Steinhaus Center, Institute of Mathematics and Computer
Research Cite this article: Magdziarz M, Szczotka W, Żebrowski P. 2013 Asymptotic behaviour of random walks with correlated temporal structure. Proc R Soc A 469: 20130419. http://dx.doi.org/10.1098/rspa.2013.0419 Received: 26 June 2013 Accepted: 30 July 2013
Subject Areas: statistical physics, applied mathematics Keywords: continuous-time random walk, Langevin equation, subdiffusion, convergence in distribution, stable distribution, ergodicity breaking Author for correspondence: Marcin Magdziarz e-mail: [email protected]
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2013.0419 or via http://rspa.royalsocietypublishing.org.
Science, Wrocław University of Technology, Wybrzeze Wyspianskiego 27, 50-370 Wrocław, Poland 2 Institute of Mathematics, University of Wrocław, Plac Grunwaldzki 2/4, 50-384 Wrocław, Poland 3 Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8, 00-956 Warsaw, Poland We introduce a continuous-time random walk process with correlated temporal structure. The dependence between consecutive waiting times is generated by weighted sums of independent random variables combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. We derive the corresponding diffusion limit and prove its subdiffusive character. Analysing the set of corresponding coupled Langevin equations, we verify the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking.
1. Introduction The standard mathematical description of diffusion processes is in terms of partial differential equations (Fokker–Planck, Kolmogorov equations) and, equivalently, in terms of Itô or Stratonovich stochastic differential equations. Studies of the diffusion processes have a glorious and exciting history. They began in 1827. In this year, Scottish botanist Robert Brown observed and investigated, for the first time, the random motion of pollen grains suspended in a liquid. It took almost 80 years to describe this chaotic (Brownian) motion of particles using mathematical equations. Einstein [1], in one of his four ‘annus mirabilis’ articles, derived an
2013 The Author(s) Published by the Royal Society. All rights reserved.
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(a) The correlated model and its scaling limit Let us begin with recalling the general definition of the CTRW process. We denote by Ti , i = 1, 2, . . ., the sequence of positive random variables, which represent the waiting times between consecutive jumps of the walker. Then, the process N(t) = max{k ≥ 0 : ki=1 Ti ≤ t} counts the number of jumps of the walker up to time t. Next, let Ji , i = 1, 2, . . ., be the sequence of random
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2. Limiting behaviour
2
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equation for Brownian motion from microscopic principles. Today, it is called the diffusion equation and has the well-known form ∂p/∂t = K(∂ 2 p/∂x2 ). This equation was first described by Fick [2]. Einstein’s explanation of diffusion was based on two fundamental assumptions: (i) the consecutive steps of the particle are independent with finite second moment and (ii) the mean time taken to perform a step is finite. However, in the 1970s researchers started to investigate situations in which the assumptions made by Einstein do not hold. Surprisingly, the way that photocopier machines work was the trigger for these developments [3]. At that time, the theory of ‘anomalous diffusion’ was born, but it has only been in the last decade that one has been able to observe its vivid development [4]. Today, an increasing number of experimentally observed processes can be described as anomalous. Starting from the signalling of biological cells to the foraging behaviour of animals, it seems that in many cases the overall motion of a particle is better described by steps and waiting times that are not independent and that can broaden power-law distributions [4]. The most popular models of anomalous dynamics are continuous-time random walks (CTRWs), which were introduced to physics by Montroll & Weiss [5]. A CTRW is a process in which the motion of a random walker is described by consecutive jumps and waiting times between them. Each pair of jumps and waiting times is drawn from some associated probability distributions. If both Einstein’s assumptions hold, the CTRW converges in distribution to the Brownian motion (Wiener process). However, if we relax one of these assumptions, then we arrive at the class of fractional dynamics and anomalous diffusions [6–8]. In this paper, we analyse CTRWs for which the second of Einstein’s assumptions does not hold. More precisely, we assume that waiting times have infinite mean. Moreover, we make an additional assumption that they are dependent. The dependence between consecutive waiting times is generated by weighted sums of independent random variables with diverging mean, combined with a reflecting boundary condition. The weights are determined by the memory kernel, which belongs to the broad class of regularly varying functions. Such correlated CTRWs are good candidates for modelling human mobility [9], financial market dynamics [10], or chaotic and turbulent flows [11]. The introduced model generalizes those studied in Chechkin et al. [12] and Tejedor & Metzler [13]. Recent progress in the field of CTRWs with correlated temporal and spatial structure can be found in Meerschaert et al. [14] and Magdziarz et al. [15,16]. The Langevin description of some classes of anomalous diffusions has been recently studied in Magdziarz [17], Magdziarz et al. [18] and Teuerle et al. [19]. We would like to emphasize that the word ‘correlated’ is used here in the broad sense and should be understood as ‘dependent’. As the waiting times considered in this paper are heavy tailed, the usual correlation coefficient between them is not defined. In §2, we define the correlated model and derive its diffusion limit. We verify that it is subdiffusive and analyse its speed of relaxation. We also check the asymptotic behaviour of a correlated CTRW for the case of an exponential kernel. In §3, we introduce the Langevin description of the limit process and extend it to the non-zero external forces. Taking advantage of the Langevin picture we analyse the speed of relaxation, Einstein relations, equilibrium distributions, ageing and ergodicity breaking. The last section is the summary of the obtained results.
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describes the position of the walker at time t. In this paper, we assume that the jumps Ji are independent, identically distributed (iid) α-stable random variables with the Fourier transform given in the well-known stretched exponential form [20] α
E[eikJi ] = e−|k| ,
0 < α ≤ 2.
(2.2)
Moreover, we introduce the dependent sequence of waiting times Ti in the following manner: i (2.3) M(i − j + 1)ξj , Ti = j=1 where M is the memory function and ξj , j = 1, 2, . . . , is the sequence of iid γ -stable random variables with the Fourier transform [20] πγ sgn(k) , 0 < γ < 1, |β| ≤ 1. (2.4) E[eikξj ] = exp −|k|γ 1 − iβ tan 2 Note that, for β = 1, random variables ξi are positive. However, for other values of the skewness parameter β, ξi are supported also on the negative half-line. Therefore, the absolute value (reflecting boundary condition) must appear in (2.3). We emphasize that taking the absolute value of sum (2.3) is one of a few different ways of reflecting the process at the origin. Another, common in the literature, way of reflecting the jump process is by subtracting its running infimum (see Bertoin [21, ch. 6]). Observe that, in definition (2.3), we use only values of function M at points that are positive integers. Hence, the sequence of waiting times Ti is in fact uniquely determined by the sequence of random variables ξj and the values of M at integer points. Thus, with no restrictions of generality, we will further assume for technical reasons that M is continuous and bounded on interval (0, 1). In order to investigate the limit behaviour of the CTRW (2.1), we need some assumptions on the scaling properties of function M, which are outlined in appendix C. In this paper, however, we will concentrate mainly on the wide class of regularly varying functions [22]. Recall that function M(t) mapping (0, ∞) into (0, ∞) is regularly varying, if for any λ ∈ R lim
t→∞
M(λt) = λρ . M(t)
The number ρ ∈ R \ {0} is the index of regular variation of M(t). A function L(t) satisfying limt→∞ L(λt)/L(t) = 1 for any λ ∈ R is called slowly varying. Regularly varying functions play a key role in calculus, differential equations, probability and number theory [22]; in addition, they have found widespread applications in other scientific fields, such as statistical physics, insurance mathematics or renewal theory (see [23] and references therein). The prominent examples of regularly varying functions are: power-law functions M(t) = tρ , logarithms M(t) = log(t) or constants M(t) = const. We emphasize that, by choosing β = 1 in (2.3) and M(t) = tρ in (2.4), we obtain the correlated model introduced in Chechkin et al. [12]. On the other hand, choosing β = 0 and M(t) = const., we arrive at the correlated CTRW derived in Tejedor & Metzler [13]. Thus, the CTRW (2.1) analysed in our paper generalizes these two models, by combining the underlying correlation mechanisms for waiting times Ti (weighted sum of ξi ’s combined with a reflecting boundary condition). Moreover, as compared with Chechkin et al. [12], we extend the choice of the kernel M(t) to the wide class of regularly varying functions. In the last part of this section, we separately analyse the case of the exponential kernel function, which does not belong to the family of regularly varying functions.
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variables representing the consecutive jumps of the walker. Consequently, the CTRW process defined as N(t) Ji (2.1) R(t) =
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In the next theorem, which is the main result of this section, we derive the asymptotic diffusion limit of the correlated CTRW (2.1).
(2.5)
1+1/γ
M(Bn )/n → 1, as n → ∞. Lα (t) is the Here, the deterministic scaling sequence {Bn } is such that Bn α α-stable Lévy motion with Fourier transform E[eikLα (t) ] = e−t|k| . Moreover, τ s (2.6) S(τ ) = (s − y)ρ dTγ (y) ds, 0
0
where Tγ (y) is the γ -stable Levy motion with Fourier transform E[eikTγ (y) ] = exp{−y|k|γ (1 − iβ tan(π γ /2) sgn(k))} independent of Lα (t), whereas S−1 (t) is the inverse of S(τ ), i.e. S−1 (t) = inf{τ > 0 : S(τ ) > t}.
Proof. See appendix E.
The scaling sequence Bn can be written in the form Bn = L(n)n1/(1+ρ+1/γ ) , where L(n) is some slowly varying function. A more general result extending theorem 2.1 is formulated as theorem D.1 in appendix D. d
The notation ‘−→’ in the above theorem means convergence in distribution. However, in the appendices, we prove even stronger results—functional convergence in the Skorohod J1 topology (see Billingsley [24]). In particular, this convergence implies convergence of all finite-dimensional distributions. Now, let us explain the structure of the limit process X(t). The α-stable Lévy motion Lα (t) appears here as the limit of rescaled cumulated jumps ni=1 Ji . Moreover, the cumulated waiting n i−1 n times i=1 Ti = i=1 | j=0 M(i − j)ξj | converge (after proper rescaling) to the process S(t). It follows that the inverse process S−1 (t) is the scaling limit of the counting process N(t). Finally, the limit of the correlated CTRW has the form of subordination X(t) = Lα (S−1 (t)). It appears that the limit process X(t) has nice scaling properties. Note first that for any a > 0 τ ua aτ s (s − y)ρ dTγ (y) ds u=s/a (ua − y)ρ dTγ (y) du = a S(aτ ) = 0
0
z=y/a 1+ρ
= a
0
0
τ u d 1+ρ+1/γ (u − z)ρ dTγ (az) du = a S(τ ). 0
0
Thus, S(τ ) is self-similar with index 1 + ρ + 1/γ . It follows from the property P[S−1 (t) ≤ τ ] = P[t ≤ S(τ )] that S−1 (t) is self-similar with index 1/(1 + ρ + 1/γ ). Finally, applying the wellknown fact that Lα (t) is 1/α-self-similar, we get that X(t) = Lα (S−1 (t)) is self-similar with index H = 1/α(1 + ρ + 1/γ ). Note that the parameter H can take any value from the infinite interval (0, ∞). Thus, depending on the choice of the parameters, the process X(t) can display scaling, which is characteristic for sub-, super- or normal diffusion [25]. However, the second moment is finite only for α = 2. Then, the mean square displacement yields E[X2 (t)] = K · t1/(1+ρ+1/γ ) , where K = E[X2 (1)] is the generalized diffusion constant. As the parameter 1/(1 + ρ + 1/γ ) takes values in the interval (0, 1/2], the mean square displacement of X(t) displays subdiffusive behaviour.
(b) Exponential kernel Now, we investigate in detail the diffusion limit of the correlated CTRW with exponential kernel M(t). This model was introduced in Chechkin et al. [12]. We show that, in the limit, the exponential
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R(nt) d −→ X(t) = Lα (S−1 (t)) as n → ∞. (Bn )1/α
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Theorem 2.1. Let Ji , i ∈ N, be the sequence of iid α-stable random variables with Fourier transform (2.2). Let Ti , i ∈ N, be the correlated sequence of waiting times defined by (2.3) with ξi given by (2.4) and M(t) regularly varying with index ρ > 0. Assume that the jumps Ji , i ∈ N, and the waiting times Ti , i ∈ N, are independent. Then, the corresponding CTRW process R(t) (2.1) satisfies
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kernel kills the dependence between waiting times. As a result, the scaling limit is the same as for the uncorrelated temporal structure.
exp{−c(i − j + 1)}ξj ,
c > 0,
(2.7)
j=0
where ξi satisfy (2.4) with β = 1. Assume that the jumps Ji , i ∈ N, and the waiting times Ti , i ∈ N, are independent. Then, the corresponding CTRW process R(t) (2.1) satisfies R(nt) d −→ Y(t) = Lα (Sˆ −1 (t)) as n → ∞. nγ /α
(2.8)
ˆ ) is the Here, Lα (t) is the α-stable Lévy motion with Fourier transform E[eikLα (t) ] = e−t|k| . Moreover, S(τ γ -stable subordinator with Fourier transform πγ ˆ sgn(k) , (2.9) E[eikS(τ ) ] = exp −τ (ec − 1)−γ |k|γ 1 − i tan 2 α
ˆ ), i.e. Sˆ −1 (t) = inf{τ > 0 : S(τ ˆ ) > t}. which is independent of Lα (t), whereas Sˆ −1 (t) is the inverse of S(τ
Proof. See appendix F.
The above result confirms that the limit process Y(t) of the correlated CTRW with an exponential kernel is equal (up to a constant) to the limit of the CTRW with uncorrelated waiting times Ti [4,8,26]. This shows that the exponential kernel kills the dependence between waiting times, and the correlation between successive rests of the walker can be observed only in the primary stage of the motion. It should be noted that the probability density function of the scaling limit Y(t) satisfies the following fractional diffusion equation [4]: ∂w(x, t) 1−γ = K0 Dt ∇ α w(x, t). ∂t 1−γ
stands for the fractional derivative of the Riemann–Liouville type, and Here, the operator 0 Dt ∇ α is the Riesz fractional derivative with the Fourier transform F {∇ μ f (x)} = −|k|α f˜(k). K > 0 is the generalized diffusion coefficient. For α = 2, the second moment of Y(t) is finite and given by E[Y2 (t)] = c · tγ , where c = E[Y2 (1)]. This result agrees with the asymptotic subdiffusive behaviour of the mean square displacement derived in Chechkin et al. [12] for large times.
3. Langevin framework (a) Coupled Langevin equations The great advantage of the obtained diffusion limit (2.5) is that it can be used to describe the corresponding correlated dynamics in the framework of coupled Langevin equations. Applying the method of Fogedby [27], we get the following set of coupled Langevin equations for position x and time t of the considered process: s ρ ˙ (3.1) x˙ (s) = Γα (s), t(s) = (s − y) dz(y) and z˙ (s) = Γγ (s). 0
Here, Γα (s) = dLα (s)/ds and Γγ (s) = dTγ (s)/ds are two independent noises. The first equation in (3.1) is the usual Langevin equation in the operational time s. The next two equations describe the relationship between the operational time s and the physical time t. Solving the above system of equations is rather straightforward. First, one solves the first equation to get the driving process
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i
Ti =
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Theorem 2.2. Let Ji , i ∈ N, be the sequence of iid α-stable random variables with Fourier transform (2.2). Let Ti , i ∈ N, be the correlated sequence of waiting times given by
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0
0
0
Finally, we obtain the solution of (3.1) in the subordination form X(t) = x(s(t)) = Lα (s(t)), where s(t) is the inverse of t(s), i.e. s(t) = inf{s > 0 : t(s) > t}. So, we arrive at exactly the same process as in (2.5). To include external force F(x) in the description of correlated dynamics, one needs to add the drift term in the first equation in (3.1). Then, we obtain the following set of equations: s F(x(s)) (3.3) + Γα (s), ˙t(s) = (s − y)ρ dz(y) and z˙ (s) = Γγ (s), x˙ (s) = mη 0 which describes the correlated dynamics in the presence of external force F(x). Here, m is the particle mass and η denotes friction. The solution to the above set of equations is obtained in the same way as (3.1). We get that the solution is equal to X(t) = x(s(t)), where s(t) is the inverse of t(s) given in (3.2) and x(s) satisfies the following stochastic differential equation: dx(s) =
F(x(s)) ds + dLα (s). mη
Based on (3.3), one can efficiently approximate numerically trajectories of X(t). It is enough to simulate the noises Γα (s) and Γγ (s), which can be done by the well-known method of simulating stable processes [20] and plugging the obtained results into (3.3).
(b) Asymptotic properties Now, let us apply previously obtained representations (3.1) and (3.3) to analyse the asymptotic properties of the correlated model. We start with the speed of single-mode relaxation. Assume that F ≡ 0 and denote by w(k, t) = E[exp(ikX(t))] the characteristic function (Fourier transform) of X(t). We have w(k, t) = E[exp(−|k|α s(t))]. Recall that s(t) is the inverse of t(s) given in (3.2). Denote s u a(s) = (u − y)ρ dTγ (y) du and b(s) = sρ+1 sup |Tγ (u)|. 0 0
u∈[0,s]
Then we have a(s) ≤ t(s) ≤ b(s) or, equivalently, exp(−|k|α a−1 (t)) ≤ exp(−|k|α s(t)) ≤ exp(−|k|α b−1 (t)). Here, a−1 (t) and b−1 (t) are the inverses of a(s) and b(s), respectively. Now, as both processes a(s) and b(s) are heavy-tailed with index γ and self-similar with index 1 + ρ + 1/γ , applying the above inequalities and Tauberian theorems, we get that w(k, t) 1/tγ for large t. The notation w(k, t) 1/tγ means that there exist positive constants c1 and c2 , such that c1 t−γ < w(k, t) < c2 t−γ for large t. Thus, the correlated model displays power-law single-mode relaxation. In what follows, we will analyse properties which require the existence of moments of the process. Therefore, from now until the end of the paper, we assume that α = 2. This means that the process Lα=2 (t) is just the standard Brownian motion. We will denote it by B(t). Thus, its moments of any order exist. In particular, E[B2 (t)] = 2Dt, where D > 0 is the diffusion constant. Consequently, the second moment of the correlated model yields E[X2 (t)] = E[B2 (s(t))] = 2DE[s(t)] = 2DDγ t1/(1+ρ+1/γ ) ,
(3.4)
which is typical for subdiffusion. Here, Dγ = E[s(1)] depends on the distribution of the correlated waiting times. It is natural to ask about the stationary solution of (3.3). As the subordinator s(t) tends to infinity as t → ∞, we obtain that the stationary solution of (3.3) has the well-known form wst (x) ∝ exp(−V(x)/Dmη). Here, V(x) = − F(x) dx denotes the external potential. Comparing
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0
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x(s) = Lα (s) acting in the operational time s. In the next step, one solves the second and third equations to get z(s) = Tγ (s) and s u s u (3.2) t(s) = (u − y)ρ dz(y) du = (u − y)ρ dTγ (y) du.
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the above expression with the classical Gibbs–Boltzmann equilibrium distribution weq ∝ exp(−V(x)/[kB T]), we obtain that the correlated model satisfies the generalized Einstein relation (3.5)
which connects the diffusion parameter D with the dissipative parameter η and the energy kB T. Further on, if we assume that the force is constant F ≡ F0 , we obtain the following expression for the first moment of the correlated process X(t): EF0 [X(t)] =
F0 F0 F0 E[s(t)] + E[B(s(t))] = E[s(t)] = Dγ t1/(1+ρ+1/γ ) . mη mη mη
Comparing the above expression with the second moment in the force-free case (3.4), we arrive at the second Einstein relation EF0 [X(t)] = (F0 /2)(E[X2 (t)]/kB T). Similar to other models of subdiffusion based on CTRWs, the studied process displays weak ergodicity breaking [28–33]. To see this, let us analyse the time-averaged mean square displacement t− 1 δ 2 (, t) = [X(τ + ) − X(τ )]2 dτ . (3.6) t− 0 Here, is the lag time and t represents the total length of the observation. For standard diffusion processes, δ 2 (, t) scales as 2D for large t. However, after some standard calculations, we get the following expression for the correlated process: E[δ 2 (, t)] ∼
2DDγ t1−1/(1+ρ+1/γ )
for t. Thus, the correlated process displays weak ergodicity breaking. Another feature of anomalous dynamics displayed by CTRW models is ageing [34,35], manifested by the temporal decay of the response of the process to a sinusoidal, time-dependent force. Applying the subordination method to the correlated process under the influence of timedependent force F(t) = F0 sin(ωt) (see [17,34] for details), we get that the corresponding first moment of X(t) has the form F0 t E[X(t)] = sin(ωu) dE[s(u)] mη 0
t F0 Dγ 1 u1/(1+ρ+1/γ )−1 sin(ωu) du. = mη 1 + ρ + 1/γ 0 Because of the factor u1/(1+ρ+1/γ )−1 inside the integral, the first moment decays as time proceeds and we observe the so-called ‘death of linear response’ [34].
4. Summary In this paper, we have introduced the CTRW process with correlated temporal structure. Consecutive waiting times were defined as weighted sums of independent random variables combined with a reflecting boundary condition. The weights, in turn, were determined by the regularly varying memory function. Our model generalizes two different correlated models introduced in Chechkin et al. [12] and Tejedor & Metzler [13]. We have analysed various asymptotic properties of the studied process. We have derived the corresponding diffusion limit and proved its subdiffusive character. It should be emphasized that the functional limit theorems proved in the appendices apply to the class of correlated processes, which is much wider than the CTRWs studied in the main part of the paper. Applying the subordination scheme we have extended the correlated model to include external forces. The derived coupled Langevin equations have been further used to analyse Einstein relations, speed of relaxation, equilibrium distribution, ergodicity breaking and ageing properties. We have separately analysed the case of exponential memory function. We have shown that in such a setting the exponential kernel kills the dependence between waiting times and the
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kB T , mη
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D=
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Appendix The structure of this appendix is as follows. Necessary definitions and notations are gathered in appendix A. In appendix B, we define two useful mappings and show when they are continuous. Next, in appendix C, we formulate two general assumptions on the sequence of jumps Jk and the sequence of innovations ξk and the function M defining the correlated time structure of the CTRW process. We also discuss in what situations these general assumptions are satisfied. Results from appendices B and C will be used in appendix D, in which we prove the general result dealing with the weak convergence of the sequence of scaled CTRW processes. In appendix E, we give the proof of theorem 2.1, which is the conclusion from appendix D. In appendix F, we prove theorem 2.2.
Appendix A. Definitions and notations Let D denote the space of functions mapping [0, ∞) into R that are right-continuous having limits from the left (cádlág functions). We equip this space with the Skorohod J1 topology. We define also two subspaces of D; namely, D↑ being the space of all cádlág functions starting from 0 that are non-negative, non-decreasing and unbounded from above, and the space Dv containing all cádlág functions of bounded variation on any interval [0, r], r > 0. Let M and Ms denote the space of Radon measures on [0, ∞) and the space of signed Radon measures on [0, ∞), respectively. For any function y ∈ D↑ there exists measure μy ∈ M such that μy ((a, b]) = y(b) − y(a) for any interval (a, b], 0 ≤ a < b. Any function z ∈ Dv may be written in the form z = z+ − z− , where z+ , z− ∈ D↑ . Hence, to any z ∈ Dv corresponds the measure μz ∈ Ms defined as μz = μz+ − μz− . r In the following appendices, we write 0 f (x)μy (dx) to denote the Lebesgue integral of f r with respect to the Radon measure μy , y ∈ D↑ , while 0 f (x) dy(x) is to be understood in the Riemann–Stieltjes sense with f and y being the integrand and the integrator, respectively. For r r r r z ∈ Dv , we write 0 f (x)μz (dx) = 0 f (x)μz+ (dx) − 0 f (x)μz− (dx). Recall that 0 f (x) dz(x) exists if f is continuous on [0, r], and then r r r + f (x) dz(x) = f (x) dz (x) − f (x) dz− (x) 0
0
=
r 0
0
f (x)μz+ (dx) −
r 0
f (x)μz− (dx) =
r 0
f (x)μz (dx).
Given functions yn , y ∈ D↑ , we say that sequence {μyn } of measures from M converges vaguely ∞ ∞ v to some measure μy ∈ M, which we denote as μyn → μy , if 0 g(u)μyn (du) → 0 g(u)μy (du) for any function g : [0, ∞) → [0, ∞), which is continuous and has compact support. It is equivalent to convergence yn (t) → y(t) for all t being continuity points of y. Similarly, given functions zn , v z ∈ Dv , we say that measures μzn ∈ Ms converge vaguely to the measure μz ∈ Ms if μz+n → μz+ v
and μz−n → μz− .
Appendix B. Continuity of two useful mappings For any x ∈ D and y ∈ D↑ , we define mapping as t def h(x, y)(t) = x(t − u)μy (du), 0
t ≥ 0.
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Funding statement. The research of M.M. was partially supported by an NCN Maestro grant.
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correlation between successive rests of the walker can be observed only in the primary stage of the motion. However, the diffusion limit is the same as for the renewal CTRW with independent waiting times.
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as n → ∞.
Proof. See the electronic supplementary material. For any x ∈ D, z ∈ Dv and κ ∈ M we define mapping t s def f (x, y, κ)(t) = x(s − u)μz (du) κ(ds), 0
0
t ≥ 0.
− Lemma B.2. Let xn , x be as in lemma B.1 and assume that sequence {(zn , z+ n , zn )} ⊂ Dv × D↑ × D↑ + + − + converges to (z, z , z ) ∈ Dv × D↑ × D↑ in the Skorohod J1 topology, where zn = zn − z− n and z = z − v − z . Moreover, let κn , κ ∈ M, such that κn → κ. Then for any r > 0
sup |f (xn , zn , κn )(t) − f (x, z, κ)(t)| → 0 as n → ∞. 0≤t≤r
Proof. See the electronic supplementary material.
Appendix C. General assumptions on a continuous-time random walk with correlated temporal structure The construction of a CTRW with correlated temporal structure is based on the sequence of random vectors {(Ji , ξi )}i≥1 defined on the common probability space (Ω, F , P) and the function M. Consecutive jumps of the walker are given by the sequence {Ji }, while waiting times between the jumps are of the form Ti = | ij=1 M(i − j + 1)ξj |. We define the counting process N(t) = max{k ≥ N(t) 0 : T1 + · · · + Tk ≤ t} and the CTRW R(t) = i=1 Ji . Below we state assumptions (A1) and (A2) on the sequence {(Ji , ξi )} and the function M, respectively. Under these assumptions, we will be able to obtain asymptotic behaviour of the sequence of scaled CTRWs with correlated temporal structure. (A1) There exist sequences {an } and {bn } such that the sequence of processes of scaled partial sums (Ln (t), Ξn (t), Ξn+ (t), Ξn− (t)) ⎛ ⎞ [nt] [nt] [nt] [nt] def ⎝ −1 = an Ji , b−1 ξi , b−1 ξi 1(ξi > 0), −b−1 ξi 1(ξi < 0)⎠ n n n i=1
i=1
i=1
i=1
converges weakly in the Skorohod J1 topology to the process (L(t), Ξ (t), Ξ + (t), Ξ − (t)) such that Ξ = Ξ + − Ξ − and the trajectories of processes Ξ + and Ξ − are the elements of D↑ , almost surely. (A2) There exists a sequence {cn } and a continuous function m mapping [0, ∞) into R, such that ∀r > 0 sup |Mn (t) − m(t)| → 0 as n → ∞, 0≤t≤r
where
def Mn (t) = c−1 n M(nt + 1),
t ≥ 0.
Before proceeding to investigate the asymptotic of the CTRW with correlated temporal structure (see appendix D), we discuss the situations in which the above assumptions hold true. The following lemma, which is an extension of [36, theorem 7.1], is a useful tool for checking whether (A1) holds.
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sup |h(xn , yn )(t) − h(x, y)(t)| → 0 0≤t≤r
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Lemma B.1. Assume that the sequence of functions xn ∈ D, n ≥ 1, converges to a continuous function x uniformly on any interval [0, r], r > 0, and let the sequence yn ∈ D↑ , n ≥ 1, converge to y ∈ D↑ in the Skorohod J1 topology. Then for any r > 0
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Lemma C.1. Let {Xn,j ≡ (X1,n,j , X2,n,j )}n,j≥1 be the array of random vectors such that {Xn,j }j≥1 is the iid sequence for any n ≥ 1. Assume that
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(i)
v
nP(Xn,1 ∈ ·) → ν0 (·) as n → ∞, where ν0 is a Lévy measure on R2 \ {0}; (ii) P(X2,n,1 > 0) → p and P(X2,n,1 < 0) → q = 1 − p
as n → ∞;
(iii) 2 lim lim sup nE(Xi,n,1 1(|Xi,n,1 | ≤ ε)) = 0,
ε0 n→∞
i = 1, 2.
Let {Yn,j ≡ (Y1,n,j , Y2,n,j , Y3,n,j , Y4,n,j )}n,j≥1 be the array of random vectors, where def
(Y1,n,j , Y2,n,j , Y3,n,j , Y4,n,j ) = (X1,n,j , X2,n,j , X2,n,j 1(X2,n,k > 0), −X2,n,j 1(X2,n,k < 0)). Define the sequence of partial sum processes Yn ≡ (Y1,n , Y2,n , Y3,n , Y4,n ) [nt] def
Yn (t) =
(Yn,k − E(Yn,k 1(Yn,k ∞ ≤ 1))),
k=1
where x∞ = maxi=1,2,3,4 |xi |, x ∈ R4 . Then (a) Yn ⇒ Y in the Skorohod J1 topology, where Y ≡ (Y1 , Y2 , Y3 , Y4 ) is the Lévy process with the Fourier transform k,Y(t) ik,x Ee = exp t (e − 1 − ik, x1(0 < x∞ ≤ 1))ν(dx) , k ∈ R4 , x=0
where Lévy measure ν is defined as def
ν(dx) = pν0 (dx1 × dx2 )δx2 (dx3 )δ0 (dx4 ) + qν0 (dx1 × dx2 )δ0 (dx3 )δ(−x2 ) (dx4 ); (b) processes Y3 and Y4 are independent and Y2 = Y3 − Y4 .
Proof. See the electronic supplementary material.
Remark C.2. One can prove results similar to [36, theorem 7.1] and lemma C.1 in situations where {Xn,k } is the array of dependent random vectors and the first coordinate of the limit process Y has a gaussian component. For more details, see [37,38]. Assumption (A2) is also quite weak. In particular, it is satisfied for functions from a broad family of regularly varying functions with index ρ > 0 (see appendix E). Assumption (A2) also holds when Mn (t) → m(t) for every t ≥ 0 and functions Mn are non-decreasing.
Appendix D. Asymptotic of a continuous-time random walk with correlated temporal structure Theorem D.1. Let the assumptions (A1) and (A2) be satisfied with sequences {bn } and {cn } such that −1 in the nbn cn → ∞ as n → ∞. Define the sequence of processes Rn (t) = a−1 n R(nbn cn t). Then Rn ⇒ L ◦ S t s Skorohod J1 topology, where S(t) = 0 | 0 m(s − y) dΞ (y)| ds. def
Proof. See the electronic supplementary material.
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=
t
s
0
0
m(s − y) dΞ (y) ds =
t t−y 0
t
t
0
y
m(s − y) ds dΞ (y)
m(u) du dΞ (y).
0
Furthermore, one can easily show that ∀t ≥ 0
t [nt] 1 def M(i) → m(u) du = g(t) as n → ∞ ncn 0 i=1
and then S(t) =
t
0 g(t − y) dΞ (y).
Appendix E. Proof of theorem 2.1 Let sequences {Jk } and {ξk } and function M be as in §2a. We will obtain theorem 2.1 as a conclusion from theorem D.1. Proof of theorem 2.1. In order to use theorem D.1, we need to check if its assumptions are satisfied, i.e. if sequence {(Jk , ξk )} fulfils condition (A1) and the regularly varying function M satisfies condition (A2). We begin with checking condition (A1). The main tool we use to verify that this condition holds will be lemma C.1; thus, we need to check its assumptions first. Before doing so, recall that {Jk } is the iid sequence with Fourier transform given by (2.2). As shown in [39, §1.2.6] we D
have that Jk ∼ Sα (1, 0, 0), n−1/α (J1 + · · · + Jn ) = J, where J ∼ Sα (1, 0, 0) is independent of {Jk } and the distribution of Jk is infinitely divisible with Lévy measure 1 −1−α x dx, x > 0, def να (dx) = 21 −1−α dx, x < 0. |x| 2 v
Then, we also have that nP(n−1/α J1 ∈ ·) → να (·) as n → ∞ (e.g. [40, theorem 3.2.2]). Similarly, the assumption that ξk have Fourier transform (2.4) implies that ξk ∼ Sγ (1, β, 0), D
n−1/α (ξ1 + · · · + ξn ) = ξ , where ξ ∼ Sγ (1, β, 0) is independent of {ξk }, the distribution of ξk is infinitely divisible with Lévy measure ⎧ 1 + β −1−γ ⎪ ⎨ dx, x > 0, x def 2 νγ (dx) = ⎪ ⎩ 1 − β |x|−1−γ dx, x < 0 2 v
and that nP(n−1/γ ξ1 ∈ ·) → νγ (·) as n → ∞. Now, we define the array of random vectors {Xn,k } ≡ {(n−1/α Jk , n−1/γ ξk )} and check that it fulfils the assumptions of lemma C.1. Take arbitrary A, B ∈ B(R) such that {(0, 0)} ∈ / A × B. Then, as sequences {Jk } and {ξk } are assumed to be independent, we have that nP(Xn,1 ∈ A × B) = nP(n−1/α J1 ∈ A)P(n−1/γ ξ1 ∈ B) να (A)1B (0), if 0 ∈ B, → νγ (B)1A (0), if 0 ∈ A. Hence condition (i) holds with ν0 (dx × dy) ≡ να (dx)δ0 (dy) + δ0 (dx)νγ (dy). Assumption (ii) is obviously satisfied with p = P(ξ1 > 0) and q = P(ξ1 < 0).
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S(t) =
11
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Remark D.2. Let the assumptions of theorem D.1 be satisfied. If we additionally assume that ξi > 0 almost surely for all i ≥ 1 and that function M is non-negative, then
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Next observe that for arbitrary small ε > 0
nE((n−1/γ ξ1 )2 1(|n−1/γ ξ1 | ≤ ε)) →
0 0 : ⎩ ⎭ ⎫ ⎬
⎧ ⎨
i=1
⎞ k 1⎝ 1 1 = Ti ≤ n1/γ t + 1⎠ = N(n1/γ t) + . max k ≥ 0 : ⎭ ⎩ n n n ⎛
i=1
We define the auxiliary sequence of CTRW processes as
1/γ t) N(n1/γ t) 1 def R(n −1 = L (t) − = L , Rn (t) = S n n n n n n−1/α
t ≥ 0.
By [36, corollary 7.1], it follows that Ln ⇒ Lα and that sequences of processes Ln , Sn are ˆ in the Skorohod independent. This, together with (F 3), implies joint convergence (Ln , Sn ) ⇒ (Lα , S) J1 topology. Then from [41, theorem 3.6], it follows that Rn ⇒ Lα ◦ Sˆ−1 in the Skorohod J1 topology.
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= (ec − 1)−1 Ξn (t) − (ec − 1)−1 Zn (t),
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j=1
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We conclude the proof of theorem 2.2 with the observation that R(nt)/nγ /α = Rnγ (t). As nγ → ∞ with n → ∞, n−γ /α R(nt) has the same limit as sequence Rn and convergence (2.8) follows.
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