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First passage behaviour of fractional Brownianmotion in two-dimensional wedge domains

J.-H. Jeon, A. V. Chechkin and R. Metzler

EPL, 94 (2011) 20008

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EPL, 94 (2011) 20008 www.epljournal.org

doi: 10.1209/0295-5075/94/20008

First passage behaviour of fractional Brownian motion

in two-dimensional wedge domains

J.-H. Jeon1, A. V. Chechkin2,3 and R. Metzler1,4(a)

1Department of Physics, Technical University of Munich - James-Franck Straße, 85747 Garching, Germany, EU2Akhiezer Institute for Theoretical Physics NSC KIPT - Akademicheskaya Str.1, 61108 Kharkov, Ukraine3 School of Chemistry, Tel Aviv University - Ramat Aviv, Tel Aviv 69978, Israel4Department of Physics, Tampere University of Technology - FI-33101 Tampere, Finland, EU

received 17 February 2011; accepted in final form 21 March 2011published online 18 April 2011

PACS 05.40.-a – Fluctuation phenomena, random processes, noise, and Brownian motionPACS 02.50.Ey – Stochastic processesPACS 87.10.Mn – Stochastic modeling

Abstract – We study the survival probability and the corresponding first passage time densityof fractional Brownian motion (FBM) confined to a two-dimensional open wedge domain withabsorbing boundaries. By analytical arguments and numerical simulation we show that in thelong-time limit the first passage time density scales as ℘Θ(t)≃ t

−1+π(2H−2)/(2Θ) in terms of theHurst exponent H and the wedge angle Θ. We discuss this scaling behaviour in connection withthe reaction kinetics of FBM particles in a one-dimensional domain.

Copyright c© EPLA, 2011

Introduction. – The first passage defines the momentat which a random quantity crosses a given threshold valuefor the first time. For instance, this could be the time whena random walker first leaves a defined region of space.Other examples include phenomena as diverse as diffusion-limited reactions [1], cyclisation of a polymer [2,3], stockmarket dynamics [4], or search problems [5]. The conceptof first passage statistics is of quite ubiquitous importancein the theory of stochastic processes [6].First passage characteristics have been investigated for

various geometries. Thus, on semi-infinite domains theprobability density of the first passage generally displaysa power-law tail, and the associated mean first passagetime diverges [6]. Recently, remarkable progress has beenachieved, relating the underlying geometry to the resultingfirst passage behaviour [7].In the case of normal diffusion, characterised by the

linear growth 〈r(t)2〉 ≃ t of the ensemble-averaged meansquared displacement, the associated first passage prob-lems are obtained by solving the associated diffusion equa-tion with given boundary conditions or, for simpler geome-tries, by the method of images [6]. For anomalous diffusionwith the non-linear growth [8]

〈r2(t)〉 ≃ t2H , 2H �= 1, (1)

(a)E-mail: metz@ph.tum.de

of the mean squared displacement, no general theoryexists to obtain the corresponding first passage timestatistics. Only for certain classes of anomalous diffusionmodels analytical approaches for the first passage prop-erties are known. Examples include the continuous timerandom walk model [9] governed by a heavy-tailed waitingtime distribution with diverging characteristic waitingtime [10–13] or by a power-law jump length distribu-tion with diverging variance (Levy flights) [14]; as well asthe diffusion on fractal media [15]. Importantly the firstpassage time behaviour differs between different stochasticprocesses despite sharing the same form (1) of the meansquared displacement with dynamic exponent 2H.Fractional Brownian motion (FBM), originally intro-

duced by Kolmogorov [16] and later re-discovered byMandelbrot and van Ness [17], is an ergodic [3,18] gener-alised Gaussian process, whose increments in one spatialdimension,

dx(t) = ξH(t)dt, (2)

are stationary and normally distributed but dependent.Here the quantity ξH(t) represents fractional Gaussiannoise with zero mean (〈ξH(t)〉= 0) and autocorrelation

〈ξH(t1)ξH(t2)〉 = 2KHH(2H − 1)|t1− t2|

2H−2

+4KHH|t1− t2|2H−1δ(t1− t2). (3)

20008-p1

J.-H. Jeon et al.

Θθ

r

x3x2x1

(b)

(a)

particle 1 particle 2 particle 3

x

y

Fig. 1: (Colour on-line) (a) Stochastic motion of a particleconfined to a two-dimensional wedge of opening angle Θ. Theparticle motion at time t follows r(t) = (r(t), θ(t)) in polarcoordinates. The excursion is terminated when the particle hitsone of the absorbing walls located at θ= 0 or Θ, the momentof first passage. (b) Diffusion-limited reaction of three particlesthat move according to FBM in one-dimensional space.

KH of dimension [KH ] = cm2/s2H is the anomalous

diffusion coefficient. The fractional Gaussian noise isnegatively (antipersistence) or positively (persistence)correlated for the cases of subdiffusion and superdiffusion,respectively. FBM has been widely used to describeanomalous diffusion phenomena including annual riverdischarge [19], stock market dynamics [20], the motionof tracer particles in crowded environments [21–24],or single-file diffusion [25]. Despite its popularity anddeceivingly simple definition in terms of fractionalGaussian noise the exact stochastic properties of FBMare not well understood. The first passage properties areknown only for the one-dimensional semi-infinite domain.The associated long-time asymptotic form of the firstpassage time density ℘(t)∼ tH−2 was conjectured by Dingand Yang from scaling arguments [26], and rigorouslyproved by Molchan [27]. Recently, it was shown that theWilemskii-Fixmann approximation, originally introducedfor the description of polymer cyclisation, producesthe first passage time behaviour and barrier crossingsof particles driven by fractional Gaussian noise [28].The difficulty to analytically access FBM’s first passageproperties is related to the fact that FBM is a stronglynon-Markovian process, which does not fall into the classof semi-martingales [29].Here we analyse the first passage time behaviour of two-

dimensional FBM confined to a wedge domain of openingangle Θ (fig. 1). This problem is of particular interest asthe reaction kinetics of three diffusive particles in one-dimensional space can be mapped on this problem [6,30].In the case of ordinary Brownian motion (H = 1/2), thewedge problem can be solved exactly using the Green’sfunction formalism [6]. Based on analytical and numericalarguments we conjecture the first passage behaviour ofFBM for arbitrary Hurst exponent in a wedge geometry.

In the following section we define multi-dimensionalFBM and the first passage in a wedge domain. We thenreview the Green’s function formalism for the Browniancase and obtain analytical forms for the first passagetime quantities. We proceed to present our main results,the asymptotic scaling forms of the first passage timebehaviour as function of H and Θ. These are numericallyconfirmed in the subsequent section. We conclude with adiscussion related to reaction kinetics of FBM particles.

Fractional Brownian motion in wedge domain. –

Consider the random walk of a particle confined to a two-dimensional wedge domain of opening angle Θ, see fig. 1.The particle starts at a position r0 inside the wedge andmoves within the domain until eventually it hits one of theabsorbing boundaries at a time t for the first time. For thisprocess, we obtain the survival probability

SΘ(t) =

∫

W

G(r, t)dr (4)

on the wedge domainW , and the first passage time density

℘Θ(t) =−dS (t)

dt. (5)

In eq. (4), G(r, t) is the probability density to find theparticle at position r at time t. Due to the presence ofthe absorbing boundaries, G(r, t) is not normalised, asexpressed by the survival probability, which decays fromSΘ(0) = 1 to limt→∞SΘ(t) = 0.Two-dimensional FBM is defined as a superposition of

independent FBMs for each Cartesian coordinate [31–33]:

r(t) =

2∑

i=1

∫ t

0

dt′ξHi (t′)xi+ r0, (6)

where xi is the unit vector in Cartesian direction i (i=x, y), and ξHi (t) is the fractional Gaussian noise. Due tothis definition it is clear that for a quadrant geometrywith Θ= π/2 the absorption to either wall decouples. Wenow first address the case of normal Brownian motion(H = 1/2), for which exact results for the survival andfirst passage distributions can be found.

First passage process for H = 1/2. – For normalBrownian motion, the diffusion in a wedge domain isdescribed in terms of the Green’s function G(r, θ; t), thatsatisfies the diffusion equation in polar coordinates,

∂

∂tG(r, θ; t) =K1/2

(

∂2

∂r2+1

r

∂

∂r+1

r2∂2

∂θ2

)

G(r, θ; t).

(7)The associated boundary conditions are G(r, 0; t) =G(r,Θ; t) = 0, representing the absorbing walls. Thesolution G is completely specified by the initial conditionG(r, θ; 0) = δ(r− r0)δ(θ− θ0)/r0. At long times t, thesolution G can be approximated by [6]

G(r, θ; t)≃π sin

(

πθΘ

)

4K1/2Θte−(r

2+r20)/(4K1/2t)Iπ/Θ

(

rr02K1/2t

)

,

(8)

20008-p2

First passage of FBM in 2D wedge domain

where Iν(z) is the modified Bessel function of the firstkind, which can be expressed by the series expansion

Iν(z) = (z/2)ν∞∑

k=0

(z2/4)k

k!Γ(ν+ k+1). (9)

The survival probability (4) in polar coordinates is then

SΘ(t) =

∫ Θ

0

∫

∞

0

rG(r, θ; t)drdθ, (10)

from which the property SΘ(0) = 1 follows directly fromthe sharp initial condition. Using the approximationsIν(z)≈ (z/2)

ν/Γ(1+ ν) and e−r2

0/(4K1/2t) ≈ 1 for the long-

time limit in eq. (8), one obtains the scaling expressions

SΘ(t)≃

(

r0√

K1/2

)π/Θ

t−π/(2Θ), (11a)

℘Θ(t)≃π

2Θ

(

r0√

K1/2

)π/Θ

t−1−π/(2Θ). (11b)

Notably, the wedge angle Θ enters the scaling exponentsinverse-proportionally; survival and first passage distribu-tions decay faster for decreasing wedge angle, as it should.We observe an interesting crossover as function of Θ: aslong as Θ� π/2 the mean first passage time

T =

∫

∞

0

t℘Θ(t)dt (12)

diverges, T →∞. In particular, in the half-space limitΘ= π, the process reduces to the first passage in a one-dimensional, semi-infinite geometry (along the y-axis),and we find the usual Levy-Smirnov scaling ℘(t)≃ t−3/2.When the wedge angle is smaller, Θ<π/2, the meanfirst passage time is finite, reflecting the much higherprobability to hit one of the two walls of the wedge. Thus,for Θ= π/2, the absorption to either of the two walls canbe viewed as two decoupled Brownian random walks inx- and y-direction. The resulting survival probability isthen given as the product of two one-dimensional survivalfunctions corresponding to Θ= π:Sπ/2(t) =Sπ(t)

2 ≃ t−1,for which the mean first passage time is logarithmicallydivergent. Remarkably, the scaling relation

SΘ/2(t)≃S2Θ(t) (13)

holds more generally for any Θ, see relations (11).

First passage process for H �= 1/2. – Let us nowaddress the case of general Hurst exponent, H �= 1/2, forwhich no analogue to the above Green’s function methodis known. While there exist dynamic equations for FBMin the literature [34,35] of the form

∂

∂tG (r, t) =K(t)∇2G (r, t) (14)

with the time-dependent diffusion coefficient K(t)∼ t2H−1, this description cannot fully specify the behav-iour of FBM in the presence of non-natural boundaryconditions. To see this, we follow the procedure of thepreceding section, and find the first passage behaviours

SG

Θ (t)∝ t−π(2H)/(2Θ), (15a)

℘G

Θ(t)∝ t−1−π(2H)/(2Θ). (15b)

While, naturally, the Brownian case H = 1/2 is consis-tent with the results of the previous section, theresults (15) are inconsistent with Molchan’s result℘(t)≃ tH−2 for a completely open wedge, Θ= π. There-fore, for arbitrary wedge angle Θ and Hurst exponentH the distributions (15) cannot be correct. In fact,the same inconsistency is found when one naivelyapplies the method of images to the free space solution(4πKHt

2H)−1/2 exp(−x2/[4KHt2H ]).

We now argue in favour of a conjecture for the correctscaling forms of the first passage time quantities. Let usstart by recollecting the known special cases, for whichanalytical results are available:i) When Θ= π, the first passage time quantities must

converge to Molchan’s result for a one-dimensional semi-infinite domain [27]:

Sπ(t)≃ tH−1, (16a)

℘π(t)≃ tH−2, (16b)

valid in the long-time limit t≫ 1.ii) When Θ= π/2, due to the independence of the x and

y motion the survival probability, by the above argument,is given as the product of two one-dimensional survivalprobabilities, Sπ/2 =S 2

π , and thus

Sπ/2(t)≃ t2H−2, (17a)

℘π/2(t)≃ t2H−3. (17b)

iii) For H = 1/2, we know the analytical results (11)with the full Θ-dependence.Assuming that the first passage time exponents are

simple combinations of the Hurst exponent H and thewedge angle Θ, to satisfy the above three special caseswe conjecture the unique scaling form

SΘ(t)≃ tπ(2H−2)/(2Θ), (18a)

℘Θ(t)≃ t−1+π(2H−2)/(2Θ). (18b)

These results imply that the survival decays faster forslower diffusion (i.e., lower value of H), in analogy withMolchan’s result and findings for the barrier crossing ofFBM [28]. Note that the inconsistent results (15) based onthe dynamic equation (14) follow from our results throughthe substitution H→ 1−H.

20008-p3

J.-H. Jeon et al.

Simulations. – To numerically verify the conjec-ture (18) for the survival probability and the first passagetime density, we performed simulations for various wedgeangles and Hurst exponents, and compared the resultswith eqs. (18). For this comparison, we here focus onthe survival probability, which generally shows betterstatistics.Our simulations procedure is as follows. A wedge of

opening angle Θ is constructed by imposing two absorbingwalls at y= 0 and y= (tanΘ)x in the x-y plane (fig. 1(a)).At t= 0, the particle is located at r0 = (r0 cos

Θ2 , r0 sin

Θ2 )

on the line bisecting the wedge. For t > 0, the particleundergoes two-dimensional FBM in discrete time indexedby n, following the rule

r(tn) =(

n∑

m=1

ξH1 (tm)+ r0 cosΘ

2,n∑

m=1

ξH2 (tm)+ r0 sinΘ

2

)

.

(19)

Here, the two independent fractional Gaussian noise func-tions ξH1 (t) and ξ

H2 (t) are generated by the Hosking

method [32,36]. The simulation is terminated when theparticle escapes the wedge domain for the first time. Thisis the survival time for the particle in the given run. Fromrepeated runs we construct the survival probability SΘ(t)as the accumulated number of particles surviving untiltime t. For each given wedge angle and Hurst exponent,the survival probability SΘ(t) was obtained from 100000runs and for initial radius r0 = 0.25.To confirm that our simulations procedure produces

correct results for the survival probability, we first considerthe two special cases of wedge angles Θ= π and π/2, andcompare them to the predicted scaling behaviours (16a)and (17a). Figure 2 shows on the left the distributions ofSΘ(t) obtained from simulations of a fully open wedge,Θ= π, for Hurst exponents H = 1/4 (subdiffusion), 1/2(normal diffusion), and 3/4 (superdiffusion). Clearly, thedistributions follow the predicted scaling behaviours forthis case corresponding to the one-dimensional, semi-infinite domain. Similarly, for the case of a rectangularwedge (Θ= π/2) on the right of fig. 2 we find excellentverification of our simulations method, compared to thepredicted behaviour Sπ/2(t)∼ t

2H−2 for all values of H.We now investigate whether our conjecture (18a) for

the survival probability also holds for other wedge angles.In our simulation, the survival probabilities for wedgeangles Θ= π/4, 3π/4, 5π/4, and 7π/4 were investigated,for Hurst exponents H = 1/8, 1/4, 1/2, 3/4, and 5/6. Asdemonstrated in fig. 3, the simulated survival probabilitiesremarkably well follow our conjecture (18a) for all wedgeangles and Hurst exponents. Particularly, the scalingexponents of the survival indeed decreases for growingvalues of the Hurst exponent H, in line with Molchan’sresult and findings for the barrier crossing behaviour ofFBM.

Discussion. – We studied the first passage proper-ties of two-dimensional FBM confined to a wedge domain

Fig. 2: (Colour on-line) Survival probability SΘ(t) for wedgeangles Θ= π (left) and Θ= π/2 (right) with the Hurst expo-nents H = 1/4, 1/2, and 3/4 (from bottom to top). The linesrepresent the expected scaling behaviours, eqs. (16a) and (17a).

with absorbing boundaries. Starting from special cases, forwhich exact long-time scaling expressions can be analyti-cally derived, we obtained functional dependencies of firstpassage time exponents on the Hurst exponent H andwedge angle Θ. By numerical analysis we confirmed thisconjecture. We believe that our results are an importantadditional brick in the construction of a clear picture of theelusive FBM process. The obtained forms (18) of the firstpassage time quantities imply that the mean first passagetime of a particle from a wedge domain across the domainwalls, T , diverges for wedge angles Θ�Θc. The criticalangle corresponds to Θc = π(1−H).From the above results one may obtain some insight into

the reaction kinetics of three particles in one-dimensionalspace, driven by FBM (fig. 1(b)). 1) Surrounded prey. Letus consider the diffusion-limited reaction of the centralparticle with either of the two comrade particles. Theproblem can be transformed such that one particle isconfined between two moving absorbing boundaries [6].If xi(t) denotes the position of the i-th particle at timet, the survival condition of the central particle is x1(t)<x2(t) and x2(t)<x3(t) for particles sharing the samediffusion properties (H and KH). We regard xi(t) asthe i-th coordinate of a single particle, that diffuses inthree-dimensional space. Then, the survival criterion isunderstood such that this particle diffuses inside a domainlimited by absorbing planes at x1 = x2 and x3 = x2. This isequivalent to the first passage problem for two-dimensionalmotion in a wedge domain, the wedge angle given bythe two intersecting planes, i.e., Θmid = π/3 (see ref. [6]).Thus, from the distribution (18a) we find that the survivalprobability for the central particle has the asymptotic formSmid(t)∼ t

3(H−1). Contrary to our naive expectation, theabove scaling exponent shows that superdiffusive particles(with H > 1/2) survive longer than ordinary Brownianparticles (H = 1/2), and subdiffusive particles have abetter chance to meet each other. These characteristics

20008-p4

First passage of FBM in 2D wedge domain

Fig. 3: (Colour on-line) Survival probability SΘ(t) for wedge angles Θ= π/4, 3π/4, 5π/4, and 7π/4, with Hurst exponentsH = 1/8, 1/4, 1/2, 3/4, and 5/6 (from bottom to top). The corresponding expected scaling behaviours (cf. eq. (18a)) aredepicted by the full lines.

will lead to significant differences in the total reactionamount, compare figs. 2 and 3. Intriguingly the meanreaction time T is finite for H <H∗ = 2/3, meaning thatreactions always occur for subdiffusive motion, similar tothe considerations in ref. [37]. 2) Chased prey. For the casethat one of the corner particles (x3, say) is chased by thetwo other diffusing particles (x1 and x2), the problem isreformulated to the task of finding the survival probabilityfor the wedge domain of angle Θend = 2π/3. Hence foridentical particles the probability that one corner particlesurvives until time t scales as Send(t)∼ t

3(H−1)/2. In thiscase the reaction is slower than that of the surroundedparticle. Note the scaling relation Smid(t)≃S 2

end(t), suchthat the reaction of a particle surrounded by two otherscorresponds to the product of two independent cornerparticle reactions.We note that closer inspection of the simulated survival

probabilities shows that in the non-Markovian case (H �=1/2) the first passage process exhibits a somewhat intri-cate scaling behaviour: the long-time behaviours (18) arepreceded by another distinct scaling law at intermediatetimes. This behaviour is particularly noticeable for thecases of wedge angle Θ� π, for which first passage eventsoccur easily at short and intermediate times. The appar-ent intermediate scaling behaves differently, depending onthe diffusion characteristics. For subdiffusion, the inter-mediate scaling exponent is smaller than the long-timeexponent πΘ (1−H) and larger for superdiffusion.The first passage behaviour displayed by FBM is differ-

ent from results derived from the diffusion equation withtime-dependent diffusion coefficient. The latter would leadto an inconsistent H-dependence, corresponding to thereplacement H→ 1−H in the correct forms (18). A simi-lar inconsistency occurs if the method of images wereapplied to construct the solution in the presence of non-natural boundary conditions. We note in passing that thesubstitution H→ 1−H in our results for Θ= π leads tothe scaling for the first passage behaviour found in theanalysis of a generalised Langevin equation in ref. [38].

The somewhat counterintuitive behaviour that smallerH implies faster decay of the first passage time density,may be connected to the fact that FBM is fuelled byexternal noise, that is not balanced by friction. That is,FBM does not obey the fluctuation-dissipation relation,in contrast to the generalised (fractional) Langevin equa-tion [34,35,39] that describes FBM in conditions closeto thermal equilibrium, when the fluctuation-dissipationtheorem is valid. We will investigate the latter behaviourin a separate work.We finally note that while for a compact process in

one dimension the first passage is identical to the firstarrival to a given position, these processes are no longernecessarily equal to each other in higher dimensions.This may also be the reason why the Wilemskii-Fixmanapproximation reproduces Molchan’s result ℘(t)≃ tH−2 inone dimension, but delivers a different result in higherdimensions. It will also be interesting to study FBMgeneralisation of first passage properties in cones of higherdimensions [40].

∗ ∗ ∗

We thank M. Lomholt and I. Sokolov for help-ful discussion. Financial support from the Academy ofFinland (FiDiPro scheme) and the European Commissionthrough MC IIF Grant No. 219966 LeFrac is gratefullyacknowledged.

Appendix

The survival probabilities presented in fig. 3 were obtainedwith the same initial starting point and number of simu-lation runs for consistency. Depending on the wedgeangle and Hurst exponent, however, some cases are moreproblematic to show reliable long-time scaling propertieswithin the time window we used. Here we present addi-tional representative results for these cases.i) Figure 4 depicts the case for Θ= π/4 with r0 = 2.0

(i.e., eight times the value used in the main text) and

20008-p5

J.-H. Jeon et al.

Fig. 4: (Colour on-line) Left: survival probability SΘ(t) forwedge angle Θ= π/4 and Hurst exponents H = 1/4 (below)and 1/2 (above). Here r0 = 2.0, and the number of simulationruns is 250000. Right: survival probability SΘ(t) at wedgeangle Θ= 3π/4 for the Hurst exponent H = 1/4. Here r0 =0.10, and the number of simulation runs is 100000. The solidlines are the expected power-laws predicted by eq. (18a).

250000 runs. Now the survival probabilities for H = 1/4and 1/2 exhibit satisfactory long-time scaling.ii) To obtain extended long-time scaling behaviour for

H = 1/4 and wedge angle Θ= 3π/4, the initial distancefrom the origin was decreased to r0 = 0.10. The obtainedresult in fig. 4 shows good agreement with eq. (18a).

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