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Anomalous diffusion and semimartingales
A. Weron and M. Magdziarz
EPL, 86 (2009) 60010
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June 2009
EPL, 86 (2009) 60010
doi: 10.1209/0295-5075/86/60010
www.epljournal.org
Anomalous diffusion and semimartingales
A. Weron and M. Magdziarz(a)
Hugo Steinhaus Center, Institute of Mathematics and Computer Science, Wroclaw University of Technology
Wyspianskiego 27, 50-370 Wroclaw, Poland, EU
received 27 February 2009; accepted in final form 5 June 2009
published online 10 July 2009
PACS
PACS
PACS
05.40.Fb – Random walks and Levy flights
02.50.Ey – Stochastic processes
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 random
time. Thus, in many cases, anomalous diffusion can be embedded in Brownian diffusion. Such
an embedding takes place if and only if the anomalous diffusion is a semimartingale process. We
also discuss the structure of anomalous diffusion models. Categorization of semimartingales can
be applied to differentiate among various anomalous processes. In particular, identification of the
type of subdiffusive dynamics from experimental data is feasible.
c EPLA, 2009
Copyright Introduction. – The beginning of the long-lasting
marriage of Brownian motion with physics dates back
to two breakthrough papers [1,2], in which both authors
Einstein and Smoluchowski explained independently the
phenomenon of Brownian motion as a result of collisions
between the suspended particles and the molecules of the
surrounding fluid. A different and innovative approach
was proposed by Langevin [3]. Even earlier, in 1900,
Bachelier [4] bound up Brownian motion with financial
modeling, by introducing the first model of stock price at
the Paris Bourse.
However, in spite of many obvious advantages, models
based on Brownian motion fail to provide satisfactory
description of many dynamical processes [5–7]. The
detailed empirical analysis of various complex systems
shows that some of the significant properties of such
systems cannot be captured by the Brownian diffusion
models. One should mention here such properties as:
nonlinear in time mean-squared displacement, long-range
correlations, nonexponential relaxation, heavy-tailed and
skewed marginal distributions, lack of scale invariance,
discontinuity of the trajectories, and many others [8]. To
capture such anomalous properties of physical systems,
some different mathematical models need to be introduced. In recent years one observes a rapid evolution
in this direction, which results in emerging of various
alternative models, such as: continuous-time random
walks [9], fractional kinetic equations [8,10,11], fractional
(a) E-mail:
[email protected]
Brownian motions [12,13], generalized Langevin equations [14], jump-diffusion models [15], subordinated
Langevin 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 models
based on Brownian diffusion. As we will show in the
following, Brownian motion is doing well also in the
world of anomalous (non-Brownian) diffusion [18]. It turns
out that the large part of the currently explored models
of anomalous transport can be represented as Brownian
motion but in the appropriate internal clock of the system.
Semimartingales. – We will start with an observation that anomalous diffusion processes are in a very
close relation with the so-called semimartingales. The
concept of semimartingales became popular in the 1970s,
see the basic papers by Föllmer [19], Bichteler [20,21],
and Dellacherie [22]. However, it was introduced earlier
by Meyer [23]. The stochastic integration with respect to
semimartingales was developed by him and his collaborators. Since then their mathematical theory attracted a
considerable attention and has found already important
applications to asset pricing in economy [15]. A semimartingale Xt uniquely decomposes as
Xt = Mt + At ,
(1)
where Mt is a local martingale (its increment process
is a fair-game, meaning that the expectation value of
increments is zero), and At is a finite variation process,
60010-p1
A. Weron and M. Magdziarz
which can be considered as the drift part of Xt [24]. Thus,
the most relevant subclass of semimartingales constitute
martingales. The concept of a martingale is the stochastic
analog of a conserved quantity. A process Mt is said to
be a martingale if it has the property that the (future)
conditional mean value is equal to the last value specified
by the condition, i.e.
Mt |Mtn , . . . , Mt1 = Mtn ,
t > tn > . . . > t1 .
In other words, Mt is a martingale if its conditional
expectation, given information up to time tn , is the value
of the process at that time. Thus, in a certain sense,
martingales are the stochastic analog of conservation
laws: expectation is conserved. The concept of martingale
can also be used effectively to characterize the reductive
properties of energy-based stochastic extension of the
Schrödinger equation [25]. For the martingale description
of physical fluctuations see [26]. Also, the time operator in
the Prigogine theory of irreversibility can be constructed
as an integral with respect to martingales [27]. The
martingale property has found widespread applications in
finance and game theory, since it formalizes the concept
of the fairness of the game.
Let us mention some well-known examples of martingales: 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 Brownian
motion, Poisson process, Lévy processes, are semimartingales. Nevertheless, it is not always the case. An example
of the process, which does not belong to this class is
the fractional Brownian motion [12,13], as well as the
fractional α-stable motion [28]
∞
H−1/α
H−1/α
dLα (u).
(t − u)+
− (−u)+
−∞
Semimartingales are the most general reasonable stochastic integrators, for which the pathwise evaluation of
stochastic Itô-type integrals is possible [20,21]. The class
of semimartingales is invariant under smooth transformation (Itô lemma), random change of time and Itô stochastic integration. Thanks to these plausible properties,
semimartingales have found widespread applications and
play a fundamental role in stochastic modelling including
for example modern modelling of financial data [15,29].
We will demonstrate in this letter that, the role of semimartingales in the description of anomalous diffusion is
significant as well. The large part of well-known anomalous
diffusion processes can be represented as time-changed
Brownian motion. Thus, by employing the I. Monroe
result [30], every time-changed Brownian motion is a
semimartingale. In other words, we obtain the following
interpretation of Monroe’s result in the context of anomalous dynamics —the Embedding Principle for anomalous diffusion: Anomalous diffusion process can be represented as time-changed Brownian motion if and only if
it is a semimartingale. The above statement confirms
the intimate relation between semimartingales and anomalous transport. Since the mathematical theory of semimartingales is well developed, it can help to identify and
understand better various properties of anomalous diffusions. The quantity called the quadratic variation is of
particular interest from the point of view of applications
of semimartingales in physics. Let X(t) be a stochastic process observed on time interval [0, T ]. Then, for
t ∈ [0, T ], the quadratic variation V (2) (t) corresponding to
X(t) is defined as
V (2) (t) = lim Vn(2) (t),
n→∞
(2)
(2)
where Vn (t) is the partial sum of the squares of increments of the process X(t) given by
Vn(2) (t) =
n
2
−1 j=0
X
2
(j + 1)T
jT
∧
t
−
X
∧
t
n
n
2
2
(3)
with a ∧ b = min{a, b}. The crucial property of the trajectories of semimartingales is the fact that their quadratic
variation is always finite. This is not the case for any
stochastic process in general. For example, quadratic variation of fractional Brownian motion (FBM) BH (t) with
Hurst index H less than 1/2, is infinite. This follows from
the fact that FBM is H–self-similar and its increments
2
satisfy (BH (t + h) − BH (t))2 ∼ h2H BH
(1). Since 2H < 1,
the sum of the squares of the increments in (3) diverges
to infinity as n → ∞. Similar reasoning can be applied
to diffusion on fractal, which is yet another example of
process with infinite quadratic variation and does not
belong to the class of semimartingales. The dissimilarities
in the behavior of quadratic variation provide a simple way
how to distinguish between different kinds of subdiffusion
(CTRW, FBM, diffusion in confined media) from experimental data [31]. One only needs to examine the behavior
of the quadratic variation of a given trajectory. Detection
of the type of anomaly is an important and timely problem, see for example [32–34] for discussion on the origins
of anomaly in the case of intracellular diffusion.
Thus, the information that a given physical process
belongs to the family of semimartingales, gives an important additional gain in understanding the behavior of the
trajectories. This can be used to identify the type of anomaly from experimental data [31]. Moreover, representation
of a given stochastic process as time-changed Brownian
motion allows to run efficient Monte Carlo simulations
of its trajectories and to perform numerical analysis of
sample paths [35].
Next, to illustrate the embedding principle, we will
consider in details several physically relevant special cases
60010-p2
Anomalous diffusion and semimartingales
50
1
B(Uα(t))
S (t)
α
U (t)
0
−1
−2
0
25
25
t
50
25
t
50
4
B(Sα(t))
Uα(t) , Sα(t)
α
0
0
25
t
2
0
−2
0
50
Fig. 1: (Colour on-line) Correspondence between the processes
Uα (t) and its inverse Sα (t). They are obtained by reflection
wrt the diagonal of the square. The trajectories of the first
process are superlinear, whereas the behavior of the second
one is sublinear. For every jump of Uα (t) we observe the
corresponding flat period of Sα (t). Here α = 0.7.
of anomalous diffusions and for each case we will identify
an appropriate subordinator.
Sub- and superdiffusion. – In the large family of
anomalous diffusion processes, two distinct classes are of
special importance. Namely, subdiffusion and superdiffusion. The first class is characterized by the sublinear in time mean-squared displacement. Its presence was
reported in condensed phases [8], ecology [36], biology [33],
and many more. The superdiffusive regime is characterized
by the superlinear in time or even infinite (Lévy flights)
mean-squared displacement. Superdiffusion is observed in
the transport of micelle systems, bacterial motion, quantum optics (see [8] and references therein) and even in the
transport of light in certain optical materials [37].
Clearly, the classical Brownian motion B(t) is not
suitable for modelling of sub- and super-diffusion, since
its fundamental feature is the linearity of the meansquared displacement. Nevertheless, by introducing the
appropriate random time (α-stable subordinator) to the
system, one is able to transform the standard diffusion
B(t) into anomalous one. Let us denote by Uα (t) the
strictly increasing α-stable Lévy process (subordinator)
with the well-known formula for its Laplace transform
α
e−kUα (t) = e−tk , 0 < α < 1 [38,39]. By
Sα (t) = inf{τ : Uα (τ ) > t}
Fig. 2: (Colour on-line) In the top panel a trajectory of
the process B(Uα (t)) is shown. The jumps of the process,
which are inflicted by the subordinator Uα (t), are typical for
superdiffusion. In the bottom panel a sample path of the
process B(Sα (t)) is visualized. We observe the characteristic
constant periods of the trajectory, which correspond to the
constant periods of Sα (t). The process is subdiffusive. Here
α = 0.7.
obtain the new process
X(t) = B(Uα (t)).
Such an operation is called subordination and it was first
introduced by Bochner [40]. The process B, called the
parent process, is directed by the new operational time
clock Uα called subordinator. It is easy to verify that
the process X(t) is superdiffusive. Namely, its Fourier
µ
transform is of the form eikX(t) = e−tk , where µ = 2α.
The trajectories of X(t) are discontinuous, see fig. 2.
We can observe long jumps, which are characteristic for
Lévy flights. Having in mind that the trajectories of the
classical Brownian particle are continuous, we infer that
the superdiffusive character of X(t) is inflicted by the
subordinator Uα (t). Moreover, for the second moment
of X(t) we have X 2 (t) = ∞, which is typical for Lévy
flights.
Fractional Fokker-Planck equations. – The probability density function (PDF) of X(t) satisfies the superdiffusive fractional Fokker-Planck equation (FFPE) [8]
∂p(x, t) ∂ µ p(x, t)
=
,
∂t
∂xµ
∂d
where the operator ∂x
µ is the Riesz fractional derivative.
Thus, by the time change t → Uα (t), Brownian motion is
transformed into superdiffusion.
By the similar subordination method, we are able to
transform B(t) into subdiffusion. Let us consider the
following time-changed process Y (t) = B(Sα (t)). Here,
Sα (t) is the previously introduced inverse α-stable subordinator. Now, the process Y (t) is subdiffusive. Its second
moment is given by Y 2 (t) ∼ tα , 0 < α < 1, thus it is
µ
we denote the left inverse of Uα (t). We call Sα (t)
the inverse α-stable subordinator. The correspondence
between Uα and Sα can be observed in fig. 1. Namely,
Sα (t) is obtained from Uα (t) by reflection wrt to a diagonal of the square. Clearly, the trajectories of Uα (t) are
superlinear, 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
(4)
60010-p3
A. Weron and M. Magdziarz
sublinear. The PDF of Y (t) obeys the subdiffusive frac- A typical framework for the treatment of anomalous
tional diffusion equation [8,41]
diffusion problems under the influence of an external force
F (x) is in terms of the FFPE [8]
2
∂p(x, t)
1−α ∂ p(x, t)
,
(5)
= 0 Dt
∂t
∂x2
∂F (x)
∂µ
∂p(x, t)
= −
+ µ 0 Dt1−α p(x, t).
1−α
∂t
∂x
∂x
where the operator D
is the Riemann-Liouville frac0
t
tional derivative. The trajectories of Y (t) are continuous
with the characteristic flat periods, fig. 2. These constant
periods are typical for subdiffusion and represent the time
intervals in which the particle gets immobilized in the
trap. Comparing figs. 1 and 2 we observe the clear correspondence between the flat periods of Sα (t) and Y (t).
Thus, we conclude that the process Sα (t) is responsible for
giving the effect of subdiffusion. Note that for α → 1 we
have that
∂
The operator ∂x
µ introduces the Lévy-flight–type behavior, whereas 0 Dt1−α causes the memory effects typical for
subdiffusion. The analysis of the corresponding Langevin
equation [35,48] shows that also in the presence of external force F (x), the underlying process can be represented as time-changed Brownian motion B(T (t)) for
some appropriate subordinator T (t). As an example, let
us consider the subdiffusive Ornstein-Uhlenbeck process
defined by the above FFPE with F (x) = −x and µ = 2.
Uα (t) = Sα (t) = t,
Then, Y (t) can be represented as B(T (t)), where T (t) =
1
−2Sα (t)
). Concluding, there always exists an appro2 (1 − e
which shows that the standard diffusion is placed exactly priate
T (t) such that the time-changed Brownian motion
on the border between sub and superdiffusion and when B(T (t)) describes sub or superdiffusion in the presence of
the operational time coincides with the real time.
external force. In general, the processes B and T do not
It is worth mentioning that the subordination is have to be independent. Let us underline that the same
reversible in the following sense: we have that
arguments as above can be applied also for the case when
the force is time-dependent F = F (t) [17].
B(Sα (Uα (t))) = B(t).
µ
Jump-diffusions. – Another significant class of anomalous diffusions constitute non-Gaussian Lévy processes,
also known as jump-diffusion models due to their discontinuous trajectories. It is one of the most important and
fundamental classes of random processes characterized
by stationarity and independence of increments. Special
examples include the Poisson, Compound Poisson, and
stable Lévy motions. Since their introduction in the 1930s,
Lévy processes were studied extensively by many mathematicians [49]. However, in recent years they attracted
considerable attention of physical community, too [50].
Continuity of paths has always played a crucial role
in the properties of classical diffusion models. Nevertheless, recent developments confirm that many transport phenomena (i.e. bacterial motion [6], transport of
light in certain optical materials [37], stock prices [15])
display a discontinuous behavior. In such cases jumpdiffusion models rather than Brownian motion provide the
proper background to describe and understand discontinuous transport. Every Lévy process L(t) can be fully characterized by its Fourier transform eikL(t) = e−tψ(k) , where
the function ψ(k) is the so-called Lévy exponent. However,
the most important fact here is that for each Lévy process
L there exists a subordinator T (t) such that L(t) can be
represented as B(T (t)) [15]. For example, to obtain the
Linnik geometric stable process [51] one has to choose
T (t) = Uα (Γt ), where Γt is the Gamma subordinator [49].
In general case, the assumption about the independence
of subordinator and Brownian motion has to be relaxed.
Concluding, the class of Lévy processes, which belongs to
External force fields. – Many physical transport the family of anomalous diffusions, can also be described
problems take place under the influence of external fields. as time-changed Brownian motion [7].
Thus, the double subordination recovers the classical
Brownian 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 subordination [45–47]. It should be noted that subordination
in the sense of integral transformation of PDFs, when
both processes (parent process and subordinator) are
independent, is not enough for some purposes (see [45]
for the calculation of first-passage times). This is related
to the fact that in some cases PDF’s do not give the
complete description of the underlying process. However,
subordination represented as the superposition of two
stochastic processes gives such full characterization of
the process. The discussion on the duality of subdiffusion
and Lévy flights in the framework of Continuous-Time
Random walks (CTRW) and propagators for the case
when subordinator is independent of Brownian motion,
can be found in [46,47]. The authors of the above papers
have demonstrated that, under assumption that the
initial ticks of the physical clock follow extremely inhomogeneously and show intervals of high condensation,
operational time leads to Lévy flights. This corresponds
to the subordination (4). If the ticks triggering the
motion of the random walker are taken from the set
of ticks of a physical clock according to a power law
waiting-time distribution, such random walk leads to the
subdiffusive dynamics and FFPE (5). This corresponds
to the subordination procedure of the form B(Sα (t)).
60010-p4
Anomalous diffusion and semimartingales
Moreover, every standard CTRW process possesses the
representation of the form B(T (t)) as well. It is not
surprising, since the family of CTRWs is closely related
to the class of Lévy processes, i.e. every Lévy process can
be obtained as a limit of the appropriate CTRW [49].
Conclusions. – In this letter we have categorized the
anomalous diffusion models (including: sub- and superdiffusion, CTRWs, and jump-diffusions) represented
as time-changed Brownian motion. We have indicated
the close and promising link between statistical physics
(anomalous transport) and the theory of semimartingales.
One should note at this point that not every anomalous
diffusion process can be represented as time-changed
Brownian motion. Such counterexamples are FBM and
random walk on fractals. Both models do not belong
to the family of semimartingales. The fact that a given
physical process is a semimartingales, gives an important
Ev
a
M nes
od c
el ent
s
Brownian
Diffusions
l
na
tio ns
ac io
Fr ot
M
Relaxation phenomena. – In recent years, the question of understanding the phenomenology of relaxation
processes in various materials has attracted considerable
attention [52–55]. It is surprising that in spite of the
variety of materials and experimental techniques, the
behavior of relaxing systems is very similar and can
be described in terms of simple relaxation functions.
Therefore, it is expected that there exists a common
nature of microscopic relaxation mechanisms with the
corresponding mathematical structure. It turns out that
an unified mathematical structure can be provided in the
framework of time-changed Brownian motion.
Let us denote by R(t) the distance reached by the particle after time t. Then, the corresponding time-domain
relaxation function Φ(t), describing the system’s survival
probability in the initial state, is defined as the Fourier
transform of R(t), i.e. Φ(t) = eikR(t) . Now, by the appropriate choice of R(t), we obtain different relaxation functions. For R(t) = B(Sα (t)), where Sα (t) is the previously introduced inverse α-stable subordinator, we get
the Mittag-Leffler relaxation function. Next, for R(t) =
B(T (t)) with T (t) = tUα (1) and Uα being the α-stable
subordinator, we win the stretched exponential relaxation
function. Moreover, subordinating the Brownian motion
by the processes obtained as the limit of certain CTRWs
leads to the Havriliak-Negami relaxation in [54], the generalized Mittag-Leffler relaxation in [53], and the ColeDavidson relaxation in [55] by employing the tempered
stable distributions.
As one can observe, randomization of the time of Brownian motion using the appropriate subordinator, provides
a unified mathematical description of all the most relevant relaxation patterns. Different choice of subordinator
results in different behavior of the anomalously relaxing
systems. It suggests that complex systems can be identified and characterized by their own internal operational
time. The determination of this operational time seems
to be the key factor in understanding and describing of
relaxation phenomena in complex systems.
Semimartingales
Fig. 3: (Colour on-line) A schematic structure (ring) of anomalous diffusion models: semimartingales (excluding Brownian
diffusions), fractional motions, and evanescent models. According to the Embedding Principle only semimartingales can be
embedded in the Brownian diffusion.
additional information on the behavior of its trajectories.
In particular, it is known that the trajectories of semimartingales are of finite quadratic variation. The different
behavior of quadratic variations can be used to construct
an efficient statistical test which differentiates between
different types of subdiffusive dynamics in experimental
data [31]. Detection of the origins of anomaly is an
important and timely problem of statistical physics.
The presented considerations indicate that complex
systems displaying anomalous type of behavior can be
identified by their subordinators representing internal
operational time. In another words only such anomalous
diffusions (semimartingales) can be embedded in Brownian diffusion. The determination of the proper subordinator is important here since it allows via Langevin equations
to apply immediately Monte Carlo methods for solution
of the corresponding FFPE [35,48], and for the numerical analysis of the trajectories. Therefore, the analysis of
the subordinators of the system is yet another promising method leading to better understanding of anomalous
behavior of complex systems.
The Embedding Principle suggests also the interesting question on a general structure of anomalous diffusion models. Namely, according to the above analysis we
have the following picture: a large class of semimartingales (excluding Brownian diffusions), fractional motions
(including fractional α-stable motion and FBM), and a
class of so-called evanescent models (including for example Langevin equations with fractional noise [56] etc.)
see fig. 3. It should be noted that fractional differential
equations are not directly related to semimartingales. For
example, FBM is not a semimartingale and its PDF solves
some fractional differential equation [57]. Coupled CTRW
might lead in the limit to semimartingales [54]. However,
this is not always the case, since FBM can be obtained
as a limit of certain coupled CTRW [58]. More research is
needed on the structure of anomalous diffusion, especially
on 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, and
K. Weron. The second author was partially supported by
the Foundation for Polish Science through the Domestic
Grant for Young Scientists (2009).
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