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Anomalous diffusion and semimartingales

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2009 EPL 86 60010

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EPL, 86 (2009) 60010 www.epljournal.org

doi: 10.1209/0295-5075/86/60010

Anomalous diffusion and semimartingales

A. Weron and M. Magdziarz(a)

Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of TechnologyWyspianskiego 27, 50-370 Wroclaw, Poland, EU

received 27 February 2009; accepted in final form 5 June 2009published online 10 July 2009

PACS 05.40.Fb – Random walks and Levy flightsPACS 02.50.Ey – Stochastic processesPACS 05.10.-a – Computational methods in statistical physics and nonlinear dynamics

Abstract – We argue that the essential part of the currently explored models of anomalous(non-Brownian) diffusion are actually Brownian motion subordinated by the appropriate randomtime. Thus, in many cases, anomalous diffusion can be embedded in Brownian diffusion. Suchan embedding takes place if and only if the anomalous diffusion is a semimartingale process. Wealso discuss the structure of anomalous diffusion models. Categorization of semimartingales canbe applied to differentiate among various anomalous processes. In particular, identification of thetype of subdiffusive dynamics from experimental data is feasible.

Copyright c© EPLA, 2009

Introduction. – The beginning of the long-lastingmarriage of Brownian motion with physics dates backto two breakthrough papers [1,2], in which both authorsEinstein and Smoluchowski explained independently thephenomenon of Brownian motion as a result of collisionsbetween the suspended particles and the molecules of thesurrounding fluid. A different and innovative approachwas proposed by Langevin [3]. Even earlier, in 1900,Bachelier [4] bound up Brownian motion with financialmodeling, by introducing the first model of stock price atthe Paris Bourse.However, in spite of many obvious advantages, models

based on Brownian motion fail to provide satisfactorydescription of many dynamical processes [5–7]. Thedetailed empirical analysis of various complex systemsshows that some of the significant properties of suchsystems cannot be captured by the Brownian diffusionmodels. One should mention here such properties as:nonlinear in time mean-squared displacement, long-rangecorrelations, nonexponential relaxation, heavy-tailed andskewed marginal distributions, lack of scale invariance,discontinuity of the trajectories, and many others [8]. Tocapture such anomalous properties of physical systems,some different mathematical models need to be intro-duced. In recent years one observes a rapid evolutionin this direction, which results in emerging of variousalternative models, such as: continuous-time randomwalks [9], fractional kinetic equations [8,10,11], fractional

(a)E-mail: marcin.magdziarz@pwr.wroc.pl

Brownian motions [12,13], generalized Langevin equa-tions [14], jump-diffusion models [15], subordinatedLangevin equations [16,17], etc.The increasing accuracy of the empirical measurements

reveals different anomalous properties of the systems,which cannot be satisfactorily described by the modelsbased on Brownian diffusion. As we will show in thefollowing, Brownian motion is doing well also in theworld of anomalous (non-Brownian) diffusion [18]. It turnsout that the large part of the currently explored modelsof anomalous transport can be represented as Brownianmotion but in the appropriate internal clock of the system.

Semimartingales. – We will start with an observa-tion that anomalous diffusion processes are in a veryclose relation with the so-called semimartingales. Theconcept of semimartingales became popular in the 1970s,see the basic papers by Follmer [19], Bichteler [20,21],and Dellacherie [22]. However, it was introduced earlierby Meyer [23]. The stochastic integration with respect tosemimartingales was developed by him and his collabo-rators. Since then their mathematical theory attracted aconsiderable attention and has found already importantapplications to asset pricing in economy [15]. A semi-martingale Xt uniquely decomposes as

Xt =Mt+At, (1)

where Mt is a local martingale (its increment processis a fair-game, meaning that the expectation value ofincrements is zero), and At is a finite variation process,

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A. Weron and M. Magdziarz

which can be considered as the drift part of Xt [24]. Thus,the most relevant subclass of semimartingales constitutemartingales. The concept of a martingale is the stochasticanalog of a conserved quantity. A process Mt is said tobe a martingale if it has the property that the (future)conditional mean value is equal to the last value specifiedby the condition, i.e.

〈Mt|Mtn , . . . ,Mt1〉=Mtn , t > tn > . . . > t1.

In other words, Mt is a martingale if its conditionalexpectation, given information up to time tn, is the valueof the process at that time. Thus, in a certain sense,martingales are the stochastic analog of conservationlaws: expectation is conserved. The concept of martingalecan also be used effectively to characterize the reductiveproperties of energy-based stochastic extension of theSchrodinger equation [25]. For the martingale descriptionof physical fluctuations see [26]. Also, the time operator inthe Prigogine theory of irreversibility can be constructedas an integral with respect to martingales [27]. Themartingale property has found widespread applications infinance and game theory, since it formalizes the conceptof the fairness of the game.Let us mention some well-known examples of martin-

gales: the position process of the classical random walker,being the the sum of fair-game increments, is a martingale.Similarly, Brownian diffusion without drift is a martingale.However, the Brownian diffusion with a drift

dXt = σ(Xt, t) dBt+µ(Xt, t) dt

is only a semimartingale illustrating the decomposition(1). Many significant processes, including Brownianmotion, Poisson process, Levy processes, are semimartin-gales. Nevertheless, it is not always the case. An exampleof the process, which does not belong to this class isthe fractional Brownian motion [12,13], as well as thefractional α-stable motion [28]

∫ ∞−∞

((t−u)H−1/α+ − (−u)H−1/α+

)dLα(u).

Semimartingales are the most general reasonable stochas-tic integrators, for which the pathwise evaluation ofstochastic Ito-type integrals is possible [20,21]. The classof semimartingales is invariant under smooth transforma-tion (Ito lemma), random change of time and Ito stochas-tic integration. Thanks to these plausible properties,semimartingales have found widespread applications andplay a fundamental role in stochastic modelling includingfor example modern modelling of financial data [15,29].We will demonstrate in this letter that, the role of semi-

martingales in the description of anomalous diffusion issignificant as well. The large part of well-known anomalousdiffusion processes can be represented as time-changedBrownian motion. Thus, by employing the I. Monroeresult [30], every time-changed Brownian motion is a

semimartingale. In other words, we obtain the followinginterpretation of Monroe’s result in the context of anom-alous dynamics —the Embedding Principle for anom-alous diffusion: Anomalous diffusion process can be repre-sented as time-changed Brownian motion if and only ifit is a semimartingale. The above statement confirmsthe intimate relation between semimartingales and anom-alous transport. Since the mathematical theory of semi-martingales is well developed, it can help to identify andunderstand better various properties of anomalous diffu-sions. The quantity called the quadratic variation is ofparticular interest from the point of view of applicationsof semimartingales in physics. Let X(t) be a stochas-tic process observed on time interval [0, T ]. Then, fort∈ [0, T ], the quadratic variation V (2)(t) corresponding toX(t) is defined as

V (2)(t) = limn→∞V

(2)n (t), (2)

where V(2)n (t) is the partial sum of the squares of incre-

ments of the process X(t) given by

V (2)n (t) =

2n−1∑j=0

∣∣∣∣X((j+1)T

2n∧ t)−X

(jT

2n∧ t)∣∣∣∣2

(3)

with a∧ b=min{a, b}. The crucial property of the trajec-tories of semimartingales is the fact that their quadraticvariation is always finite. This is not the case for anystochastic process in general. For example, quadratic vari-ation of fractional Brownian motion (FBM) BH(t) withHurst index H less than 1/2, is infinite. This follows fromthe fact that FBM is H–self-similar and its incrementssatisfy (BH(t+h)−BH(t))2 ∼ h2HB2H(1). Since 2H < 1,the sum of the squares of the increments in (3) divergesto infinity as n→∞. Similar reasoning can be appliedto diffusion on fractal, which is yet another example ofprocess with infinite quadratic variation and does notbelong to the class of semimartingales. The dissimilaritiesin the behavior of quadratic variation provide a simple wayhow to distinguish between different kinds of subdiffusion(CTRW, FBM, diffusion in confined media) from experi-mental data [31]. One only needs to examine the behaviorof the quadratic variation of a given trajectory. Detectionof the type of anomaly is an important and timely prob-lem, see for example [32–34] for discussion on the originsof anomaly in the case of intracellular diffusion.Thus, the information that a given physical process

belongs to the family of semimartingales, gives an impor-tant additional gain in understanding the behavior of thetrajectories. This can be used to identify the type of anom-aly from experimental data [31]. Moreover, representationof a given stochastic process as time-changed Brownianmotion allows to run efficient Monte Carlo simulationsof its trajectories and to perform numerical analysis ofsample paths [35].Next, to illustrate the embedding principle, we will

consider in details several physically relevant special cases

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Anomalous diffusion and semimartingales

0 25 500

25

50

t

Uα(t

) , S

α(t)

Sα(t)

Uα(t)

Fig. 1: (Colour on-line) Correspondence between the processesUα(t) and its inverse Sα(t). They are obtained by reflectionwrt the diagonal of the square. The trajectories of the firstprocess are superlinear, whereas the behavior of the secondone is sublinear. For every jump of Uα(t) we observe thecorresponding flat period of Sα(t). Here α= 0.7.

of anomalous diffusions and for each case we will identifyan appropriate subordinator.

Sub- and superdiffusion. – In the large family ofanomalous diffusion processes, two distinct classes are ofspecial importance. Namely, subdiffusion and superdif-fusion. The first class is characterized by the sublin-ear in time mean-squared displacement. Its presence wasreported in condensed phases [8], ecology [36], biology [33],and many more. The superdiffusive regime is characterizedby the superlinear in time or even infinite (Levy flights)mean-squared displacement. Superdiffusion is observed inthe transport of micelle systems, bacterial motion, quan-tum optics (see [8] and references therein) and even in thetransport of light in certain optical materials [37].Clearly, the classical Brownian motion B(t) is not

suitable for modelling of sub- and super-diffusion, sinceits fundamental feature is the linearity of the mean-squared displacement. Nevertheless, by introducing theappropriate random time (α-stable subordinator) to thesystem, one is able to transform the standard diffusionB(t) into anomalous one. Let us denote by Uα(t) thestrictly increasing α-stable Levy process (subordinator)with the well-known formula for its Laplace transform〈e−kUα(t)〉= e−tkα , 0<α< 1 [38,39]. By

Sα(t) = inf{τ : Uα(τ)> t}we denote the left inverse of Uα(t). We call Sα(t)the inverse α-stable subordinator. The correspondencebetween Uα and Sα can be observed in fig. 1. Namely,Sα(t) is obtained from Uα(t) by reflection wrt to a diag-onal of the square. Clearly, the trajectories of Uα(t) aresuperlinear, whereas the behavior of Sα(t) is sublinear.Now, randomizing the time of the Brownian motion

B(t) by using the independent random process Uα(t), we

0 25 50−2

0

2

4

t

B(S

α(t))

0 25 50−2

−1

0

1

t

B(U

α(t))

Fig. 2: (Colour on-line) In the top panel a trajectory ofthe process B(Uα(t)) is shown. The jumps of the process,which are inflicted by the subordinator Uα(t), are typical forsuperdiffusion. In the bottom panel a sample path of theprocess B(Sα(t)) is visualized. We observe the characteristicconstant periods of the trajectory, which correspond to theconstant periods of Sα(t). The process is subdiffusive. Hereα= 0.7.

obtain the new process

X(t) =B(Uα(t)). (4)

Such an operation is called subordination and it was firstintroduced by Bochner [40]. The process B, called theparent process, is directed by the new operational timeclock Uα called subordinator. It is easy to verify thatthe process X(t) is superdiffusive. Namely, its Fouriertransform is of the form 〈eikX(t)〉= e−tkµ , where µ= 2α.The trajectories of X(t) are discontinuous, see fig. 2.We can observe long jumps, which are characteristic forLevy flights. Having in mind that the trajectories of theclassical Brownian particle are continuous, we infer thatthe superdiffusive character of X(t) is inflicted by thesubordinator Uα(t). Moreover, for the second momentof X(t) we have 〈X2(t)〉=∞, which is typical for Levyflights.

Fractional Fokker-Planck equations. – The proba-bility density function (PDF) ofX(t) satisfies the superdif-fusive fractional Fokker-Planck equation (FFPE) [8]

∂p(x, t)

∂t=∂µp(x, t)

∂xµ,

where the operator ∂dµ

∂xµis the Riesz fractional derivative.

Thus, by the time change t→Uα(t), Brownian motion istransformed into superdiffusion.By the similar subordination method, we are able to

transform B(t) into subdiffusion. Let us consider thefollowing time-changed process Y (t) =B(Sα(t)). Here,Sα(t) is the previously introduced inverse α-stable subor-dinator. Now, the process Y (t) is subdiffusive. Its secondmoment is given by 〈Y 2(t)〉 ∼ tα, 0<α< 1, thus it is

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A. Weron and M. Magdziarz

sublinear. The PDF of Y (t) obeys the subdiffusive frac-tional diffusion equation [8,41]

∂p(x, t)

∂t= 0D

1−αt

∂2p(x, t)

∂x2, (5)

where the operator 0D1−αt is the Riemann-Liouville frac-

tional derivative. The trajectories of Y (t) are continuouswith the characteristic flat periods, fig. 2. These constantperiods are typical for subdiffusion and represent the timeintervals in which the particle gets immobilized in thetrap. Comparing figs. 1 and 2 we observe the clear corre-spondence between the flat periods of Sα(t) and Y (t).Thus, we conclude that the process Sα(t) is responsible forgiving the effect of subdiffusion. Note that for α→ 1 wehave that

Uα(t) = Sα(t) = t,

which shows that the standard diffusion is placed exactlyon the border between sub and superdiffusion and whenthe operational time coincides with the real time.It is worth mentioning that the subordination is

reversible in the following sense: we have that

B(Sα(Uα(t))) =B(t).

Thus, the double subordination recovers the classicalBrownian diffusion.The first reasonable usage of subordination in physics

dates back to [42], see also [43,44]. The time since,physicists developed a considerable intuition on subor-dination [45–47]. It should be noted that subordinationin the sense of integral transformation of PDFs, whenboth processes (parent process and subordinator) areindependent, is not enough for some purposes (see [45]for the calculation of first-passage times). This is relatedto the fact that in some cases PDF’s do not give thecomplete description of the underlying process. However,subordination represented as the superposition of twostochastic processes gives such full characterization ofthe process. The discussion on the duality of subdiffusionand Levy flights in the framework of Continuous-TimeRandom walks (CTRW) and propagators for the casewhen subordinator is independent of Brownian motion,can be found in [46,47]. The authors of the above papershave demonstrated that, under assumption that theinitial ticks of the physical clock follow extremely inho-mogeneously and show intervals of high condensation,operational time leads to Levy flights. This correspondsto the subordination (4). If the ticks triggering themotion of the random walker are taken from the setof ticks of a physical clock according to a power lawwaiting-time distribution, such random walk leads to thesubdiffusive dynamics and FFPE (5). This correspondsto the subordination procedure of the form B(Sα(t)).

External force fields. – Many physical transportproblems take place under the influence of external fields.

A typical framework for the treatment of anomalousdiffusion problems under the influence of an external forceF (x) is in terms of the FFPE [8]

∂p(x, t)

∂t=

[−∂F (x)

∂x+

∂µ

∂xµ

]0D1−αt p(x, t).

The operator ∂µ

∂xµintroduces the Levy-flight–type behav-

ior, whereas 0D1−αt causes the memory effects typical for

subdiffusion. The analysis of the corresponding Langevinequation [35,48] shows that also in the presence of exter-nal force F (x), the underlying process can be repre-sented as time-changed Brownian motion B(T (t)) forsome appropriate subordinator T (t). As an example, letus consider the subdiffusive Ornstein-Uhlenbeck processdefined by the above FFPE with F (x) =−x and µ= 2.Then, Y (t) can be represented as B(T (t)), where T (t) =12 (1− e−2Sα(t)). Concluding, there always exists an appro-priate T (t) such that the time-changed Brownian motionB(T (t)) describes sub or superdiffusion in the presence ofexternal force. In general, the processes B and T do nothave to be independent. Let us underline that the samearguments as above can be applied also for the case whenthe force is time-dependent F = F (t) [17].

Jump-diffusions. – Another significant class of anom-alous diffusions constitute non-Gaussian Levy processes,also known as jump-diffusion models due to their discon-tinuous trajectories. It is one of the most important andfundamental classes of random processes characterizedby stationarity and independence of increments. Specialexamples include the Poisson, Compound Poisson, andstable Levy motions. Since their introduction in the 1930s,Levy processes were studied extensively by many mathe-maticians [49]. However, in recent years they attractedconsiderable attention of physical community, too [50].Continuity of paths has always played a crucial rolein the properties of classical diffusion models. Never-theless, recent developments confirm that many trans-port phenomena (i.e. bacterial motion [6], transport oflight in certain optical materials [37], stock prices [15])display a discontinuous behavior. In such cases jump-diffusion models rather than Brownian motion provide theproper background to describe and understand discontinu-ous transport. Every Levy process L(t) can be fully charac-terized by its Fourier transform 〈eikL(t)〉= e−tψ(k), wherethe function ψ(k) is the so-called Levy exponent. However,the most important fact here is that for each Levy processL there exists a subordinator T (t) such that L(t) can berepresented as B(T (t)) [15]. For example, to obtain theLinnik geometric stable process [51] one has to chooseT (t) =Uα(Γt), where Γt is the Gamma subordinator [49].In general case, the assumption about the independenceof subordinator and Brownian motion has to be relaxed.Concluding, the class of Levy processes, which belongs tothe family of anomalous diffusions, can also be describedas time-changed Brownian motion [7].

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Anomalous diffusion and semimartingales

Moreover, every standard CTRW process possesses therepresentation of the form B(T (t)) as well. It is notsurprising, since the family of CTRWs is closely relatedto the class of Levy processes, i.e. every Levy process canbe obtained as a limit of the appropriate CTRW [49].

Relaxation phenomena. – In recent years, the ques-tion of understanding the phenomenology of relaxationprocesses in various materials has attracted considerableattention [52–55]. It is surprising that in spite of thevariety of materials and experimental techniques, thebehavior of relaxing systems is very similar and canbe described in terms of simple relaxation functions.Therefore, it is expected that there exists a commonnature of microscopic relaxation mechanisms with thecorresponding mathematical structure. It turns out thatan unified mathematical structure can be provided in theframework of time-changed Brownian motion.Let us denote by R(t) the distance reached by the parti-

cle after time t. Then, the corresponding time-domainrelaxation function Φ(t), describing the system’s survivalprobability in the initial state, is defined as the Fouriertransform of R(t), i.e. Φ(t) = 〈eikR(t)〉. Now, by the appro-priate choice of R(t), we obtain different relaxation func-tions. For R(t) =B(Sα(t)), where Sα(t) is the previ-ously introduced inverse α-stable subordinator, we getthe Mittag-Leffler relaxation function. Next, for R(t) =B(T (t)) with T (t) = tUα(1) and Uα being the α-stablesubordinator, we win the stretched exponential relaxationfunction. Moreover, subordinating the Brownian motionby the processes obtained as the limit of certain CTRWsleads to the Havriliak-Negami relaxation in [54], the gener-alized Mittag-Leffler relaxation in [53], and the Cole-Davidson relaxation in [55] by employing the temperedstable distributions.As one can observe, randomization of the time of Brown-

ian motion using the appropriate subordinator, providesa unified mathematical description of all the most rele-vant relaxation patterns. Different choice of subordinatorresults in different behavior of the anomalously relaxingsystems. It suggests that complex systems can be identi-fied and characterized by their own internal operationaltime. The determination of this operational time seemsto be the key factor in understanding and describing ofrelaxation phenomena in complex systems.

Conclusions. – In this letter we have categorized theanomalous diffusion models (including: sub- and super-diffusion, CTRWs, and jump-diffusions) representedas time-changed Brownian motion. We have indicatedthe close and promising link between statistical physics(anomalous transport) and the theory of semimartingales.One should note at this point that not every anomalousdiffusion process can be represented as time-changedBrownian motion. Such counterexamples are FBM andrandom walk on fractals. Both models do not belongto the family of semimartingales. The fact that a givenphysical process is a semimartingales, gives an important

Semimartingales

Fractional

Motions Ev

anescent

Models

BrownianDiffusions

Fig. 3: (Colour on-line) A schematic structure (ring) of anom-alous diffusion models: semimartingales (excluding Browniandiffusions), fractional motions, and evanescent models. Accord-ing to the Embedding Principle only semimartingales can beembedded in the Brownian diffusion.

additional information on the behavior of its trajectories.In particular, it is known that the trajectories of semi-martingales are of finite quadratic variation. The differentbehavior of quadratic variations can be used to constructan efficient statistical test which differentiates betweendifferent types of subdiffusive dynamics in experimentaldata [31]. Detection of the origins of anomaly is animportant and timely problem of statistical physics.The presented considerations indicate that complex

systems displaying anomalous type of behavior can beidentified by their subordinators representing internaloperational time. In another words only such anomalousdiffusions (semimartingales) can be embedded in Brown-ian diffusion. The determination of the proper subordina-tor is important here since it allows via Langevin equationsto apply immediately Monte Carlo methods for solutionof the corresponding FFPE [35,48], and for the numeri-cal analysis of the trajectories. Therefore, the analysis ofthe subordinators of the system is yet another promis-ing method leading to better understanding of anomalousbehavior of complex systems.The Embedding Principle suggests also the interest-

ing question on a general structure of anomalous diffu-sion models. Namely, according to the above analysis wehave the following picture: a large class of semimartin-gales (excluding Brownian diffusions), fractional motions(including fractional α-stable motion and FBM), and aclass of so-called evanescent models (including for exam-ple Langevin equations with fractional noise [56] etc.)see fig. 3. It should be noted that fractional differentialequations are not directly related to semimartingales. Forexample, FBM is not a semimartingale and its PDF solvessome fractional differential equation [57]. Coupled CTRWmight lead in the limit to semimartingales [54]. However,this is not always the case, since FBM can be obtainedas a limit of certain coupled CTRW [58]. More research isneeded on the structure of anomalous diffusion, especiallyon the evanescent models.

60010-p5

A. Weron and M. Magdziarz

∗ ∗ ∗The authors acknowledge many useful discussions with

J. Klafter, I. M. Sokolov, N. W. Watkins, andK. Weron. The second author was partially supported bythe Foundation for Polish Science through the DomesticGrant for Young Scientists (2009).

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