Radiophysics Faculty, Nizhniy Novgorod State …arXiv:0810.1492v1 [cond-mat.stat-mech] 8 Oct 2008 L´evy Flight Superdiﬀusion: An Introduction A. A. Dubkov♯, B. Spagnolo⋆, and

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Levy Flight Superdiffusion: An Introduction

A. A. Dubkov♯, B. Spagnolo⋆, and V. V. Uchaikin♭

♯ Radiophysics Faculty, Nizhniy Novgorod State University23 Gagarin Ave., 603950 Nizhniy Novgorod, Russia∗⋆ Dipartimento di Fisica e Tecnologie Relative and CNISM-INFM,

Group of Interdisciplinary Physics†, Universita di Palermo,

Viale delle Scienze, I-90128, Palermo, Italy‡ and♭ Department of Theoretical Physics, Ulyanovsk State University

42 L. Tolstoy str., 432970 Ulyanovsk, Russia§

(Dated: October 9, 2008)

After a short excursion from discovery of Brownian motion to the Richardson ”law

of four thirds” in turbulent diffusion, the article introduces the Levy flight superdif-

fusion as a self-similar Levy process. The condition of self-similarity converts the

infinitely divisible characteristic function of the Levy process into a stable char-

acteristic function of the Levy motion. The Levy motion generalizes the Brown-

ian motion on the base of the α-stable distributions theory and fractional order

derivatives. The further development of the idea lies on the generalization of the

Langevin equation with a non-Gaussian white noise source and the use of func-

tional approach. This leads to the Kolmogorov’s equation for arbitrary Markovian

processes. As particular case we obtain the fractional Fokker-Planck equation for

Levy flights. Some results concerning stationary probability distributions of Levy

motion in symmetric smooth monostable potentials, and a general expression to

calculate the nonlinear relaxation time in barrier crossing problems are derived.

Finally we discuss results on the same characteristics and barrier crossing problems

with Levy flights, recently obtained with different approaches.

Keywords: Levy process, Levy motion, Levy flights, stable distributions, fractional differential

equation, barrier crossing

I. INTRODUCTION

Two kinds of motions can easily be ob-served in Nature: smooth, regular motion,like Newtonian motion of planets, and ran-dom, highly irregular motion, like Brownianmotion of small specks of dust in the air. Thefirst kind of motion can be predicted and con-sequently can be described in the frame ofdeterministic approach. The second one de-

∗e-mail: [email protected]†http://gip.dft.unipa.it‡e-mail: [email protected]§e-mail: [email protected]

mands the statistical approach.The first man who noted the Brown-

ian motion was the Dutch physician, JanIngen–Housz in 1794, who, while in the Aus-trian court of Empress Maria Theresa, ob-served that finely powdered charcoal floatingon an alcohol surface executed a highly er-ratic random motion. A similar observationwas made by the Scottish botanist RobertBrown [Brown, 1828]. He observed undera microscope the continuous irregular mo-tion of small particles (sized in some mi-crometers and less). The particles moved bydisordered trajectories, their motion did notweaken, did not depend on chemical proper-

http://arXiv.org/abs/0810.1492v1

ties of a medium, strengthened with increas-ing medium temperature, with a diminutionof its viscosity and sizes of particles. But R.Brown considered the motion of the parti-cles (not being atoms of course) as their ownproperty and said nothing about atoms ormolecules.

It should pass almost 8 decades before twophysicists Albert Einstein [Einstein, 1905]and Marian von Smoluchowski [Smolu-chovski, 1906] found the physical explanationof Brownian motion. It was based on consid-eration of thermal motion of molecules sur-rounding the Brownian particle. The historyof the further study of Brownian motion is as-sociated with names of Langevin [Langevin,1908], Perrin [Perrin, 1908], Fokker [Fokker,1914], Planck [Planck, 1917], Uhlenbeck,Ornstein [Uhlenbeck & Ornstein, 1930],Chandrasekhar [Chandrasekhar, 1943] andother well-known physicists. However, forthe first time the diffusion equation appearedin the thesis of Louis Bachelier [Bachelier,1900], a student of A. Poincare. His thesis,entitled ”The theory of speculations”, was de-voted to the study of random processes inmarket prices evolution.

It is astonishing, how the same diffusionequation can describe the behavior of neu-trons in a nuclear reactor, the light in at-mosphere, the stock market values rate on fi-nancial exchange, particles of flower dust sus-pended in a fluid and so on. The fact thatcompletely different by nature phenomenaare described by identical equations is a di-rect indication that the matter concerns notthe concrete mechanism of the phenomenon,but rather the same common quality of wholeclass of similar phenomena. The statementof this quality in terms of physical laws andmathematical postulates or definitions allowsto liberate a given pattern from details, whichare not influencing essentially the physicalprocess, and to explore the obtained modelthrough general laws. This is a typical situa-tion for statistical physics and applied math-ematics. The new approaches proposed by

Einstein, Smoluchowski and Langevin to de-scribe the Brownian motion, in fact, openthe door to model a great variety of natu-ral phenomena. At the same time for mathe-maticians, whose achievements built the the-ory of random processes, the first object ofits application became the Brownian motion.The major contribution to the mathemat-ical theory of Brownian motion has beenbrought by N. Wiener [Wiener, 1930], whohas proved that the trajectories of Brown-ian process almost everywhere are continuousbut are not differentiable anywhere. Alongwith Wiener the mathematical aspects ofBrownian motion were treated by Markov,Doob, Kac, Feller, Bernstein, Levy, Kol-mogorov, Stratonovich, Ito and others [Doob,1953; Kac, 1957; Feller, 1971; Levy, 1925,1965; Kolmogorov, 1941; Stratonovich, 1963,1967, 1992; Ito, 1944, 1946, 1965].

Two important properties are intrinsic tothe homogeneous Brownian motion: the dif-fusion packet initially concentrated at a pointtakes later the Gaussian form, whose widthgrows in time as t1/2. This kind of diffusionwas called the normal diffusion.

Twenty years later Einstein, Smolu-chowski and Langevin works, L. Richard-son published the article [Richardson, 1926]where he presented empirical data being incontradiction with the normal diffusion: thesize ∆ of an admixture cloud in a turbu-lent atmosphere grows in time proportionallyto t3/2, that is much faster then in the nor-mal case (t1/2). This turbulent diffusion wasthe first example of superdiffusion processes,when ∆ ∝ tγ with γ > 1/2.

The phenomenon has been interpretedas a diffusion process with a variable dif-fusivity D(r) ∝ r4/3. This Richardson’s”law of four thirds” was grounded theo-retically by Russian mathematicians A. N.Kolmogorov [Kolmogorov, 1941] and A. M.Obukhov [Obukhov, 1941] as a consequenceof the self-similarity hypothesis of locallyisotropic turbulence. However, the fact thatthe diffusivity should depend not on the co-

2

ordinates (turbulent medium is supposed tobe homogeneous in average), but on the scaleor distance l between a pair of diffusing parti-cles, creates essential difficulties both to finda solution to the equation and for its inter-pretation.

As Monin showed in Ref. [Monin, 1955],the same law of the diffusive packet widen-ing with time can be obtained in the frame-work of the homogeneous Markovian pro-cesses family, when the characteristic func-tion of the spatial distribution of the diffusivesubstance

P (k, t) =⟨eikX(t)

⟩=∫ +∞

−∞eikxP (x, t)dx

(1)obeys the equation

∂P (k, t)

∂t= −C|k|αP (k, t) (2)

with C being a positive constant. Under ini-tial condition P (x, 0) = δ(x) we obtain fromEq. (2)

P (k, t) = exp{−C|k|αt}, (3)

where α = 2/3. This is nothing but thecharacteristic function of the α-stable Levydistribution, and the random process itselfis the Levy motion (Levy flights). Later,anomalous diffusion in the form of Levyflights has been discovered in many otherphysical, chemical, biological, and finan-cial systems [Shlesinger et al., 1993; Met-zler & Klafter, 2000, 2004; Metzler et al.,2007; Brockmann & Sokolov, 2002; Eliazar &Klafter, 2003; Barkai, 2004; Chechkin et al.,2006; Mandelbrot, 1997; Mantegna, 1991].

Levy flights are stochastic processes char-acterized by the occurrence of extremely longjumps, so that their trajectories are not con-tinuous anymore. The length of these jumpsis distributed according to a Levy stablestatistics with a power law tail and diver-gence of the second moment. This pecu-liar property strongly contradicts the ordi-nary Brownian motion, for which all the mo-ments of the particle coordinate are finite.

The presence of anomalous diffusion can beexplained as a deviation of the real statisticsof fluctuations from the Gaussian law, giv-ing rise to the generalization of the centrallimit theorem by Levy and Gnedenko [Levy,1925, 1937; Gnedenko & Kolmogorov, 1954].The divergence of the Levy flights varianceposes some problems as regards to the physi-cal meaning of these processes. However, re-cently the relevance of the Levy motion ap-peared in many physical, natural and socialcomplex systems. The Levy type statistics,in fact, is observed in various scientific areas,where scale invariance phenomena take placeor can be suspected. Among many interest-ing examples we cite here chaotic dynamics ofcomplex systems [Zaslavsky, 2005; Solomonet al., 1993, 1994], diffusion and annihilationreactions of Levy flights with bounded long-range hoppings [Albano, 1991], front dynam-ics in reaction-diffusion systems with Levyflights [del-Castillo-Negrete et al., 2003], frac-tional diffusion [West et al., 1997; Chaves,1998], thermodynamics of anomalous diffu-sion [Zanette & Alemany, 1995], dynamicalfoundation on noncanonical equilibrium [An-nunziato et al., 2001], quantum fractional ki-netics [Kusnezov et al., 1999], trapping dif-fusion [Vazquez et al., 1999] , Levy diffusionprocesses as a macroscopic manifestation ofrandomness [Grigolini et al., 1999; Bolognaet al., 1999], diffusion by flows in porousmedia [Painter, 1996], two-dimensional Levyflights [Desbois, 1992], persistent Levy mo-tion [Chechkin & Gonchar, 2000], self-avoiding Levy flights [Grassberger, 1985],Levy flights with quenched noise ampli-tudes [Kutner & Maass, 1998], cooling downLevy flights [Pavlyukevich, 2007], branchingannihilating Levy flights [Albano, 1996], ran-dom Levy flights in the kinetic Ising andspherical models [Bergersen & Racz, 1991;Xu et al., 1993], plane rotator in presence ofa Levy random torque [Caceres, 1999], fluc-tuations and transport in plasmas [Chechkinet al., 2002b; Lynch et al., 2003], transport instochastic magnetic fields [Zimbardo & Vel-

3

tri, 1995], Levy flights in the Landau-Tellermodel of molecular collisions [Carati et al.,2003], subrecoil laser cooling [Bardou et al.,1994, 2002; Reichel et al., 1995; Schaufler etal., 1997, 1999], scintillations and Levy flightsthrough the interstellar medium [Boldyrev& Gwinn, 2003], Levy flights in cosmicrays [Wilk & Wlodarczyk, 1999], anoma-lous diffusion in the stratosphere [Seo &Bowman, 2000], long paleoclimatic time se-ries of the Greenland ice core measure-ments [Ditlevsen, 1999a], seismic series andearthquakes [Posadas et al., 2002; Sotolongo-Costa et al., 2000], signal processing [Nikias& Shao, 1995], time series statistical analysisof DNA [Scafetta et al., 2002], primary se-quences of proteinlike copolymers [Govorunet al., 2001], spatial gazing patterns of bac-teria [Levandowsky et al., 1997], Levy-flightspreading of epidemic processes [Janssen etal., 1999], contaminant migration by biotur-bation [Reible & Mohanty, 2002], flights ofan albatross [Viswanathan et al., 1996; Ed-wards et al., 2007], fractal time in animalbehaviour of Drosophila and animal locomo-tion [Cole, 1995, Seuront et al., 2007], finan-cial time series [Mandelbrot, 1963; Bouchaud& Sornette, 1994; Mantegna & Stanley, 1996,1998; Chowdhury & Stauffer, 1999], humanstick balancing and Levy flights [Cabrera &Milton, 2004] and human memory retrievalas Levy foraging [Rhodes & Turvey, 2007].Experimental evidence of Levy processes wasalso observed in the motion of single ionin one-dimensional optical lattice [Katori etal., 1997] and in the particle evolution alongpolymer chains [Sokolov et al., 1997; Lomholtet al., 2005], and in self–diffusion in systemsof polymerlike breakable micelles [Ott et al.,1990].

Levy flights are a special class of Marko-vian processes, therefore the powerful meth-ods of the Markovian analysis are in force inthis case. We mean a possibility to investi-gate the stationary probability distributionsof superdiffusion, the first passage time andthe residence time characteristics, the spec-

tral characteristics of stationary motion, etc.Of course, this type of diffusion has a lotof peculiarities different from those observedin normal Brownian motion. The main dif-ference from ordinary diffusion consists inreplacing the white Gaussian noise sourcein the underlying Langevin equation with aLevy stable noise.

In the first part of the present paper, wegive a short introduction to the Levy motion.Being a generalization of the Brownian diffu-sion, it takes an intermediate place betweenBrownian motion and Levy processes (i.e.infinitely divisible processes, see [Bertoin,1996]) in the random process hierarchy sys-tem. The Levy motion is introduced as aself-similar Levy process.

The second part of this paper is devotedto the stationary probabilistic characteristicsand the problem of barrier crossing for Levyflights. We use functional approach to derivethe generalized Kolmogorov equation directlyfrom Langevin equation with the Levy pro-cess [Dubkov & Spagnolo, 2005]. In particu-lar case of Levy stable noise source we obtainthe Fokker-Planck equation with fractionalspace derivative. Starting from this equationwe find the exact stationary probability dis-tribution (SPD) of fast diffusion in symmet-ric smooth monostable potentials for the caseof Cauchy stable noise. Specifically, we con-sider potential profiles U(x) = γx2m/(2m),with odd and even m, useful to describe thedynamics of overdamped anharmonic oscil-lator driven by Levy noise. We find thatfor Levy flights in steep potential well, withpotential exponent 2m greater or equal tofour, the variance of the particle coordinateis finite [Dubkov & Spagnolo, 2007]. Thisgives rise to a confined superdiffused mo-tion, characterized by a bimodal stationaryprobability density, as previously reported inRefs. [Chechkin et al., 2002a, 2003a, 2004,2006]. Here we analyze the SPD as a func-tion of a dimensionless parameter β, which isthe ratio between the noise intensity D andthe steepness γ of the potential profile. We

4

find that the SPDs remain bimodal with in-creasing β parameter, that is with decreasingthe steepness γ of the potential profile, or byincreasing the noise intensity D.

The particle escape from a metastablestate, and the first passage time densityhave been recently analyzed for Levy flightsin Refs. [Ditlevsen, 1999b; Rangarajan &Ding, 2000a, 2000b; Buldyrev et al., 2001;Chechkin et al., 2003b, 2005, 2006, 2007; Dy-biec & Gudowska-Nowak, 2004; Dybiec et al.,2006, 2007; Bao et al., 2005; Ferraro & Za-ninetti, 2006; Imkeller & Pavlyukevich, 2006;Imkeller et al., 2007; Koren et al., 2007]. Themain focus in these papers is to understandhow the barrier crossing behavior, accordingto the Kramers law [Kramers, 1940], is mod-ified by the presence of the Levy noise. Fi-nally we discuss briefly some results on thebarrier crossing events with Levy flights, re-cently obtained with different approaches.

II. LEVY PROCESSES

To see better a place of the diffusion pro-cesses under consideration among other ran-dom processes we shall remind some defi-nitions. We restrict ourselves to the one-dimensional case when X, x ∈ (−∞,∞) andt ≥ 0.

A random process {X(t), t ≥ 0} is a set ofrandom variables X (t), given on the sameprobability space and corresponding to anypossible time t ≥ 0.

A random process {X(t), t ≥ 0} is calleda Markovian process, if for any n ≥ 1 andt1 < t2 < . . . < tn < t P(X(t) < x |X(t1) =x1, . . . , X(tn) = xn) = P(X(t) < x|X(tn) =xn). The Markovian property is interpretedas independence of future from the past forthe known present. P. Levy states this prop-erty by the sentence ”the past influences thefuture only through the present” and under-lines analogy to Huygens’ principle (”it ispossible to say, that it is Huygens’ principlein calculation of probabilities” [Levy, 1965]).

A random process {X(t), t ∈ T} is called

the process with independent increments if forany n ≥ 1 and t1 < t2 < . . . < tn < t randomvariables X (0) , X (t1)−X (0) , . . . , X (tn)−X (tn−1) are mutually independent. The ran-dom variable X (0) is called the initial state(value) of the process, and its probability dis-tribution is called initial distribution of theprocess. P. Levy named such processes addi-tive. Obviously, they belong to the class ofMarkovian processes.

A random process with independent incre-ments is called homogeneous or stationary, ifthe random variables X (t+ τ) − X (t) havedistributions which are independent on t:

P {X (t+ τ) −X (t) < x} = F (x, τ ) . (4)

P. Levy named such processes linear, re-marking that among them ”there are alsoprocesses distinct from Brownian”. Now,Bertoin [Bertoin, 1996] and Sato [Sato, 1999]use the term Levy processes for the processeswith stationary independent increments.

One can paraphrase the definition by say-ing that {X(t), t ≥ 0} is a Levy process if, forevery t, τ ≥ 0, the increment X(t+ τ)−X(t)is independent on the process {X(t′), 0 ≤ t′ <t} and has the same law as X(τ). In partic-ular, X(0) = 0.

We will denote the Levy process L(t). Asit follows from the evident decomposition

L(t) = L(t

n

)+[L(

2t

n

)− L

(t

n

)](5)

+ . . .+

[L(nt

n

)− L

((n− 1)t

n

)],

the random variable L(t) can be divided intothe sum of an arbitrary number of indepen-dent and identically distributed random vari-ables. In other words, the probability distri-bution of L(t) belongs to the class of infinitelydivisible distributions [de Finetti, 1929, 1975;Khintchine, 1938; Khintchine & Levy, 1936;Levy, 1937; Gnedenko & Kolmogorov, 1954;Feller, 1971; Mainardi & Rogosin, 2006].Hence, we can express the second characteris-tics, i.e. the logarithm of characteristic func-

5

tion of the random variable L(t) in the Levy–Khinchine form [Feller, 1971]

φ (k, t) ≡ ln P (k, t) = ln⟨eikL(t)

⟩

=∫ +∞

−∞

(eikx − 1 − ik sin x

) ρ(x, t)x2

dx, (6)

where ρ(x, t) is the canonical measure density(with respect to the first argument).

For two consecutive non-overlapping timeintervals t1 and t2 we have

L(t1 + t2)d= L(t1) + L(t2), (7)

where L(t1) and L(t2) are mutually inde-

pendent random variables and the symbold=

means the equality of distributions of the cor-responding random variables. Therefore,

P (k, t1 + t2) = P (k, t1)P (k, t2) (8)

or

φ(k, t1 + t2) = φ(k, t1) + φ(k, t2). (9)

According to Eqs. (6) and (9) we have

ρ(x, t1 + t2) = ρ(x, t1) + ρ(x, t2). (10)

The differentiable solution of Eq. (10), re-garding t, is only linear one

ρ (x, t) = tρ (x) . (11)

So, from Eq. (6) we obtain

φ (k, t) = t∫ +∞

−∞

(eikx − 1 − ik sin x

) ρ(x)x2

dx,

(12)where the kernel ρ (x) ≥ 0. Note that thelast term in the bracket, −ik sin x, serves toensure the convergence of the integral andcan be omitted if the integral converges it-self. Choosing

ρ(x) = δ(x), (13)

and taking into account that

eikx −1− ik sin x = −k2x2/2+o(x2), x→ 0,(14)

we arrive at the normalized Brownian mo-tion B(t) (Wiener process) with characteris-tic function

P (k, t) = exp{−tk2/2}. (15)

III. SELF-SIMILARITY (SCALING)

The self-similarity (scaling is a synonymof self-similarity) is a special symmetry of asystem (process) revealing that the modifica-tion of the scales of one variable can be com-pensated by the homothetic transformationof the others. For example, if the state of asystem is characterized by function u (x, t),where x is the coordinate, t is the time,the requirement of invariance with respectto scale transformations x → x′ = kx andt→ t′ = lt, looks like

u (x, t) = kαlδu (kx, lt) , (16)

where k and l are positive, and α and δ are ar-bitrary numbers. By choosing kα = l = m/t,where m > 0 is a parameter of similarity,we obtain a self-similar form for the functionu(x, t)

u (x, t) = (m/t)1+δ u((m/t)1/α x,m

). (17)

In our case such a function is the probabilitydensity function P (x, t). The normalizationcondition

∫ +∞

−∞P (x, t) dx = 1 (18)

and the principle of self-similarity (17) give1 + δ = 1/α and lead to the representation(m = 1)

P (x, t) = t−1/αg(α)(xt−1/α

), (19)

whereg(α) (x) = P (x, 1) . (20)

It must be emphasized that we can define theself–similarity of a random process X(t) withstationary increments as usually (see, for ex-ample, Ref. [Chechkin et al., 2002c])

X (t+ κτ) −X (t)d=κH [X (t+ τ) −X (t)] .

In such a case H = 1/α. In terms of charac-teristic functions we have

P (k, t) = g(α)(kt1/α

), (21)

6

with g(α) obeying the equation

g(α)(k (t1 + t2)

1/α)

= g(α)(kt

1/α1

)g(α)

(kt

1/α2

)

(22)which follows from Eq. (8).

Let Y (α) be a random variable describedby the characteristic function

g(α) (k) =⟨exp

{ikY (α)

}⟩. (23)

Obviously,

g(α)(kt1/α

)=⟨exp

{ikY (α)t1/α

}⟩(24)

determines the random variable t1/αY (α) sat-isfying the relation

(t1 + t2)1/α Y (α) d

= t1/α1 Y

(α)1 + t

1/α2 Y

(α)2 , (25)

where Y(α)1 and Y

(α)2 are independent copies

of random variable Y (α). This relation is adefinition property of strictly stable randomvariables with a characteristic index α. Wearrive, therefore, at the very important sub-family of the Levy processes called the Levymotion (often, the term ”Levy flights” is usedas a synonym).

IV. STABLE RANDOM VARIABLES

To find an explicit expression for the sta-ble characteristic functions, we can proceedby two equivalent ways: (i) using the generalrepresentation of infinitely divisible charac-teristic functions (12) or (ii) using the stabil-ity property (22). We choose the latter way.

Let us introduce the second characteristic

ψ(α) (k) = ln g(α) (k) , (26)

so the property (22) of a strict stability takesthe form

ψ(α) (λ1k) + ψ(α) (λ2k) = ψ(α) (λk) , (27)

whereλ = (λα

1 + λα2 )1/α . (28)

Extending this relation to the sum of ar-bitrary number n of identically distributed(λ1 = λ2 = . . . = λn = 1) terms, we obtain

nψ(α) (k) = ψ(α)(n1/αk

). (29)

According to the property

ψ(α) (−k) =[ψ(α) (k)

]∗(30)

it is enough to determine the function ψ(α) (k)for positive arguments k > 0. Taking into ac-count its continuity in a neighborhood of theorigin and the initial condition of the charac-teristic function

ψ(α) (0) = 0, (31)

we obtain that∣∣∣ψ(α) (k)

∣∣∣ = const · kα (k > 0, α > 0)

(32)and

ψ(α) (k) = −kα (c0 − ic1) , (33)

where c0 and c1 are arbitrary real constants.Since the characteristic function satisfies therequirement

∣∣∣g(α)(k)∣∣∣ ≤ 1, (34)

thenRe

{ψ(α)(k)

}≤ 0 (35)

and the real constant c0 should be positive.On the other hand, from Eqs. (23) and (26)we have

g(α)′(0) = i〈Y (α)〉, g(α)′′(0) = −〈[Y (α)]2〉(36)

and

ψ(α)′′(0) = −〈[Y (α)]2〉 + 〈Y (α)〉2

≡ −Var{Y (α)

}≤ 0. (37)

Calculating the second derivative fromEq. (33), we obtain

ψ(α)′′(k) = −(c0 − ic1)α(α− 1)kα−2. (38)

7

As one can see from Eq. (38), we have: (i) forα = 2 the variance is finite (thus the constantc1 should be equal to zero since the varianceis real); (ii) for α < 2 and k → 0 we obtainthe infinite variance (in this case c1 does notplay any role), and (iii) for α > 2 and k → 0the derivative gives zero. This means that inthe expression for the second moment

〈[x(α)]2〉 =∫ +∞

−∞x2g(α)(x)dx,

the function g(α)(x) ceases to be a probabil-ity density, when the characteristic index ex-

ceeds the boundary value α = 2. We cometo the conclusion that a range of values ofthe characteristic index α is the interval (0, 2]closed on the right. Because of c0 > 0 and−∞ < c1 < +∞, the constants of Eq. (33)can be put in the form

c0 = 1, c1 = β tg (απ/2) , −1 ≤ β ≤ 1.(39)

Therefore, the characteristic functiong(α,β)(k) of the strictly stable probabilitydistribution g(α,β)(x), with parameters α andβ, is given by the formula

g(α,β)(k) = exp{−|k|α

[1 − iβ tan

(απ

2

)sgn k

]}, (40)

where sgn x is the sign function. The charac-teristic index α (with α < 2) determines thedecreasing rate of the large values probabilityfor stable distributions

P{∣∣∣Y (α,β)

∣∣∣ ≥ ∆}∝ ∆−α, ∆ → ∞. (41)

The parameter β characterizes the asymme-try of the distributions: for β = 0 the sta-ble distribution is symmetric. The class ofthe symmetric stable distributions, besidesthe above-mentioned Gaussian distribution,includes also the Cauchy distribution

g(1,0) (x) =1

π (1 + x2)(42)

with the characteristic function

g(1,0) (k) = exp {− |k|} . (43)

For α < 1 the distributions with extremevalues of asymmetry β are located on a semi-axes only: positive if β = 1 or negative ifβ = −1. One of these well-known one-sidedistribution is the Levy - Smirnov distribu-tion

g(1/2,1) (x) =1√2πx−3/2 exp

(− 1

2x

), x ≥ 0.

(44)

The detailed exposition of properties ofstable random variables and their distribu-tions can be found in the books [Khintchine,1938; Levy, 1965; Gnedenko & Kolmogorov,1954; Feller, 1971; Bertoin, 1996; Sato, 1999;Uchaikin & Zolotarev, 1999b]. We shall un-derline here only the fact that all membersof the set of stable distributions are charac-terized by presence of ”heavy” (power-type)tails and, as a consequence, of infinite vari-ance, and that concerns all of them, exceptthe Gaussian (normal) distribution. Fromthe point of view of the whole ”noble family”,the Gaussian distribution should be lookingdefiantly abnormal, monstrous, ugly ducklingamong white swans. For us (at least, formany of us) the infinite variance associateswith an infinite error (what is not the truth),with an infinite energy or with something elsewhat does obviously not have any physicalsense.

Professional physicists will recognize inCauchy density a natural profile of a radia-tion line or the cross-section formula for reso-nances in nuclear reactions, and they will re-member the Holtzmark distribution describ-ing fluctuations of electric field strength cre-ated by Poisson ensemble of point-like ions

8

in plasma and the fluctuations of the gravita-tion field of stellar systems. But at this stagetheir acquaintance with the stable laws usu-ally comes to the end. This should be causedby the circumstance that the stable densi-ties, as a rule, are not expressed in terms of

elementary functions: the above mentionedformulas fully exhaust a set of ”convenient”distributions. There are some more distribu-tions which are expressed through the knownspecial functions, like the following one (seeRef. [Garoni & Frankel, 2002])

g(2/3,0) (x) =1

2√

3π|x|−1 exp

(2

27x−2

)W−1/2,1/6

(4

27x−2

), (45)

where Wµ,ν (x) is the hypergeometric Whit-taker function. However, in the age of com-puters the lack of simple formulas has no sostrong importance. In fact, if simple or com-plex formulas can be processed by a com-puter, never mind, it computes fast and verywell, provided that we check the full process.

The properties of ”anomalous” stable dis-tributions are really remarkable. If we shallsum up n independent random variables,distributed under the same stable law, thebreadth of this new distribution will growproportionally to n1/α, and the breadth ofdistribution of arithmetic mean will grow asn1/α−1. For α = 1, when not only the vari-ance diverges, but also the expectation valuedoes not exist, the width of arithmetic meandistribution remains constant! And if the re-sults of your measuring are distributed bythe Cauchy law, the repetition of measuringcannot decrease the ”statistical error” in noway. If α < 1, increasing the number of termsin the sum of results in widening of ”samplemean” distribution! Clearly, the law of largenumbers does not work here because the ex-pectation value does not exist.

In summary we remark that the whole setof stable laws appear as limiting distributionsin the generalized central limiting theorem:any other laws cannot be limiting ones. Thisis, certainly, their most important advantage.

V. STABLE PROCESSES AND LEVY

MOTION

Having designated the random realiza-tion of the processes under considerationas L(α,β)(t), we write the condition of self-similarity as

L(α,β)(t) = t1/αY (α,β), (46)

where Y (α,β) is the strictly stable randomvariable, with parameters α and β. The setof such processes is sometimes called stableprocesses.

The random process {X (t) , t ∈ T} iscalled stable (strictly stable), if all its finite-dimensional distributions are stable (strictlystable). This definition generalizes the con-cept of Gaussian process, not restricted bythe requirements of homogeneity and inde-pendence of increments. Thus, however, it isnecessary to introduce the concept of mul-tivariate stable distribution or multivariatestable vector.

The random vector Y = (Y1, . . . , Ym) iscalled stable random vector in ℜm, if forany positive numbers λ1, λ2 there are posi-tive number λ and vector c ∈ ℜm, such that

λ1Y′ + λ2Y

′′ d=λY + c, (47)

where Y′,Y′′ are independent copies of therandom vector Y.

The stable random vector Y is calledstrictly stable if the last equality, with c = 0,holds true for any λ1 and λ2.

9

The stable random vector Y is called sym-metric stable random vector, if it satisfies therelation

P{Y ∈ A} = P{−Y ∈ A} (48)

for any Borel set A ⊂ ℜm. Similarly to theone-dimensional case, the symmetric vectoris strictly stable (the inverse statement, cer-tainly, is not true).

Recall the standard definition. The ran-dom process {L(α,β)(t), t ≥ 0} is called (stan-dard) α -stable Levy-motion with parameters0 < α ≤ 2, −1 ≤ β ≤ 1, if

1. L(α,β) (0) = 0 almost certainly;

2.{L(α,β) (t) , t ≥ 0

}is a process with in-

dependent increments;

3. L(α,β) (t+ τ) − L(α,β) (t)d= τ 1/αY (α,β)

at any t and τ .

For the sake of brevity we shall call it L(α,β)–process, then the Wiener process will be des-ignated as L(2,0)-process.

For better understanding of the principaldifference between L-processes with α = 2and α < 2, consider the third property forthe Wiener process, namely, the Lindebergcondition which reflects the continuity of itstrajectories. According to the continuity cri-terion of a random process [Loeve, 1963], forα = 2 and τ → 0 we have

P{∣∣∣L(2,0) (t+ τ ) − L(2,0) (t)

∣∣∣ ≥ ∆}

τ=

P{∣∣∣Y (2,0)

∣∣∣ ≥ ∆/√τ}

τ=

1√πτ

∞∫

∆/√

τ

e−z2/4dz. (49)

Evaluating the indeterminate form byl’Hopital’s rule, we obtain for τ → 0

1√π

d

dτ

∫ ∞

∆/√

τe−z2/4dz =

∆

2√πτ−3/2e−∆2/4τ → 0.

(50)

From Eq. (41) and using the property 3, forα < 2 and τ → 0, we have

P{∣∣∣L(α,β) (t+ τ) − L(α,β) (t)

∣∣∣ ≥ ∆}

τ=

P{τ 1/α

∣∣∣Y (α,β)∣∣∣ ≥ ∆

}

τ

=P{∣∣∣Y (α,β)

∣∣∣ ≥ ∆τ−1/α}

τ∝ ∆−α > 0. (51)

Thus, L(2,0) is the only L(α,β) process pos-sessing continuous trajectories. As it wasshown in Ref. [Seshadri & West, 1982], Levy

index α is the fractal dimensionality of theLevy process trajectories.

The size of a diffusion package for α < 2

10

grows with time faster than√t, namely, pro-

portionally to t1/α, and its shape differs fromthe Gaussian law. The variance is infinite,but nothing interferes in using any other mea-sure of width, for example, the width on thehalf of peak height, or the width of the in-terval containing some fixed probability. Forexample, if we consider the fractal momentof Levy process increments

⟨∣∣∣L(α,β) (t) − L(α,β) (0)∣∣∣δ⟩

=∫ +∞

−∞P (x, t)|x|δdx,

which is finite for 0 < δ < α, we immediatelyobtain from Eq. (19)

⟨∣∣∣L(α,β) (t) − L(α,β) (0)∣∣∣δ⟩∼ tδ/α,

and, as a result (see Ref. [Chechkin et al.,2002c])

⟨∣∣∣L(α,β) (t) − L(α,β) (0)∣∣∣δ⟩1/δ

∼ t1/α.

We consider now the stochastic model ofLevy flight superdiffusion.

VI. FRACTIONAL EQUATION FOR LEVY

FLIGHT SUPERDIFFUSION

If α = 2, the Levy motion becomes theBrownian motion with characteristic function

P (2,0)(k, t) = e−tk2

, (52)

obeying the differential equation

∂P (2,0)(k, t)

∂t= −k2P (2,0)(k, t), (53)

under initial condition

P (2,0)(k, 0) = 1. (54)

Factor −k2 is the Fourier image of the one-dimensional Laplace operator △1 = ∂2/∂x2.The inverse transformation yields the partialdifferential equation

∂P (2,0)(x, t)

∂t=∂2P (2,0)(x, t)

∂x2, (55)

with initial condition

P (2,0)(x, 0) = δ(x). (56)

For the symmetric Levy motion with an arbi-trary α, the corresponding expression of thecharacteristic function reads

P (α,0)(k, t) = e−t|k|α, (57)

and

∂P (α,0)(k, t)

∂t= −|k|αP (α,0)(k, t). (58)

Taking into account that −|k|α is the Fourier

image of the Riesz fractional operator △α/21 =

∂α/∂|x|α, we arrive at the fractional differen-tial equation

∂P (α,0)(x, t)

∂t=∂αP (α,0)(x, t)

∂|x|α . (59)

By means of the direct Fourier transforma-tion, one can be convinced of the validity oftwo following integral representations of theRiesz derivative

∂αf(x)

∂|x|α = − 1

K(α)

∫ +∞

−∞

f(x) − f(ξ)

|x− ξ|1+αdξ = − 1

K(α)

∫ +∞

0

2f(x) − f(x− ξ) − f(x+ ξ)

ξ1+αdξ.

(60)

Here

K(α) =π

Γ(α + 1) sin(πα/2). (61)11

Finally, in the case of the asymmetric Levymotion, the equation for probability distribu-tion becomes

∂P (α,β)(x, t)

∂t= D(α,β)

x P (α,β)(x, t). (62)

This equation contains the Feller fractionalspace derivative D(α,β)

x , which is determinedby the relation

D(α,β)x f(x) = −A(α, β)

K(α)

∫ +∞

−∞

1 + β sgn (x− ξ)

|x− ξ|1+α[f(x) − f(ξ)]dξ

= −A(α, β)

K(α)

∫ +∞

0

2f(x) − (1 + β)f(x− ξ) − (1 − β)f(x+ ξ)

ξ1+αdξ , (63)

where

A(α, β) = 1 + β2tg(απ/2). (64)

A more detailed consideration of fractionaldifferential equation for description of Levymotion can be found in Refs. [Saichev & Za-slavsky, 1997; Uchaikin, 1999, 2000, 2002,2003a, 2003b; Uchaikin & Zolotarev, 1999;Metzler & Klafter, 2000; Mainardi et al.,2001; Metzler & Nonnenmacher, 2002; Lenziet al., 2003; Gorenflo & Mainardi, 2005;Sokolov & Chechkin, 2005; Zaslavsky, 2005].

VII. LEVY WHITE NOISE

Let us come back to the Levy processes.The time derivative of the Levy process

ξ (t) = dL(t)/dt ≡ L(t) (65)

is a stationary random process and has anal-ogy to the Gaussian white noise, which is thetime derivative of the Wiener process. TheLevy process, in fact, is a generalized Wienerprocess.

Now we derive the characteristic func-tional Θt[u] of this Levy delta-correlatednoise. By definition, we have

Θt[u] =⟨exp

{i∫ t

0u (τ) ξ (τ) dτ

}⟩=⟨exp

{i∫ t

0u (τ) dL (τ)

}⟩

=

⟨exp

{i lim

δτ→0

n∑

k=1

u (ϑk) [L (τk) − L (τk−1)]

}⟩(66)

= limδτ→0

⟨n∏

k=1

exp {iu (ϑk) [L (τk) − L (τk−1)]}⟩

= limδτ→0

n∏

k=1

P (u(ϑk),∆τk),

where ϑk is some internal point of time interval (τk−1, τk) , δτ = maxk

∆τk, ∆τk = τk −τk−1

12

(τ0 = 0, τn = t), and P (u(ϑk),∆τk) is thecharacteristic function of increments. To ob-tain Eq. (66) we used the statistical inde-

pendence of increments of Levy process L (t).Further from Eqs. (12) and (66) we obtain

Θt[u] = limδτ→0

n∏

k=1

exp

{∆τk

∫ +∞

−∞

eiu(ϑk)x − 1 − iu (ϑk) sin x

x2ρ (x) dx

}

= exp

{limδτ→0

n∑

k=1

∆τk

∫ +∞

−∞

eiu(ϑk)x − 1 − iu (ϑk) sin x

x2ρ (x) dx

}

= exp

{∫ t

0dτ∫ +∞

−∞

eiu(τ)x − 1 − iu (τ) sin x

x2ρ (x) dx

}. (67)

Now we are going to derive a useful func-tional formula for the Levy white noise. Theformula to split the correlation between aGaussian random vector field ξ (r, t) and itsarbitrary functional R[ξ] was for the firsttime obtained by Furutsu [Furutsu, 1963] andNovikov [Novikov, 1965]. For Gaussian ran-dom process ξ (t) with zero mean it reads

〈ξ (t)R [ξ]〉 =∫K (t, τ)

⟨δR [ξ]

δξ (τ)

⟩dτ, (68)

where K (t, τ) = 〈ξ (t) ξ (τ)〉 is the correla-tion function of Gaussian noise ξ (t). We use

the generalization of Furutsu–Novikov for-mula (68), obtained by Klyatskin [Klyatskin,1974], for arbitrary functional Rt[ξ] of a non-Gaussian random process ξ (τ), defined onthe observation interval τ ∈ (0, t),

〈ξ (t)Rt[ξ + z]〉 =Φt [u]

iu (t)

∣∣∣∣∣u= δ

iδz

〈Rt [ξ + z]〉 .

(69)Here z (t) is arbitrary deterministic function,and Φt [u] = ln Θt [u]. From Eq. (67) we ob-tain the following expression for variationaloperator in Eq. (69)

Φt [u]

iu (t)=∫ +∞

−∞

eiu(t)x − 1 − iu (t) sin x

iu (t) x2ρ (x) dx =

∫ +∞

−∞

ρ (x)

x2dx∫ x

0[eiu(t)y − cos y]dy. (70)

Substituting this equation in Eq. (69) we arrive at

〈ξ (t)Rt [ξ + z]〉 =∫ +∞

−∞

ρ (x)

x2dx∫ x

0

[exp

{y

δ

δz (t)

}− cos y

]〈Rt [ξ + z]〉 dy . (71)

By inserting the operator of functional differentiation into the average in Eq. (71) and by

13

putting z = 0, we get finally

〈ξ (t)Rt [ξ]〉 =∫ +∞

−∞

ρ (x)

x2dx∫ x

0

[⟨exp

{y

δ

δξ (t)

}Rt [ξ]

⟩− 〈Rt [ξ]〉 cos y

]dy . (72)

VIII. DERIVATION OF KOLMOGOROV’S

EQUATION

Let us consider now the Langevin equationwith the Levy white noise source ξ(t)

X = f (X, t) + g (X, t) ξ (t) . (73)

By differentiating with respect to time thewell-known expression for probability density

of the random process X(t)

P (x, t) = 〈δ (x−X (t))〉 , (74)

and taking into account Eq. (73), we obtain

∂P

∂t= − ∂

∂x(f(x, t)P ) − ∂

∂xg(x, t) 〈ξ (t) δ (x−X (t))〉 . (75)

By using functional differentiation rulesand following the same procedure used in

Ref. [Hanggi, 1978], from Eq. (73) we get

δ

δξ (t)δ (x−X (t)) = − ∂

∂xg (x, t) δ (x−X (t)) . (76)

Thus, the variational operator δ/δξ (t) withrespect to the function δ (x−X (t)) is equiv-alent to the ordinary differential opera-tor −∂/∂x (g (x, t) ). Taking into account

Eqs. (72), (75), and (76), we obtain, after in-tegration, the following Kolmogorov’s equa-tion for nonlinear system (73) driven by Levywhite noise [Dubkov & Spagnolo, 2005]

∂P

∂t= −∂ [f(x, t)P ]

∂x+∫ +∞

−∞

ρ (z)

z2

[exp

{−z ∂

∂xg (x, t)

}− 1 + sin z

∂

∂xg (x, t)

]dz P. (77)

We analyze further some different kernelfunctions ρ (x) to obtain particular cases of

Kolmogorov’s equation (77), related to dif-ferent non-Gaussian white noise sources.

14

(a) As a first simple case we consider aGaussian white noise ξ(t). The correspond-ing kernel function is ρ (x) = 2Dδ (x), whereD is the noise intensity. After substitutingthis kernel in Eq. (77), we obtain the ordi-nary Fokker-Planck equation

∂P

∂t= − ∂

∂x(fP ) +D

∂

∂xg∂

∂x(gP ) . (78)

(b) For additive driving noise ξ (t),g (X, t) = 1 in Eq. (73), and the exponen-tial operator in Eq. (77) reduces to the spaceshift operator. As a result, we find

∂P

∂t= − ∂

∂x[f (x, t)P ] +

∫ +∞

−∞

ρ (z)

z2

[P (x− z, t) − P (x, t) + sin z

∂P (x, t)

∂x

]dz. (79)

Equation (79) is similar to the Kolmogorov-Feller equation for purely discontinuousMarkovian processes [Saichev & Zaslavsky,1997; Kaminska & Srokowski, 2004; Dubkov& Spagnolo, 2005]

∂P

∂t= ν

∫ +∞

−∞w (x− z)P (z, t) dz−νP (x, t) ,

(80)where w (x) is the probability density ofjumps step, and ν is the constant mean rateof jumps. By putting f (X, t) = 0, omitting

the term with sinz, and comparing Eq. (79)with Eq. (80) we find the kernel function forsuch a case

ρ (x) = νx2w (x) . (81)

For non–Gaussian additive driving forceξ(α) (t), with symmetric α-stable Levy dis-tribution, the kernel function reads ρ (x) =Q |x|1−α. As a result, Eq. (79) takes the fol-lowing form

∂P

∂t= − ∂

∂x[f (x, t)P ] +Q

∫ +∞

−∞

P (z, t) − P (x, t)

|x− z|α+1 dz (82)

and describes the anomalous diffusion in formof symmetric Levy flights.

In accordance with the definition of Rieszderivative (60), Eq. (82) can be written as

∂P

∂t= − ∂

∂x[f (x, t)P ] +D

∂αP

∂ |x|α (83)

where (see Eq. (61))

D = K (α)Q =πQ

Γ(α + 1) sin (πα/2). (84)

For the first time, the fractional Fokker-Planck equation (83) for Levy flights in the

potential profile U (x) (with −U ′ = f(x, t)was obtained directly from Langevin equa-tion

X = −U ′ (X) + ξ(α) (t) , (85)

by replacing the white Gaussian noise ξ (t) ≡ξ(2) (t) with the symmetric Levy α-stablenoise ξ(α) (t), in Refs. [Ditlevsen, 1999b;Yanovsky et al., 2000; Garbaczewski &Olkiewicz, 2000] (see also [Schertzer et al.,2001]). However, some attempts were under-taken before in Refs. [Fogedby, 1994a, 1994b,

15

1998; Jespersen et al., 1999]. Recently us-ing a different approach it was derived in[Dubkov & Spagnolo, 2005].

IX. STATIONARY PROBABILITY

DISTRIBUTIONS FOR LEVY FLIGHTS

First of all, we can try to evaluate thestationary probability distribution Pst (x) ofLevy flights in the potential profile U(x) fromEq. (83). Of course, this evaluation is impos-sible for any potential profile, but the poten-tial U(x) should satisfy some constraints. Itis better to apply Fourier transform to theintegro-differential equation (83) and to writethe equation for the characteristic function

P (k, t) =⟨eikX(t)

⟩=∫ +∞

−∞eikxP (x, t) dx.

(86)After simple manipulations we find (seeEq (58))

∂P

∂t= −ik

∫ +∞

−∞eikxU ′(x)P (x, t) dx−D |k|α P .

(87)For smooth potential profiles U (x), expand-ing in power series in a neighborhood of thepoint x = 0, we can rewrite Eq. (87) in theoperator form

∂P

∂t= −ikU ′

(−i ∂∂k

)P −D |k|α P . (88)

In particular, for stationary characteristicfunction, from Eq. (88) we get

U ′(−i ddk

)Pst − iD |k|α−1 sgn k · Pst = 0 ,

(89)where sgn k is the sign function. Unfortu-nately, one cannot solve analytically Eq. (89)for arbitrary potential U (x) and arbitraryLevy exponent α.

Let us consider, as in Ref. [Chechkin et al.,2002a], the symmetric smooth monostablepotential U (x) = γx2m/ (2m) (m = 1, 2, . . .).The Eq. (89), therefore, transforms into thefollowing differential equation of (2m− 1)-order

P(2m−1)st +(−1)m+1 β2m−1 |k|α−1 sgn k·Pst = 0 ,

(90)

where β = 2m−1

√D/γ. As it was proved by

analysis of Eq. (90) for small arguments kin Ref. [Chechkin et al., 2002a], the station-ary probability distribution Pst (x) has non-unimodal shape and power tails

Pst (x) ∼1

|x|2m+α−1 , |x| → ∞ . (91)

According to Eq. (91), we have a confinementof Levy flights (finite variance of particle’scoordinate) in the case when

2m > 4 − α. (92)

Because of the Levy index α ∈ (0, 2], aconfinement takes place for all values of α,starting from quartic potential (m = 2). Ex-act solution of Eq. (90) can be only obtainedfor the case of Cauchy noise source ξ(1) (t)(α = 1). Due to the symmetry of the char-acteristic function Pst (−k) = Pst (k), we canreduce Eq. (90) to a linear differential equa-tion with constant parameters

P(2m−1)st − (−1)m β2m−1Pst = 0 (k > 0) .

(93)From the corresponding characteristic equa-tion

λ2m−1 = (−1)m β2m−1, (94)

we select the roots with negative real part,which are meaningful from physical point ofview. The general solution of Eq. (93), there-fore, reads

16

Pst (k) =[(m−1)/2]∑

l=0

Al exp

{−β |k| cos

π (m− 2l − 1)

2m− 1

}· cos

(β |k| sin π (m− 2l − 1)

2m− 1− ϕl

),

(95)

where the quadratic brackets in the upperlimit of the sum [(m − 1)/2] denote the in-teger part of the enclosed expression. Theunknown constants Al and ϕl can be calcu-lated from the conditions

Pst (0) = 1, P(2j−1)st (+0) = 0, (96)

where j = 1, 2, . . . , m − 1. Now substitutingEq. (95) in Eq. (96) we obtain

[(m−1)/2]∑

l=0

Al cosϕl = 1,[(m−1)/2]∑

l=0

Al cos

[π (2j − 1) (m+ 2l)

2m− 1− ϕl

]= 0 (j = 1, 2, . . . , m− 1).

(97)

Making the reverse Fourier transform inEq. (95) we find the stationary probability

distribution (SPD) of the particle coordinate

Pst (x) =β

π

[(m−1)/2]∑

l=0

Al

x2 cos[

π(m−2l−1)2m−1

+ ϕl

]+ β2 cos

[π(m−2l−1)

2m−1− ϕl

]

x4 − 2x2β2 cos π(4l+1)2m−1

+ β4. (98)

The parabolic potential profile U (x) = γx2/2corresponds to a linear system (73). In thissituation, from Eqs. (97) and (98) we easilyobtain the following obvious result

Pst (x) =β

π (x2 + β2), (99)

that is due to the stability of the Cauchy dis-tribution (99), the probabilistic characteris-tics of driving noise increments (see Eq. (42))and Markovian process X (t) are similar (seealso Ref. [West & Seshadri, 1982]).

For quartic potential (m = 2), from the setof equations (97), we find A0 = 2/

√3 and

ϕ0 = π/6. Substituting these parameters inEq. (98) we obtain

Pst (x) =β3

π (x4 − x2β2 + β4), (100)

which coincides, for β = 1, with the resultobtained in Refs. [Chechkin et al., 2002a,2006]. The plots of stationary probabilitydistributions (100) for Levy flights in sym-

17

metric quartic potential, for different valuesof the parameter β, are shown in Fig. 1.

-2 -1 1 2

0.6

1

0

1

2

3

x

Pst(x)

FIG. 1 Stationary probability distributions for

Levy flights in symmetric quartic potential

U(x) = γx4/4 for different values of dimen-

sionless parameter β: (1) β = 0.5, (2) β = 1,

(3) β = 1.5.

The superdiffusion in the form of Levyflight gives rise to a bimodal stationary prob-ability distribution, when the particle movesin a monostable potential. This bimodal dis-tribution is a peculiarity of Levy flights. Infact the ordinary diffusion of the Brownianmotion is characterized by unimodal SPD.The SPD of superdiffusion has two maximaat the points x = ±β/

√2, with the value

(Pst)max = 4/ (3πβ). Since the value of theminimum is Pst (0) = 1/ (πβ), the ratio be-tween maximum and minimum value is con-stant and equal to 4/3. The variance of theparticle coordinate, obtained from Eq. (1) isfinite: 〈X2〉st = β2. As a result, the prob-ability distribution becomes more wide with

increasing parameter β = 3

√D/γ, that is with

decreasing the steepness γ of the quartic po-tential profile, or with increasing the noiseintensity D.

A detailed analysis of the solution of thedifferential equation (90), for arbitrary Levy

index α and quartic potential (m = 2) wasperformed in Refs. [Chechkin et al., 2002a,2004]. In Fig. 2 the profiles of SPD (ob-tained by an inverse Fourier transformation)in symmetric quartic potential are shown forthe different Levy indices from α = 1, at the

FIG. 2 Forms of stationary probability distribu-

tion in the symmetric quartic potential for dif-

ferent Levy indices, from α = 1 till α = 2. From

Ref. [Chechkin et al., 2002a].

top of the figure, up to α = 2 at the bot-tom [Chechkin et al., 2002a]. It is seen thatthe bimodality is most strongly expressed forα = 1 (Cauchy stable noise source). By in-creasing the Levy index, the bimodal profilesmoothes out, and, finally, it turns into a uni-modal one at α = 2, recovering the Boltz-mann distribution.

Carrying out analogous procedure we ob-tain the stationary probability distributionsfor the cases m = 3, 4, 5 [Dubkov & Spagnolo,2007]

Pst (x) =β5

π (x2 + β2) (x4 − 2β2x2 cosπ/5 + β4),

18

Pst (x) =β7

π (x4 − 2β2x2 cos π/7 + β4) (x4 + 2β2x2 cos 2π/7 + β4), (101)

Pst (x) =β9

π (x2 + β2) (x4 − 2β2x2 cosπ/9 + β4) (x4 + 2β2x2 cos 4π/9 + β4).

The plots of distributions (101), for dif-ferent values of parameter β, are respectivelyshown in Figs. 3–5.

It must be emphasized that according toFigs. 3–5, these distributions remain bimodaland have the same tendency with increas-ing β, but the ratio between maximum andminimum increases with increasing m. FromEqs. (100) and (101) we see that the secondmoment of the particle coordinate is finite form ≥ 2, which confirms the inequality (92).This means that there is a confinement ofthe particle motion due to the steep potentialprofile, even if the particle moves accordingto a superdiffusion in the form of Levy flights.The presence of two maxima is a peculiarityof the superdiffusion motion. Because of thefast diffusion due to Levy flights, the parti-cle reaches very quickly regions near the po-tential walls on the left or on the right withrespect to the origin x = 0. Then the par-ticle diffuses around this position, until a

-2 -1 1 2

1

0

0.6

Pst(x)

x

1

2

3


Levy flights in symmetric potential U(x) =

γx6/6 for different values of dimensionless pa-

rameter β: (1) β = 0.5, (2) β = 1, (3) β = 1.5.

new flight moves it in the opposite direction

-2 -1 1 2

0.6

1.2

0

Pst(x)

x

1

2

3




rameter β: (1) β = 0.5, (2) β = 1, (3) β = 1.5.

to reach the other potential wall. As a re-sult, the particle spends a large time in somesymmetric areas with respect to the pointx = 0, differently from the Brownian diffu-sion in monostable potential profiles. Thesesymmetric areas lie near the maxima of thebimodal SPD. For fixed D and m, these max-ima are closer or far away the point x = 0 de-pending on the greater or smaller steepness γof the potential profile. This corresponds to agreater or smaller confinement of the particlemotion. Of course, such confinement is morepronounced for greater m, that is for steeperpotential profiles.

On the basis of Eqs. (99)–(101) and theknown behavior of density tails (91), we canwrite the general expressions for stationaryprobability distribution in the case of po-tential U (x) = γx2m/ (2m) with odd m =2n+ 1 [Dubkov & Spagnolo, 2007]

19

-2 -1 1 2

0.40.6

1

1.2

0

Pst(x)

x

1

2

3




rameter β: (1) β = 0.5, (2) β = 1, (3) β = 1.5.

Pst (x) =β4n+1

π (x2 + β2)

n−1∏

l=0

1

x4 − 2β2x2 cos [π (4l + 1) / (4n+ 1)] + β4, (102)

and even m = 2n

Pst (x) =β4n−1

π

n−1∏

l=0

1

x4 − 2β2x2 cos [π (4l + 1) / (4n− 1)] + β4. (103)

The strong proof of non-unimodality ofthe SPD for symmetric monostable potentialU (x) = |x|c /c in the case c > 2 was givenin Ref. [Chechkin et al., 2004]. Indeed, fromEq. (83) we have for SPD

d

dx

[|x|c−1 sgn x · Pst

]+D

dαPst

d |x|α = 0. (104)

As a result, from Eqs. (104) and (60) at thepoint x = 0 we obtain

dαPst

d |x|α∣∣∣∣∣x=0

=∫ +∞

−∞

Pst (−z) − Pst (0)

|z|α+1 = 0.

(105)

Because of the symmetry of the SPD Pst (x),Eq. (105) gives

∫ ∞

0

Pst (z) − Pst (0)

zα+1= 0. (106)

For unimodal probability distribution withthe maximum at the origin, the integral inthe left side of Eq. (106) should be negative,which contradicts Eq. (106).

The estimation of bifurcation time fortransition from unimodal initial distributionto bimodal stationary one for the quartic po-tential (c = 4) was done in Refs. [Chechkinet al., 2004, 2006]. The dependence of this

20

bufurcation time t12 from Levy index α isplotted in Fig. 6. Also authors proved an ex-

FIG. 6 Bifurcation time t12 versus Levy index

α for quartic potential. Black dots: bifurca-

tion time deduced from the numerical solution of

the fractional Fokker-Planck equation. Dashed

and solid lines: two subsequent approximations.

From Refs. [Chechkin et al., 2004, 2006].

istence of a transient trimodal state betweeninitial unimodal and final bimodal ones. Thisevolution, shown in Fig. 7, can be only ob-served for monostable potential with c > 4and for fixed values of the Levy index α.The corresponding bifurcation times of tran-sitions t13 (unimodal → trimodal) and t32(trimodal → bimodal) versus Levy index α,with potential exponent c = 5.5, are plottedin Fig. 8.

X. BARRIER CROSSING

The problem of escape from metastablestates investigated by Kramers [Kramers,1940] is ubiquitous in almost all scientificareas [Hanggi et al., 1990; Spagnolo et al.,2007]. Since many stochastic processes donot obey the Central Limit Theorem, the cor-responding Kramers escape behavior will dif-fer. An interesting example is given by the α-stable noise-induced barrier crossing in longpaleoclimatic time series [Ditlevsen, 1999a].Another new application is the escape fromtraps in optical or plasma systems [Fajans &

FIG. 7 The evolution of the probability dis-

tribution for α = 1.2 and c = 5.5 from

unimodal through trimodal to bimodal. From

Refs. [Chechkin et al., 2004, 2006].

FIG. 8 Bifurcation times t13 and t32 versus Levy

index α for the power potential with the exponent

c = 5.5. From Ref. [Chechkin et al., 2004].

Schmidt, 2004].The main tools to investigate the barrier

crossing problem for Levy flights are the firstpassage times, crossing times, arrival timeand residence times. We should emphasizethat the problem of mean first passage time(MFPT) meets with some difficulties becausefree Levy flights represent a special class of

21

discontinuous Markovian processes with infi-nite mean squared displacement. First of all,the fractional Fokker-Planck equation (83) isintegro-differential, and the conditions at ab-sorbing and reflecting boundaries differ fromthe usual conditions for ordinary diffusion.Superdiffusion motion is characterized by thepresence of jumps, and, as a result, a parti-cle can reach instantaneously the boundaryfrom arbitrary position. One can mentionsome erroneous results for Levy flights ob-tained in Ref. [Gitterman, 2000], because au-thor used the traditional conditions at twoabsorbing boundaries (see the related corre-spondence [Yuste & Lindenberg, 2004; Git-terman, 2004]). The numerical results for thefirst passage time of free Levy flights con-fined in a finite interval were presented inRef. [Dybiec et al., 2006]. The complexityof the first passage time statistics (mean firstpassage time, cumulative first passage timedistribution) was elucidated together with adiscussion of the proper setup of correspond-ing boundary conditions, that correctly yieldthe statistics of first passages for these non-Gaussian noises. In particular, it has beendemonstrated by numerical studies that theuse of the local boundary condition of vanish-ing probability flux in the case of reflection,and vanishing probability in the case of ab-sorbtion, valid for normal Brownian motion,no longer apply for Levy flights. This in turnrequires the use of nonlocal boundary condi-tions. Dybiec with co-authors found a non-monotonic behavior of the MFPT for two ab-sorbing boundaries, with the maximum beingassumed for α = 1 (see Fig. 9), in contrastwith a monotonic increase for reflecting andabsorbing boundaries.

According to the Kramers law, the proba-bility distribution of the escape time from apotential well with the barrier of height U0,has the exponential form

p (t) =1

Tcexp

{− t

Tc

}(107)

with mean crossing time

FIG. 9 Mean first passage time versus Levy in-

dex α of confined motion between two absorb-

ing boundaries driven by stable symmetric Levy

white noise. From Ref. [Dybiec et al., 2006].

Tc = C exp{U0

D

}, (108)

where C is some positive prefactor and Dis the noise intensity. The problem howthe stable nature of Levy flight processesgeneralizes the barrier crossing behavior ofthe classical Kramers problem was investi-gated, both numerically and analytically, inRef. [Chechkin et al., 2003b, 2005, 2006,2007]. Authors considered Levy flights in abistable potential U (x) by numerical solu-tion of Eq. (85). It was shown that althoughthe survival probability decays again expo-nentially as in Eq. (107), the mean escapetime Tc has a power-law dependence on thenoise intensity D

Tc ≃C(α)

Dµ(α), (109)

where the prefactor C and the exponent µ de-pend on the Levy index α. Using the Fouriertransform, i.e. Eq. (88), the mean escaperate was found for large values of 1/D inthe case of Cauchy stable noise (α = 1) in

22

the framework of the constant flux approx-imation across the barrier. The probabilitylaw and the mean value of escape time froma potential well for all values of the stabil-ity index α ∈ (0, 2), in the limit of smallLevy driving noise, were also determined inthe paper [Imkeller & Pavlyukevich, 2006] bypurely probabilistic methods. Escape timeshad the same exponential distribution (107),and the mean value depends on the noise in-tensity D in accordance with Eq. (109) withµ(α) = 1 and pre-factor C depending on αand the distance between the local extremaof the potential.

The barrier crossing of a particle drivenby symmetric Levy noise of index α and in-tensity D for three different generic typesof potentials was numerically investigated inRef. [Chechkin et al., 2007]. Specifically: (i)a bistable potential, (ii) a metastable poten-tial, and (iii) a truncated harmonic potential,were considered. For the low noise intensityregime, the result of Eq. (109) was recovered.As it was shown, the exponent µ(α) remainsapproximately constant, µ ≈ 1 for 0 < α < 2;at α = 2 the power-law form of Tc changesinto the exponential dependence (108). Itexhibits a divergence-like behavior as α ap-proaches 2. In this regime a monotonous in-crease of the escape time Tc with increasing α(keeping the noise intensity D constant) wasobserved. For low noise intensities the escapetimes correspond to barrier crossing by multi-ple Levy steps. For high noise intensities, theescape time curves collapse for all values of α.At intermediate noise intensities, the escapetime exhibits non-monotonic dependence onthe index α as in Fig. 9, while still retains theexponential form of the escape time density.

The first arrival time is an appropriate pa-rameter to analyze the barrier crossing prob-lem for Levy flights. If we insert in frac-tional Fokker-Planck equation (83) a δ-sink ofstrength q (t) in the origin, we obtain the fol-lowing equation for the non-normalized prob-ability density function P (x, t)

∂P

∂t=

∂

∂x[U ′ (x)P ] +D

∂αP

∂ |x|α − q (t) δ (x) ,

(110)

from which by integration over all space wemay define the quantity

q (t) = − d

dt

∫ +∞

−∞P (x, t) dx, (111)

which is the negative time derivative of thesurvival probability. According to definition(111), q (t) represents the probability den-sity function of the first arrival time: oncea random walker arrives at the sink it isannihilated. As it was shown in the paper[Chechkin et al., 2003b] for free Levy flights(U (x) = 0), the first arrival time distributionhas a heavy tail

q (t) ∼ t1/α−2 (112)

with exponent depending on Levy indexα (1 < α < 2) and differing from universalSparre Andersen result [Sparre Andersen,1953, 1954] for the probability density func-tion of first passage time for arbitrary Marko-vian process

p (t) ∼ t−3/2. (113)

In the Gaussian case (α = 2), the quantity(112) is equivalent to the first passage timeprobability density (113). From a randomwalk perspective, this is due to the fact thatindividual steps are of the same increment,and the jump length statistics therefore en-sures that the walker cannot hop across thesink in a long jump without actually hittingthe sink and being absorbed. This behaviorbecomes drastically different for Levy jumplength statistics: there, the particle can eas-ily cross the sink in a long jump. Thus, beforeeventually being absorbed, it can pass by thesink location numerous times, and thereforethe statistics of the first arrival will be differ-ent from that of the first passage. The result

23

(113) for Levy flights was also confirmed nu-merically in the paper [Koren et al., 2007].

At last, the nonlinear relaxation time tech-nique is also suitable for investigations ofLevy flights temporal characteristics. Ac-cording to definition, the mean residence timein the interval (L1, L2) reads

T (x0) =∫ ∞

0dt∫ L2

L1

P (x, t| x0, 0) dx, (114)

where x0 is the initial position of all parti-cles (x0 ∈ (L1, L2)) and P (x, t| x0, 0) is theprobability density of transitions. We do notneed to think about the boundary conditionsin this case because we are concerned withthe overall time spent by the particle in thefixed interval. Changing the order of integra-tion in Eq. (114) we obtain

T (x0) =∫ L2

L1

Y (x, x0, 0) dx, (115)

where Y (x, x0, s) is the Laplace trans-form of the transient probability densityP (x, t| x0, 0)

Y (x, x0, s) =∫ ∞

0P (x, t| x0, 0) e−stdt.

(116)

Making the Laplace transform in the frac-tional Fokker-Planck equation (83) andtaking into account the initial conditionP (x, 0|x0, 0) = δ (x− x0), we get

d

dx[U ′ (x) Y ] +D

dαY

d |x|α − sY = −δ (x− x0) .

(117)

If we put s = 0 in Eq. (117) and make theFourier transform we obtain

U ′(−i ddk

)Y − iD |k|α−1 sgn (k) Y =

eikx0

ik,

(118)where

Y (k, x0) =∫ +∞

−∞Y (x, x0, 0) e+ikxdx. (119)

After solving Eq. (118) we can calculate themean residence time as (see Eqs. (115) and(119))

T (x0) =1

2πi

∫ +∞

−∞

e−ikL1 − e−ikL2

kY (k, x0) dk.

(120)

Equations (118) and (120) are useful tools toanalyze the temporal characteristics of Levyflights in different potential profiles U (x).

XI. CONCLUSIONS

In this tutorial paper, after some shorthistorical notes on normal diffusion and su-perdiffusion, we introduce the Levy flights asself-similar Levy processes. After the def-inition of the strictly stable random vari-ables, the subfamily of the Levy motionis introduced with the fractional differen-tial equation for Levy flight superdiffusion.We used then functional analysis approachto derive the fractional Fokker-Planck equa-tion directly from Langevin equation withsymmetric α-stable Levy noise. This ap-proach allows to describe anomalous diffu-sion in the form of Levy flights. We obtainedthe general formula for stationary probabil-ity distribution of superdiffusion in symmet-ric smooth monostable potential for Cauchydriving noise. All distributions have bimodalshape and become more narrow with increas-ing steepness of the potential or with de-creasing noise intensity. We found that thevariance of the particle coordinate is finitefor quartic potential profile and for steeperpotential profiles, that is a confinement ofthe particle in a superdiffusion motion in theform of Levy flights. As a result, we can eval-uate the power spectral density of a station-ary motion. We have also discussed recentlyobtained analytical and numerical results fortime characteristics of Levy flights. Specialattention was given for some difficulties withformulation of the correct boundary condi-tions for mean first passage time problem.

24

As it was shown, the arrival and residencetimes are more appropriate characteristics forinvestigations of Levy flights in different po-tential profiles.

Acknowledgments

This work has been supported by MIUR,CNISM, and by Russian Foundation for Ba-sic Research (projects 07-01-00517 and 08-02-01259).

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