Proc. NatL Acad. Sci. USA Vol. 79, pp. 4501-4505, July 1982 Applied Mathematical Sciences
Fractal dimensionality of Levy processes (mean first-passage times/maxima moments) V. SESHADRI AND BRUCE J. WEST Center for Studies of Nonlinear Dynamics, La Jolla Institute, P. 0. Box 1434, La Jolla, California 92038 Communicated by Elliott Montroll, April 7, 1982
ABSTRACTT We determine the fractal dimensionality D ofthe space. The mean first-passage time, however, is not a sharply trajectories of a class of translationally invariant Markov pro- defined quantity because the first-passage time distribution is cesses. We also provide two simple operational measures to esti- very broad (13, 14). Maxima moments, on the other hand, are mate D. a sensitive measure of the fractal dimensionality. Recently, considerable attention has been devoted to diverse Levy distributions physical phenomena exhibiting a clustered behavior in space or time domains (14). Examples of such clustered behavior Translationally invariant stationary Markov processes in con- appear in processes in fluid mechanics (1-4), solid state physics tinuous space are described by probability densities that satisfy (ref. 1, p. 16; ref. 5), astrophysics (ref. 9, p. 170), data trans- the Bachelier-Smoluchowski-Chapman-Kolmogorov chain mission systems (ref. 1, p. 93), and Brownian motion (ref.. 1, p. condition, 201). There is apparently a close connection between such clus- tered behavior in space or time and the Hausdorff-Besicovitch (fractal; ref. 1, p. 16) dimensionality of these processes. P(X2 - x1,t) = J P(X2 - x, t - r) (x - x1,)dx [2.1] The fractal or Hausdorff-Besicovitch dimensionality D of a set may be defined as follows. If a finite part El of the set is where P(x2 - xj,t) is the probability that the value ofthe process divided into N identical parts, each of which is geometrically X changes from xi to x2 in time t. The characteristic function similar to El with the similarity ratio r, then D = tn N/en(1/r). 4(k,t) is defined as the Fourier transform of the -probability Analytic estimates of the fractal dimensionality of dynamical density-i.e., processes has so far been carried out only for Brownian motion Jx (ref. 1, p. 201). In this paper, we examine the fractal dimen- 0(k,t)= dx eia P(x,f) . [2.2] sionality of the trajectory of a class of translationally invariant _x~~~~~~~~~~~~~~~ Markov processes and also provide two operational measures For processes satisfying the chain rule (Eq. 2.1), #(k,t) obeys for estimating time. the product rule. A simple measure of the fractal dimensionality of a model dynamical process that exhibits clustering has been developed 4(k,t) = 4(kt - r) 4(kr). [2.3] by Hughes et aL (10). The model is a discrete random walk on Montroll and West (12) noticed that, since dI(k,t) satisfies rule a lattice with transition probabilities drawn from a distribution 2.3, it is an infinitely divisible stable distribution (11). which does not have a finite variance-i.e., a Levy distribution The most general form of ck(kt) for infinitely divisible stable (11). Hughes et al. were able to associate the short-scale be- distributionswas obtainedby L6vy (15) and Khinchine and Levy havior ofthe structure function to the fractal dimensionality of (16). For symmetric processes, the most general form (11) is the walk. In the continuous space-time limit their model pro- given by cess reduces to a special case of the translationally invariant processes considered herein. The distribution functions oftranslationally invariant Markov O(kt) = exp [-bt lkijI + icw(lc ).)k [2.4] processes that we consider satisfy the Bachelier- Smoluchowski-Chapman-Kolmogorov (BSCK) chain condition In Eq. 2.4, A., b, and c are constants obeying the restrictions and are called LUvy processes (12). Levy processes in one, two, 0. A-2,p b .0 and -1 c . 1. The function w(k,pI) is de- and three dimensions* are represented by their characteristics fined by function 4(Ilkl) - exp{-lklj}. The exponent. p. of the character- istic function determines the essential physical properties ofthe w(k,p) = tan (irp/2) if p. # I process. In particular, we show that Ad is the fractal dimen- sionality ofthe trajectories of the process. =-2 nlkbtl ifpA= 1. [2.5] In this paper we propose the use of mean first-passage times ir and maxima moments as operational measures of the fractal The class of processes whose characteristic functions satisfy dimensionality of the trajectories of Levy processes. The idea Eq. 2.4 are called Levy processes. The most important feature of using mean first-passage times as a measure of the fractal ofLevy distributions is that, except for p. = 2, the distributions dimensionality has its seed in the heuristic connection, estab- P(x,t) do not possess finite moments of all orders. This can be lished by Mandelbrot, between the fractal dimensionality of seen most easily from the fact (12) that, for t > 0, Brownian motion and the total time spent within a region of * We shall henceforth denote the Euclidean dimension either by the The publication costs ofthis articlewere defrayed in part by page charge symbol E or simply by the term "dimension." The fractal or Haus- payment. This article must therefore be hereby marked "advertise- dorff-Besicovitch dimension is always indicated either as such or by ment" in accordance with 18 U. S. C. §1734 solely to indicate this fact. the symbol D. 4501 Downloaded by guest on September 29, 2021 4502 Applied Mathematical Sciences: Seshadri and West Proc. NatL Acad. Sci. USA 79 (1982)
lim P(xt) - 4btfr(A) sin (vri/2)/lrxM+l. [2.6] possible to construct simple evolution equations ofthe diffusion type for the probability density-i.e., equations involving all a defined by Thus, moments Pxt) . Such first-derivative equations, in general, contain in- oe at (xIli) = J lxi" P(x,t)dx [2.7] tegral operators in the position variables. For example, in one dimension when pt # 2, the evolution equation can be shown are finite for a < A and are infinite for a > ti. In particular, (17) to be of the form the variance is infinite. aP(xt) b In two and three dimensions we restrict our attention to cen- =- sin(nm/2)i E(/ + 1) trosymmetric Levy distributions defined by the equation a3t r t ('c = = P(Y,t) 4(k,t) 4(1kl,t) exp[-btlkitj [2.8] x + c - dy{1 sign(y x)} x-Yp [2.14] where IkI = (kx2 + ky2)1/2 in two dimensions and Ikj = (k 2 + co ky + kz2ft'2 in three dimensions, respectively. The radially and In the next section, we demonstrate how formally simpler evo- spherically symmetric probability distribution functions in two lution equations possessing higher time derivatives can be ob- and three dimensions are defined, through their respective tained for certain values of the parameters A and c. Fourier transforms, via the relations Differential evolution equations P(r,t) = 2-I e btkMo(kr)kdk; 2E, Under certain conditions, differential evolution equations can be obtained for the probability density which, in general, in- and volve higher lime derivatives. 'We first consider the one-di- mensional case. P(r,t) = e btk~sin krkdk; 3E, [2.10] Let ;. be a rational number equal to m/n where m and n are 2y7r integers and the parameter c in Eq. 2.4 is selected to be zero. successively n times respectively. These L6vy distributions P(rt) are similar to the When m is even, we differentiate Eq. 2.4 one-dimensional distribution P(x,t) in that for pA c 2 they also with respect to time and inverse Fourier transform the resulting do not possess finite moments ofall orders. It can be shown via equation to obtain anen+YWZ nEam an asymptotic analysis similar to the one earned outby Montroll -P(xlt = (l)y+ /2bnmP(xt)P~x~f).m [3.1] and West (12) that, for pA c 2, in two and three dimensions, a-1t" ax", P(r,t) has the asymptotic form To ensure that the solutions of Eq. 3.1 are real and positive- bt [2] P(r,t) - (2)r r-E-t; E = 2,3. i.e., they are probability densities-they must satisfy n initial conditions P(x,0), P(') (x~t)110...,Pn-'\x~t)1t=0 (the super- From Eq. 2.11 it is clear that, for A < 2, all moments of order scripts denote the order ofdifferentiation with respect to time). a defined by Though these sufficient initial conditions may be formally ob- tained from Eq. 2.4, they are, in general, as hard to evaluate (r") = (2r)E- f r"P(r,) rE1 dr; E = 2,3, [2.12] as the inverse Fourier transform of the function O(k,t) itself. When m is odd, we differentiate Eq. 2.4 successively 2n are finite for a < pa and infinite for a 2 times and, after inverse Fourier transforming, obtain The important feature of the asymptotic behavior (Eqs. 2.6 a271 a2m and 2.11) of LUvy distributions for 0 s pA c 2 is the power-law -P( ,t) = (-l)mb2" P(x,t) . [3.2] tail. From these asymptotic forms one can see that the proba- bility P(r > rj) that the process r > iq, for large values of q, is Analogous to the even m case, solutions of Eq. 3.2 must satisfy (0 2n initial conditions. 4) = (2 )E- j P(r t) rEl dr; E = 1,2,3, When c # 0, it can be shown that closed differential equa- tions can be obtained for pa = rn/n $ 1, with value of c given by 1 -~const - . [2.13] c = tan[irp/2nj]/tan[mv/2n];
j- ±1, ±2, . a.0 p= +l, ±,...± r. [3.3] Thus, the asymptotic behavior of the process r is that of a hy- perbolic random variable. As pointed out by Mandelbrot such The above construction can be extended to higher spatial hyperbolic distributions preserve self-similarity and have tra- dimensions. for jectories with fractal dimensionality pA (ref. 1, p. 133). Thus, Eqs. 3.1 and 3.2 are not particularly useful computing L[vy processes with exponent have trajectories with fractal P-x,t) itself because they have to be solved subject to compli- use dimensionality pA. cated initial conditions. However, we have been able to Even though the Levy distributions are characterized by ap- them to compute properties such as the mean first-passage parently simple Eq. 2.4 and 2.10, there are twro major diffi- times and maxima moments. This utilization is discussed in the culties in understanding the physical properties of these dis- next two sections. tributions (12). First, the probability density P(xyt) or P(rt) can be evaluated in a closed form only for special choices of the First-passage times for lvy processes parameters and c. Apart from the diffusion and the Cauchy cases, a few other one-dimensional cases have been discussed Mandelbrot has discussed qualitatively the connection that ex- by Zolotarev (see ref. 12). The second difficulty is that it is not ists between the fractal dimensionality of the Brownian process Downloaded by guest on September 29, 2021 Applied Mathematical Sciences: Seshadri and West Proc. Natl. Acad. Sci. USA 79 (1982) 4503 and the average amount oftime spent by the process in a given region ofspace. For persistent processes (ref. 9, p. 570; ref. 10) T J1(4)= F(f4t) dt the average amount oftime spent by the process in a given re- gion of space is infinite and a quantitative connection with the - 42/n L 22±2/n cc [4.4] fractal dimensionality is hard to establish. A close quantitative 1+2/n e 2/1)l + I connection exists, however, between the mean first-passage -'reo(2t + b time (for definitions, see refs. 9 and 14) to escape a given region The most important feature of the behavior of the first-pas- of space and the Hausdorff-Besicovitch dimensionality. For sage time T1() is embodied in the first factor f2/n, 2/n -pu. example, for Brownian motion (aL = 2) starting from the origin, The second factor is a monotonic increasing factor of n. When the mean first-passage time T1 (4) to reach n = 1, we recover the result for the diffusion process (13)-i.e., i the points ± f lE, T1(4) = 42/2b. [4.5] ii a circle of radius f 2E, The exponent 2 of 4 is the same as the fractal dimensionality iii a sphere ofradius 4 3E of the Brownian trail. When n = 2, we get the result for the is proportional to g2 (13). Thus, the mean first-passage time Cauchy process, scales with 4 with exponent 2 which is also the fractal dimen- 8G sionality of the Brownian trajectories (in Mandelbrot's termi- TI(f)= f b2 [4.6] nology, the Browmian line to E trail). We show below that mean first-passage times ofLvy processes also exhibit a similar char- where G is the Catalan's constant (=0.915956 ...). The expo- acteristic scaling behavior with respect to 4. In fact, the expo- nent of 4 is consistent with Mandelbrot's identification of the nent of 4 is precisely p. which suggests that the exponent of 4 fractal dimensionality of the Cauchy trail as unity. is the Hausdorff-Besicovitch dimensionality. We establish this The scaling behavior, proved above for one dimension, can scaling relation by using exact boundary value techniques for be extended to higher dimensions. For example, in two di- p = 2/n, n = 1. oo and an approximate technique for ar- mensions the mean first-passage time T1(4) to a circle ofradius bitrary values oft. 4, and in three dimensions to a sphere ofradius 4, can be com- Exact Mean First-Passage Times. In general it is difficult puted from the radially and spherically symmetric cumulative to calculate the exact mean first-passage times ofarbitrary Mar- distribution functions F(4,t) given in ref. 13, respectively. The kov processes. However, when the equation ofevolution for the scaling result is Tj(t.) 4-b, = 2/n,2 as obtained above. probability density is a differential equation possessing only a Approximate T1(4, An exact evaluation of Tl(4) for arbitrary Laplacian in spatial variables, the first-passage problem can be p. is, in general, complicated for two reasons. First, if p. is ir- cast as a boundary value problem. Let us first consider one- rational there are no differential evolution equations. Second, dimensional processes. even ifp. is rational but cannot be expressed as 2/n, the evo- The exact first-passage time to reach the point x = 4 start- lution equation possesses spatial derivatives of order higher ingfrom the origin, can be computed for processes which satisfy than 2. For such cases, the boundary value technique as used (a = 2/n): above no longer applies. Hencel we proceed by using an ap- proximation technique as follows. n P(x,t) = ()n+l n xt) [4.1] The approximation procedure rests on the assumption that one can replace the conditional probability density f(xt) by P(x,t) in Eq. 4.3. Such a replacement yields the correct scaling Letf(x,t) be the probability that the process has not crossed the behavior ofT1(4) but not necessarily the correct coefficient. For boundaries x = + 4, given that it starts at the origin at time t transient processes, such a replacement yields a coefficient of = 0. From the theory of first-passage times (9) f(x,t) satisfies SF which is larger than the correct one but is finite. For per- Eq. 4.1 with the boundary conditions fx,0) = 6(x) andf(±4,t) sistent processes, the approximate coefficient turns out to be = 0. In addition, the n initial conditions on the derivatives of this, consider the one-dimensional case f(x,t) must be specified to ensure thatf(x,t) is real and positive. infinite. To see The solution f(xt), using the procedure detailed in ref. 13, is then given by T1(4) = F(4,t) dt - A dt AfP(xt)dx . [4.7] O o -{~~~ f(x,) = For a Levy process, we substitute for P(x,t) the Fourier trans- 1 + 1)1 form of Eq. 2.4 to obtain in terms of the scaled variables x = (>2e lxrx] [e {2e b = 24]exp L~~~~24 ~ [4.2] x/4, k= k4, t (bt)l/A, In Eq. 4.2 only the real positive eigenvalue in the boundary T1(4) b- [4.8] value problem has been chosen so thatf(x,) is real and positive. The cumulative distribution function F(4,t) is then x [iOfccdrjcuf dke~fi'e-^(1 + @icok,p) )]
F(4, t) = dxf(x,t) Thus, Eq. 4.8 has the correct scaling behavior. The coefficient
2 of 4M, however, is finite only when p. K 1, as can be seen by ( [) (2 + 1ir examining the singularities in the integrand ofEq. 4.8 after in- _ bt . [4.3] 2 -L 1)'xp tegrating over t. The integral is finite only for transient pro- bfreo' (2 f 1) - 2f cesses (p. < 1) and is infinite for persistent processes (pL > 1) This result immediately yields the mean firstpassage time to (ref. 9, p. 570; ref. 10).
reach 4, An entirely analogous argument can be presentedfor the two- Downloaded by guest on September 29, 2021 4504 Applied Mathematical Sciences: Seshadri and West Proc. Nad Acad. Sci. USA 79 (1982) and three-dimensional cases. In two and three dimensions, we where replace F(tt) by
F( J,t) P(r,t) rE1 dr; E = 2,3. [4.9] (2 + I 4a t1' b J 0- a-I ( 1)7 IJ) (24)A + (#2e 1)Y'L [5.5] In both these cases the approximation procedure carried out above yields the correct scaling behavior Tj(s) -M. The ap- It is immediately clear from the inverse Laplace transform of proximate coefficient of is finite for 0 g < 2 in two di- Eq. 5.4 that mensions but is finite for all ,a, 0 s g 2 in three dimensions. the approximation procedure can always be applied to Thus, (t> = Za F( 1A [5.6] transient processes. The results obtained above are for continuous Markov pro- The coefficient Aa, however, is finite only when a < A, as can cesses. Weiss and Rubin (18) have considered discrete random be demonstrated by performing the sum in Eq. 5.5 by a contour walk processes with transition probabilities with infinite vari- integration. ance and have obtained scaling results for mean first-passage In two and three dimensions the scaling ofZ(t) with t-i. e., times, similar to Eqs. 4.4 and 4.8. In the continuum limit their Z,,(t) - t"-can be established by a similar procedure. Below results agree with those given in this section. The following point should be noted, however. In discrete random walk we present a heuristic argument valid in each of the three di- p., 0 -S < 2 show models, with transition probabilities with infinite variance, mensions and also valid for all values of to that diverges for a A there exists a difference equation ofevolution on the lattice. In A. We by splitting the integral Eq. 5.1 as the continuum limit of these models, no closed differential proceed equation with dUlt exists. This is precisely the reason for con- structing closed differential equations with higher order time Zd,,(t)= af d a-i[[1 -F ) derivatives (cf. Eq. 4.1). zat
Maxima moments for LUvy processes f+da d a- [1 -F (4t)] [5.7] A disadvantage of using the mean first-passage time as an in- where F(4,t)-is the appropriately defined cumulative distribu- a process is it is not dicator of the fractal dimensionality of that tion function in one, two; and three dimensions (cf. Eqs. 4.3 a sharp measure (14). An alternative property of a Levy process and 4.9). In Eq. 5.7 -q is arbitrarily large. The first integral is a measure is the which is sensitive ofthe fractal dimensionality finite for any finite value of Band the divergence ofZ4x(t), ifany, in a given time t a distribution of the maxima attained (for def- arises from the second integral. For any fixed time t, for suf- inition of maxima, see ref. 14). For Brownian motion = 2) (A.t ficiently large values of ?? we may replace [1 - F(4t)] by all maxima moments are finite and well defined. However, this is not the case for L6vy processes with < 2. In this latter case, 1t JrE P(r,t) dr; E = 1,2,3. [5.8] maxima of certain order only are finite. This is to be expected since the distribution P(x~t) or P(rt) for ta < 2 has a power-law hail. The asymptotic behavior of P(rt) given by Eqs. 2.6 and 2.11 We define the fractional moment of order a of the maxima immediately leads to the result
distribution as - lim [1 - F(4,t)1 l/{ [5.9] 00 Za(t) = af df {a- [1 - f(4t)] . [5.1] in each ofthe three cases. Substitution ofEq. 5.9 into the second integral of Eq.. 5.7 yields the result that maxima moments of a are finite iff a < ja, and are infinite for a : pk. For Levy processes, Za(t) is finite iff a < ,a, IL < 2, and is infinite order otherwise. Equally interesting is the scaling behavior of Za(t)i. e., Conclusions
Za(ttOt , a<< U [5.2] Herein we summarize our main results and discuss some oftheir physical implications: These results can be demonstrated in an exact fashion when the (i) Levy processes in one, two, and three dimensions obey equation of evolution has the Laplacian form (Eq. 4.1). the asymptotic law P(r > 7v) .Thus,T the exponent is We first consider the one-dimensional case. Substituting the the fractal dimensionality of the Levy trajectories. expression from Eq. 4.3 into Eq. 5.1 and Laplace transforming (ii) Differential evolution equations for the probability den- both sides, we obtain (p, = 2/n) sity have been obtained for L6vy processes with certain param- eter values. These evolution equations are, in general, not of the diffusion type and involve higher-order derivatives in space as well as time. For LUvy processes with second degree spatial derivatives, we have obtained exact mean first-passage times )'1 (2 t+ [5.3] and maxima moments. (iii) The mean first-passage time to a boundary has been In terms ofthe scaled variable (6171)lM {we can-write Eq. shown to scale as in one, two, and three dimensions, 0 c S.3as 2. Thus, the scaling of the mean first-passage times can serve as an operational measure ofthe fractal dimensionality of A the process. Z (e) [5.4]J .-)=61+at (iv) The maxima moment Z.(t)'in a time interval t has been Downloaded by guest on September 29, 2021 Applied Mathematical Sciences: Seshadri and West Proc. Nad Acad. Sci. USA 79 (1982) 4505
shown to scale as tr/M in one, two, and three dimensions and /i = 2, D = 3/2 which agrees with known results. When A are finite as long as a < ,' and are infinite when a 2 a, i < decreases from 2 to 1/2, D decreases monotonically from 3/2 2. These moments are sensitive measures of A and hence ofthe to 0. For A -- 1/2, D = 0. fractal dimensionality of the process. (v) The fractal dimensionality of other properties of Levy The authors acknowledge the many stimulating conversations with processes can also be discussed in terms of the characteristic Prof. Katia Lindenbergand Prof. E. XW. Montroll. The financial support function. We present heuristic arguments for determining the ofthis research by the Office ofNaval Research and the Air Force Office fractal dimensionality of the zero-crossing set and the graphs of Scientific Research is also acknowledged. of such processes. Zero-Crossing Set. A simple scaling calculation shows that 1. Mandelbrot, B. B. (1977) Fractals, For-r, Chance and Dimension the probability of occupation of the origin of a Lvy process is (Freeman, San Francisco) P(0,t) - tC"Q The cumulative lime spent at the origin between 2. Frise, U., Sulem, P. L. & Nelkin, M. (1978)1J. Fluid Mech. 87, 719. times 0 and t is proportional to t1-"t*, up to an additive constant. 3. Siggia, E. D. & Aref. H. (1980) Ann. N.Y. Acad. Sci. 367, 368. We associate the fractal dimensionality ofthe zero-set with the 4. Ruelle, D. & Takens, F. (1971) Comm. Math. Phys. 20, 167. scaling of the time spent at the origin for t > 0-i.e., D = 1 5. Monin, A. S. (1978) Sov. Phys. Usp. 21, 429. - 1/,. This association is valid for A 2. 1. This result is con- 6. Rabinovich, M. 1. (1978) Sov. Phys. Usp. 21, 443. sistent with that of the Brownian process for which jt = 2, D 7. Scher, H. & Montroll, E. (1975) Phys. Rev. B 12, 2455. = 1/2 and also that ofthe Cauchy process for which = D 8. Scher, H. & Montroll, E. (1973)J. Stat. Phys. 9, 101. A 1, 9. Feller, W. (1966) An Introduction to Probability Theory and Its = 0. For all values of