On the First Passage Time and Leapover Properties of Lévy Motions

Transcription

1 On the First Passage Time an Leapover Properties of Lévy Motions T. Koren a, A.V. Chechkin b, an J. Klafter a a School of Chemistry, Tel Aviv University, Tel Aviv Israel b Institute for Theoretical Physics NSC KIPT, Akaemicheskaya st. 1, Kharkov Ukraine Abstract We investigate two couple properties of Lévy stable ranom motions: The first passage times (FPTs) an the first passage leapovers (FPLs). While, in general, the FPT problem has been stuie quite extensively, the FPL problem has harly attracte any attention. Consiering a particle that starts at the origin an performs ranom jumps with inepenent increments chosen from a Lévy stable probability law λ, ( x), the FPT measures how long it takes the particle to αβ arrive at or cross a target. The FPL aresses a ifferent question: Given that the first passage jump crosses the target, then how far oes it get beyon the target? These two properties are investigate for three subclasses of Lévy stable motions: (i) symmetric Lévy motions characterize by Lévy inex α (0 < α < 2) an skewness parameter β = 0, (ii) one-sie Lévy motions with 0< α < 1, β = 1, an (iii) two-sie skewe Lévy motions, the extreme case, 1< α < 2, β = -1. PACS number(s): Fb, Jc, Ey Keywors: Lévy motion, Lévy stable istributions, Brownian motion, first passage time, leapover. 1

2 I. Introuction Lévy motions constitute a funamental family of ranom motions generate by stochastic processes with stationary an inepenent increments. Examples of the Lévy family inclue, among others, the Brownian, Cauchy an Lévy-Smirnov processes. Since their introuction [1] Lévy motions have been investigate extensively both theoretically an experimentally. In fact, Lévy type statistics [2] turne out to be ubiquitous an is observe in various areas incluing: physics (chaotic ynamics, turbulent flows [3,4]), biology (heartbeats [5], firing of neural networks [6]), seismology (recorings of seismic activity [7]), electrical engineering (signal processing [8-10]), an economics (financial time series [11-13]). For many years Brownian motion has serve as the ominant moel of choice for ranom noise in continuous-time systems. Its remarkable statistical properties, on the one han, an its amenability to mathematical analysis, on the other, have le Brownian motion to become the moel of continuous-time ranom motion an noise. However, Brownian motion is just a single example of the Lévy family. Furthermore, it is a very special an misrepresenting member of this family. Amongst the Lévy family, the Brownian member is the only motion with continuous sample-paths. All other motions have iscontinuous trajectories, exhibiting jumps. Moreover, the Lévy family is characterize by selfsimilar motions. Brownian motion is the only selfsimilar Lévy motion possessing finite variance all other selfsimilar Lévy motions have an infinite variance. In the present work we stuy two couple properties of Lévy motions. The first one is the first passage time (FPT): How long it takes a ranom walker starting at the origin an performing inepenent jumps istribute accoring to a Lévy stable probability law to cross or arrive at a fixe target point. The secon property is the first passage leapover (FPL): How large is the walker s leap over the target; namely, given that the first passage jump crosses the target, then how far oes it get beyon the target? The FPL refers to the first laning of those jumps (flights) that crosse the target for the first time. While the istribution of FPTs has been investigate quite extensively for some types of Lévy motions [14-20], the relate problem of the FPLs has not attracte much attention. The main goal of the present work is to investigate the FPTs an FPLs for three subclasses of Lévy ranom motions: (i) symmetric Lévy motions (ii) one-sie Lévy motions, an (iii) two-sie skewe Lévy motions, the extreme case. We present numerical 2

3 results, fin the asymptotic behavior of the FPT an FPL probability ensity functions (PDFs) an compare the results with theoretical finings. The paper is organize as follows. In Sec.II we remin the reaer of the general expressions for Lévy stable istributions, the meaning of the parameters characterizing them, an explain the numerical algorithm. In Secs. III, IV an V we stuy symmetric, one-sie an skewe Lévy motions, respectively. We en with conclusions. II. Lévy motions Let us assume that a one-imensional ranom Lévy motion x(t) starts at x = 0 at t = 0, an that there is a target locate at x = a > 0. For the numerical stuies of the FPT an FPL properties we use a simple iscrete-time representation of the Lévy stable process, n xn ( ) = ξ. (1) j= 1 The FPT τ an the FPL l are efine as follows: j τ = n, l = x. (2) n Here n is the number of steps to first cross or arrive at the target, an x( n ) is the istance of the first crossing over the target from the origin. See figure 1 for a schematic escription of a leapover event. 0 x(n) l Figure 1. Schematic escription of the leapover problem. A particle locate initially at the origin x=0 performs a ranom walk accoring to a Lévy stable PDF. During the n-th step it crosses for the first time the target locate at x =. The n-th step correspons to the crossing time τ. The particle then lans at x( n) = + l, where l efines the leapover istance beyon the target. 3

4 Here the increments {ξ j } are chosen from a Lévy stable PDF λ, ( x; µσ, ) expresse in terms of its characteristic function λ, ( k; µσ, ) [21-23], αβ αβ an k ikx αβ, αβ, 2π λ ( x; µσ, ) = e λ ( k; µσ, ), (3) where α α k λ αβ, ( k; µ, σ) = exp σ k 1 iβ ϖ( k, α ) + iµ k k πα tan, if α 1 2 ϖ( k, α) = 2 ln k, if α = 1. π, (4) In general, the characteristic function an, respectively, the Lévy stable PDF are etermine by the parameters: α, β, µ an σ. The exponent α [0,2] is the Lévy inex, β [ 1,1] is the skewness parameter, µ is the shift parameter, which is a real number, an σ > 0 is a scale parameter. The inex α an the skewness parameter β play a major role in our consierations, since the former efines the asymptotic ecay of the PDF, whereas the latter efines the asymmetry of the istribution. These two properties of Lévy stable PDFs will be iscusse in more etail below for each of the subclasses of Lévy stable ranom motions uner consieration 1. The shift an scale parameters play a lesser role in the sense that they can be eliminate by proper scale an shift transformations, 1 x µ λ αβ, ( x; µ, σ) = λαβ, ( ;0,1). (6) σ σ Due to this property, in the present paper for brevity we enote the Lévy stable PDF λ, ( x; µσ, ) by λ, ( x). We note the important symmetry property of the PDF, namely αβ αβ (5) 1 In the representation given by Eq.(4) the sign of β is chosen so that the process with β = 1, 0 < α < 1 has positive increments, see Sec.IV below. 4

5 λα, β( x) = λα, β( x). (7) For instance, the asymptotics of the PDF λ α 1,1 ( x ), x 0 ( x < > 0) or x have the same behavior as λ α 1, 1 ( x ), x 0 ( x < < 0) or x. The property given by Eq. (7), obviously, leas to certain symmetry in formulating the FPT an FPL problems. We also note that only in three particular cases can the PDF λ, ( x) be expresse in terms of elementary functions: (i) α = αβ 2 (Gaussian istribution); in this case β is irrelevant; (ii) α = 1, β =0 (Cauchy istribution); an (iii) α = 1/2, β = 1 (Lévy-Smirnov istribution). In general, close analytical forms of stable law PDFs are given via the Fox H-functions [23,24]. In what follows we investigate the FPT an FPL properties for three ifferent Lévy motions, that is for three ifferent statistics of the increments {ξ j }: (i) symmetric Lévy stable PDFs, α (0,2], β = 0 ; (ii) one-sie Lévy stable PDFs, α (0,1), β = 1, an (iii) two-sie extremal Lévy stable PDF, α (1, 2), β = 1. The importance of Lévy motions with extremal values of skewness parameter β is ue to the fact that any Lévy stable process x(t) can be written in the form x 1 (t) x 2 (t) where x 1 an x 2 are inepenent Lévy processes possessing the same Lévy inex α an β = ±1 [25]. Examples of the three types of Lévy stable PDFs are shown in figure 2 for some particular values of α. Figure 2. Lévy stable PDFs with ifferent Lévy inices α an skewness parameters β. Examples of the three iscusse subclasses are presente: symmetric ( α = 1.5, β = 0, soli line), one-sie ( α = 0.75, β = 1, ashe line) an two-sie skewe, the extreme case ( α = 1.5, β = 1, asheotte line). 5

6 In the simulations the sets of increments { ξ j } are obtaine by using the metho of Chambers et. al. [21,22,26-28]. The process efine by Eq. (1) is repeate until x( n ) becomes larger or equal to the position of the target x = leaing to two ranom values, τ an l, Eq. (2). The process in Eq. (1) then starts anew. In orer to plot the asymptotic behavior of the FPT an FPL PDFs with a reasonable statistical accuracy, this proceure was repeate 1.5x x10 5 times for each ranom motion with fixe α an β. III. Symmetric Lévy motion We start by consiering the particular case of Lévy motions erive from symmetric Lévy stable PDFs. The symmetric case correspons to 0< α < 2 an β = 0, with the characteristic function λ α,0 ( k) = exp( ). (8) k α The Cauchy PDF is a special case which correspons to α = 1, ( x 2 ) λ 1,0 ( x) = 1/ π 1 +. (9) The symmetric Lévy stable PDF behaves asymptotically as [29,30] where λ α, α ( x) C ( α)/ x, x ±, (10) 1 C1 ( α) = sin ( πα /2) Γ (1 + α ). (11) π Typical trajectories of symmetric motions are shown in figure 3 for ifferent Lévy inices α. 6

7 Figure 3. Trajectories obtaine from numerical simulations with symmetric Lévy stable PDFs. As α becomes smaller larger jumps are more probable. In all cases the target is locate at = 200. Note the ifferent scales for the ifferent α values. The target location is shown by the full line an the leapover istance by the broken line. It is quite obvious that the smaller α is the larger are the jumps an, as one might expect, the larger is also the size of leapover. To obtain the FPT law for symmetric Lévy motions we recall the funamental result known as the -3/2 law or the Sparre-Anersen theorem [31,32]. The latter states that for any iscrete-time ranom walk process with inepenent steps chosen from a 3/2 continuous, symmetric arbitrary istribution, the FPT PDF ecays asymptotically as ~ n, where n is the number of steps. Since in our problem the number of steps is equivalent to the current time, we obtain the asymptotics of the FPT PDF in the symmetric Lévy case, 3/2 τ p α ( τ ). (12) The FPT PDF ecays with the same exponent inepenent of the Lévy inex α, 0< α 2 [14,16,33]. In particular for Brownian motion one obtains the first passage PDF which again ecays as 2 pα ( τ ) = exp 4πDτ 4Dτ 3/2 τ 3, (13) for τ 2 >> /(4 D), where D is the iffusion coefficient [14]. The - 3/2 exponent leas to the ivergence of the mean first passage time (MFPT): τ = τ p ( τ ) τ =. (14) α 0 7

8 The -3/2 behavior was obtaine analytically an corroborate numerically for the special case in which an absorbing bounary is place close to the location of the starting point of the Lévy motion [33]. Also note that in Ref. [33] the prefactor of the asymptotics given by Eq. (12) was obtaine uner the assumption of an exponential waiting time istribution of the single jumps. In figure 4 we show the FPT PDF on a log-log scale (natural logarithmic) for ifferent values of the Lévy inex α an the parameter (target location). The simulations nicely emonstrate the universal nature of Eq. (12). Figure 4. FPT PDFs p α ( τ ) of symmetric Lévy motions with ifferent values of α. Plotte is τ p α ( τ ) vs τ on a log-log scale. The soli lines correspon to the slope -1/2 with the maximum relative error of 4%. The Sparre-Anersen behavior observe. 3/2 p α ( τ )~ τ is clearly 8

9 Figure 5. FPL PDFs fα ( l ) of symmetric Lévy motions with ifferent values of α. The target is locate at ifferent positions. Plotte is l f ( l) vs l on a log-log scale. The fits correspon to a the asymptotic behavior of of 5%. f ( l )~ l α ( α /2+ 1) for the FPL PDF with the maximum relative error As mentione, less is known about leapover properties. Of course for Brownian motion there is no leapover since the searcher approaches the target in a continuous way. The situation might be ifferent for Lévy motions. For symmetric Lévy motions with inex α our simulations support the following power-law ecay of the FPL PDF in the limit of large l, 1 α /2 f ( l ) l, 0< α < 2, (15) α as shown in figure 5. For a phenomenological explanation of Eq. (15) we use the superiffusive nature of a typical isplacement δ x δ x of the Lévy particle uring time intervalτ, 1/ α ~ x τ, (16) where τ has a PDF whose asymptotics is given by Eq. (12). Neglecting the constant shift in the leapover, which oes not influence the far asymptotics, we write 1/ α l = x τ. (17) 9

10 Applying τ fα( l) = pα( τ ), (18) l then, from Eqs. (17) an (12), by the simple change of variables, we arrive at the asymptotic behavior of the FPL PDF, Eq. (15). Surprisingly, the tail of the FPL PDF ecays slower than the tail of the PDF of the increments, Eq. (10). However, our simple argument oes not provie the prefactor which might be smaller than that of the tail of the increments PDF. IV. One-sie Lévy motion In this Section we procee with the class of one-sie Lévy motions, which are also referre to as Lévy suborinators [17]. The increments of the one-sie motions are nonnegative, an the Lévy stable PDF of the increments is non-zero on the positive semi-axis only. This class is characterize by the following Lévy inices an skewness parameter, respectively: 0< α < 1, β = 1. Thus, the characteristic function, Eq. (4), reas as α k πα λα,1( k) = exp k 1 i tan( ) k 2 or after transformation,, (19) α k isign( k) πα /2 λ α,1( k) = exp e. (20) cos ( πα / 2) The one-sie Lévy stable PDF is often characterize also via its Laplace transform [33], sx α,1 λα,1 0 λ () s = xe () x, (21) a representation which is equivalent to Eqs. (19) an (20), α s λ α,1() s = exp. (22) cos ( πα / 2) From Eq. (19) (or alternatively, Eqs. (20) or (22)) it follows that the increments PDF has the following asymptotic behavior ( x 0) [35,36], 1 µα ( )/2 µα ( ) λα,1( x) C2 ( α) x exp C3 ( α) x, (23) 10

11 where 1/ [ 2(1 α )] α C ( α) = 2 ( πα ) cos / 2 2π 1 α [ α ] 1/ 2(1 ), (24) 3 α /(1 α) ( ) 1/(1 α ) C ( α) = 1 α α cos πα /2, (25) an α µα ( ) =. (26) 1 α For x asymptotics of the increments PDF have the same behavior as that of the symmetric motion has, Eq. (10). The absolute values in Eqs. (24)-(26) are introuce for the sake of section V. A particular case of the one-sie Lévy stable PDFs is the Lévy Smirnov istribution, for which α = 0.5, β = 1 [37], λ 1/2,1 1 3/2 1 x exp, x> 0 ( x) = 2π 2x 0, x < 0. This probability law appears, e.g., in the theory of crossing of a barrier by a Brownian particle, see Ref. [34], page 174. Trajectories of one-sie Lévy motions are shown in figure 6. (27) Figure 6. Trajectories of one-sie Lévy motions corresponing to ifferent Lévy stable PDFs (0 < α < 1; β = 1). In both cases the target locate at full line an the leapover istance by the broken line = 10. The target location is shown by the

12 The FPT an FPL problems for the one-sie Lévy motion were consiere in Ref. [17]. We recall those results which are essential to our numerical analysis: 1. There exists a probabilistic relationship between the jumps ξα,1 ( x) of the one-sie Lévy motion an the FPTs τ. No close form coul be erive for the FPT PDF for a general inex α. The center case in the one-sie Lévy motion, α = 0.5, is the only case among the family of one-sie Lévy motions where an explicit expression for the FPT PDF has been erive. For further iscussion we refer the reaers to Ref. [17]. For the particular case α = 0.5, the FPT PDF follows from Eq. (27), 2 ( τ ) p1/2 ( τ ) = A exp B, (28) with A = 2/( π ), B= 1/(2 ). 2. The MFPT is given for a general inex α, 0< α < 1, τ α = cos / 2 1 ( πα ) Γ ( + α ) which scales with the target istance from the origin. Unlike the symmetric case in section III an the two sie skewe to be iscusse in section V, here the MFPT is finite, (29) 3. The FPL PDF follows asymptotically the law of the increments, α sin( πα ) 1 fα ( l) = ~ ; l >> a. α α+ π l l l 1 ( + ) Comparisons between numerical simulations an the analytical results of the FPTs an FPLs are presente in figures 7-9, supporting the expressions in Eqs. (28-30). (30) 12

13 Figure 7. Comparison between analytical results corresponing to Eq. (28), an simulation results of the FPT for ifferent target locations. Note that the inset isplays log ( log( p ( τ ) / A) ) versus logτ supporting Eq. (28). 1/2 Figure 8. The MFPT for α = 0.5 as a function of the target location showing a goo agreement with Eq. (29) (the soli line). The relative error in the slope is 5%. 13

14 Figure 9. FPL PDFs fα ( l ) of one-sie Lévy motions with ifferent values of α an ifferent target positions. Plotte is l fα ( l) vs l on a log-log scale. The soli lines correspon to the fits supporting asymptotic behavior of f ( l )~ l α ( α + 1) with the maximum relative error of 1.8%. V. Two-sie skewe Lévy motion, the extreme case Another interesting subclass of the Lévy motions is the extreme case of the two-sie but skewe Lévy motion of inex α, 1< α < 2, an skewness parameter β = 1. This is a two-sie motion which possesses large jumps in the negative irection an small ones in the positive irection (see figure 2); thus, crossing continuously the target point locate on the positive semiaxis, > 0 (recall that at t = 0 the process begins at x = 0). Mathematically, continuous crossing implies that x( τ ) = with probability 1. However, in simulations, ue to time iscretization small leapovers occur which are negligible when compare with the large leapover values for α, 1< α < 2 in figure 5. Therefore, the leapover l is practically zero, as in the Brownian case. The PDF of the increments in this case has power-law asymptotics on the negative semi-axis an falls off in an exponential way on the positive semi-axis [35,36], 14

15 1+ α C1 ( α)/ x, x α, 1( x) 1 µα ( )/2 µα ( ) 2 α 3 α λ C ( ) x exp C ( ) x, x +. (31). Figure 10. Trajectories obtaine for ifferent two-sie skewe istributions (1 < α < 2; β = 1). In this case the leapover is practically 0. The target locate at 3 = 10. where C 1, C 2 an C 3 an µ ( α ) are given by Eqs. (11), (24), (25) an (26), respectively. Typical trajectories are shown in figure 10 for ifferent α values isplaying large jumps away from the target an small ones towars it. As to the FPT problem for skewe Lévy motions of inex α, a theorem was proven by Skorokho [25] accoring to which the FPT τ is given by a one-sie Lévy stable PDF of the Lévy inex α = 1/ α, 1/2< α < 1, the skewness parameter β = 1, an the scale parameter σ = a cos cos 2 2 α ' πα πα 1/ α. (32) 15

16 Namely, the FPT PDF is given by: where, 1 pα( τ) = k exp ( ikτ) pα( k), 2π (33) α' α' πα ' pα( k) λα',1 ( k;0, σ) = exp σ k 1 isign( k ) tan 2, (34) so that asymptotically, 1 p α ( τ )~ 1/ α + 1 τ. (35) The FPT PDF ecays as a power law with the exponent 1/ α 1 (compare with Eq. (12) an Eq. (28)). Since 1 < α < 2 the MFPT which correspons to p α ( τ ) in Eq. (35) iverges. The FPT PDF can not be written in a close form, an numerical integration of Eqs. (33) an (34) is neee. We use the Quasi Monte Carlo (QMC) metho which converge faster than the regular Monte Carlo in our case [38,39]. The results of the numerical simulations an numerical integrations for the FPT PDFs are presente in figures Figure 11. Plotte on a log-log scale is τ p α ( τ ) vs τ of Lévy motions with 1< α < 2, β = -1. 1/ α The soli lines correspon to the behavior τ p ( τ )~ τ which confirms the ecay in Eq. (35). The maximum relative error is 1.25%. 16 α

17 Figure 12. FPT PDF p α ( τ ) of Lévy motions with 1< α < 2, β = -1 obtaine from simulations an compare with numerical integrations (QMC) at long (a) an short (b) times, respectively. The agreement between the numerical integration of equations (33) an (34) an the simulations is clearly seen in whole omain of FPT. Interestingly, the FPT PDFs of the symmetric Lévy motions ecay asymptotically slower than the skewe case (1 < α < 2, β = 1) iscusse here. Therefore, the latter istributions, characterize by long jumps away from the target, lea to the seemingly paraoxical result of being a better strategy for reaching the target. Finally, we speculate on the generalization of our FPL PDF of Lévy motions with 0< α < 1 by allowing the skewness parameter β to vary ( 0< β < 1). The results presente in figure

18 Figure 13. A log-log plot of FPL PDFs fα ( l ) of Lévy motions with 0< α < 1 an varying skewness 0< β < 1. The soli lines fit to 6%. f ( l )~ l ( α/2( β+ 1) + 1) a with the maximum relative error of VI. Conclusions In summary, we have presente calculation results of the first passage time an leapover PDFs for three subclasses of the Lévy motions, characterize by the ifferent values of the Lévy inex α an the skewness parameter β : 1. For the symmetric motion, 0 < α < 2, β =0, we emonstrate the universal ecay of the FPT PDF with the exponent -3/2. We foun that the FPL PDF has a power-law asymptotics with the exponent 1 α /2; thus, the ecay is slower than that of the PDF of the increments of the Lévy motions. 2. For the one-sie Lévy motions with positive increments, 0< α < 1, β =1, we provie etaile numerical simulation of the FPT problem for the particular case α = 0.5, an confirme numerically the power-law ecay of the FPL PDF with the exponent 1 α (that is, the same ecay as for the PDF of the Lévy increments) for ifferent Lévy inices α an ifferent target positions. 18

19 3. For the extreme case of the two-sie skewe Lévy motions 1< α < 2, β = 1, we emonstrate that the FPT PDF is actually the one-sie Lévy stable PDF of Lévy inex α = 1/ α an skewness parameter β = 1, thus the PDF ecays asymptotically as a power law with the exponent 1 1/α larger than the exponent of -3/2 obtaine in the symmetric case. 4. Finally, base on numerical simulations of the general case of Lévy inex α, 0< α < 1, an skewness parameter β, 0 β 1, we suggest a power-law ecay of the FPL PDF with the exponent 1 α( β + 1)/2, which reuces to the results for the symmetric, β = 0, an the onesie, β = 1, cases, respectively. Acknowlegements The authors acknowlege iscussions with Io Eliazar, Michael Anersen Lomholt an Ralf Metzler. AVC acknowleges hospitality of the School of Chemistry, Tel Aviv University. 19

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