EFTA01147164.pdf

J. Phys. I France 7 (1997) 431-444 MARCH 1997. PACE 431 Convergent Multiplicative Processes Repelled from Zero: Power Laws and Truncated Power Laws Didier Sornette (1,20 ) and Rama Cont (t) (t ) Laboratoire de Physique de la Matière Condensée ("), Université des Sciences, BP 70, Parc Valrose, 06108 Nice Cedex 2, France (2) Department of Earth and Space Science, and Institute of Geophysics and Planetary Physics, University of California, Los Angeles, California 90095, USA (Received 2 September 1996, received in final form 12 November 1996, accepted 20 November 1996) PACS.05.40.4-j — Fluctuation phenomena, random processes, and Brownian motion PACS.64.60.Ht — Dynamic critical phenomena PACS.05.70.Ln — Nonequilibrium thermodynamics, irreversible processes Abstract. —Levy and Solomon have found that random multiplicative processes we = )41M...À/ (with A5 > 0) lead, in the presence of a boundary constraint, to a distribution P(tue) in the form of a power law w7(1+14). We provide a simple exact physically intuitive derivation of this result based on a random walk analogy and show the following: 1) the result applies to the asymptotic (t oo) distribution of w, and should be distinguished from the central limit theorem which is a statement on the asymptotic distribution of the reduced variable *(log tee — (log tve)); 2) the two necessary and sufficient conditions for P(we ) to be a power law are that (log Xj) < 0 (corresponding to a drift we i 0) and that we not be allowed to become too small. We discuss several models, previously thought unrelated, showing the common underlying mechanism for the generation of power laws by multiplicative processes: the variable log we undergoes a random walk repelled from —oc, which we describe by a Fokker-Planck equation. 3) For all these models, we obtain the exact result that p is solution of (À") = 1 and thus depends on the distribution of A. 4) For finite t, the power law is cut-off by a log-normal tail, reflecting the fact that the random walk has not the time to scatter off the repulsive force to diffusively transport the information far in the tail. Résumé. — Levy et Solomon ont montré qu'un processus multiplicatif du type w, = At )42... (avec Al > 0) conduit, en présence d'une contrainte de bord, à une distribution P(tel ) en loi de puissance w7(14."). Nous proposons une dérivation simple, intuitive et exacte de ce résultat basée sur une analogie avec une marche aléatoire. Nous obtenons les résultats suivants: 1) le régime de loi de puissance décrit la distribution asymptotique de we aux grands temps et doit être distingué du théorème limite central décrivant la convergence de la variable réduite *(log w, — (log tue)) vers la loi Gaussienne; 2) les deux conditions nécessaires et suffisantes pour que P(we ) soit une loi de puissance sont (log ?y) < 0 (correspondant à une dérive vers zéro) et la contrainte que we soit empêchée de trop s'approcher de zéro. Cette contrainte peut être mise en oeuvre de manière variée, généralisant à une grande classe de modèles le cas d'une barrière réfléchissante examiné par Levy et Solomon. Nous donnons aussi un traitement approximatif, devenant exact dans (')Author for correspondence (e-mail: sornetteenaxos.unice.fr) (" ) CNRS URA 190 © Les Éditions de Physique 1997 EFTA01147164 432 JOURNAL DE PHYSIQUE I N°3 la limite oh la distribution de A est etroite ou log-normale en terme d'equation de Fokker- Planck. 3) Pour tour cos modeles, nous obtenons le rosultat general exact que l'exposant p est la solution do ('equation (A") = 1. µ est done non-universel et depend de la speciftcite de la distribution de A. 4) Pour des t finis, la loi de puissance est tronquee par une queue log-normale due a une exploration finie de la march° aleatoire. 1. Introduction Many mechanisms can lead to power law distributions. Power laws have a special status due to the absence of a characteristic scale and the implicit (to the physicist) relationship with critical phenomena, a subtle many-body problem iu which self-similarity and power laws emerge from cooperative effects leading to non-analytic behavior of the partition or characteristic function. Recently, Levy and Solomon (1J have presented a novel mechanism based on random multi- plicative processes: Wt+L = Atwt, (1) where At is a stochastic variable with probability distribution mat) and we express we in units of a reference value wo which could be of the form en, with r constant. All our analysis below then describes the distribution of tut normalized to tut„ in other words in the "reference frame" moving with wo. At the end, we can easily make reappear the scale wo by replacing everywhere w by w/tutv Taken literally with no other ingredient, expression (1) leads to the log-normal distribution [2-4]. Indeed, taking the logarithm of (1), we can express the distribution of log w as the convolution of t distributions of log A. Using the cumulant expansion and going back to the variable we leads, for large times t, to 1 1 1 vt)2 , (2) ntut) = =Itte t. exti l—M (1°g wt where v = (log A) E focc dA log AII(A) and D = ((log 61 )2) — (log A) . Expression (2) can be rewritten 1 1 ea(iti)vt (3) P(ivt ) a ptl÷P(ur) with 1 , we Awe) = 2Dt tog —. (4) evt Since µ(we) is a slowly varying function of we, this form shows that the log-normal distribution can be mistaken for an apparent power law with an exponentµ slowly varying with the range we which is measured. Indeed, it was pointed out [51 that for we C C(v÷2", µ(wt) < 1 and the log-normal is undistinguishable from the 1/we distribution, providing a mechanism for 1/f noise. However, notice that µ(wt ) —> oo far in the tail we a e("+")}1 and the log-normal distribution is not a power law. The ingredient added by Levy and Solomon 11] is to constrain we to remain larger than a minimum value wo > 0. This corresponds to put back we to we as soon as it would become smaller. To understand intuitively what happens, it is simpler to think in terms of the variables xe = log we and I = log A, here following [1]. Then obviously, the equation (1) defines a random walk in x-space with steps I (positive and negative) distributed according to the density EFTA01147165 IsT°3 CONSTRAINED CONVERGENT MULTIPLICATIVE PROCESSES 433 (x) 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 O 1 2 3 4 5 7 X/XO Fig. 1. — Steady-state exponential profile of the probability density of presence of the random walk with a negative drift and a reflecting barrier. distribution w(f) = eill(el). The distribution of the position of the random walk is similarly defined: P (xt, t) = et' P(eit , t). • If v 5- (I) = (log A) > 0, the random walk is biased and drifts to +oo. As a consequence, the presence of the barrier has no important consequence and we recover the log-normal distribution (2) apart from minor and less and less important boundary effects at xo = log wo, as t increases. Thus, this regime is without surprise and does not lead to any power law. We can however transform this case in the following one v (/) < 0 with a suitable definition of the moving reference scale wo en such that, in this frame, the random random drifts to the left. But the barrier has to stay fixed in the moving frame, corresponding to a moving barrier in the unsealed variable wt. • If v (I) < 0, the random walk drifts towards the barrier. The qualitative picture is the following (see Figs. 1 and 2): a steady-state (t oo) establishes itself in which the net drift to the left is balanced by the reflection on the reflecting barrier. The random walk becomes trapped in an effective cavity of size of order DRY with an exponential tail (see below). Its incessant motion back and forth and repeated reflections off the barrier and diffusion away from it lead to the build-up of an exponential probability (concentration) profile (and no more a Gaussian). The probability density function of the walker position x is then of the form CI" with µ tt ltd/D. As x is the logarithm of the random variable to, then one obtains a power law distribution for w of the form w-(1.+P). We first present an intuitive approximate derivation of the power law distribution and its exponent, using the Fokker-Planck formulation in a random walk analogy. In Section 2.2, the problem is formulated rigorously and solved exactly in Section 2$. Sections 2.3 and 2.4 are generalization of the process (1). The explicit calculation of the exponent of the power law distribution is done using a Wiener-Hopf integral equation, showing that it is controlled by extreme values of the process. EFTA01147166 434 JOURNAL DE PHYSIQUE I N°3 t 0441 00 5 1.46 end 0 b(t) 5 1 100 X(t) 10- 60 - drift -WI 40 - 20 200 400 t 600 800 1000 0 r o DIM a) b) Fig. 2. — a) A typical trajectory of the random walker at large times showing the multiple reflections off the barrier. b) The time evolution of the Reston variable defined by the equation (19) with at uniformly taken in the interval [0.48;1.48] leading to p xs 1.47 according to (17) and b, uniformly taken in the interval [0;1]. Notice the intermittent large excursions. 2. The Random Walk Analogy In the xi = log wt and I t = log At variables, expression (1) reads = xt + (5) and describes a random walk with a drift (I) < 0 to the left. The barrier at to = log wo ensures that the random walk does not escape to —oo. This process is described by the Master equation [1] +00 P(x,t + 1) = x(1)P(x (6) 2.1. PERTURBATIVE ANALYSIS. — 11:i get a physical intuition of the underlying mechanism, we now approximate this exact Master equation by its corresponding Fokker-Planck equation. Usually, the Fokker-Planck equation becomes exact in the limit where the variance of 7r(1) and the time interval between two steps go to zero while keeping a constant finite ratio defining the diffusion coefficient [6]. In our case, this corresponds to taking the limit of very narrow tr(I) distributions. In this case, we can expand P(x — I, t) up to second order OP , 1,2 02P P(x — 1, t) P(x,t)— t + —4 — Ox ') 2 Ox2 IC") leading to the Fokker-Planck formulation OP(x,t) _ (x,t) _ OP(x,t) 64P(x,t) V +D (7) at Ox Ox axe where v = (1) and D = (Ii) — (1)2 are the leading cumulants of Il(log A). j(x, 0 is the flux defined by OP(x,t) j(x,t) = vP(x,t) D (8) Ox EFTA01147167 N°3 CONSTRAINED CONVERGENT MULTIPLICATIVE PROCESSES 435 Expression (7) is nothing but the conservation of probability. It can be shown that this description (7) is generic in the limit of very narrow rr distributions: the details of ir are not important for the large t behavior; only its first. two cumulants control the results 16). v and D introduce a characteristic "length" a = In the overdamped approxima- tion, we can neglect the inertia of the random walker, and the general Langeviu equation e x In Wr di F + Faun reduces to dx dt = v ÷7/(t)' (9) which is equivalent to the Fokker-Planck equation (7). q is a noise of zero mean and delta correlation with variance D. This form exemplifies the competition between drift v = —Iv' and diffusion µ(t). The stationary solution of (7), alta — 0, is immediately found to be Pc,,,(2)= A — it (10) with _ Ivl µ= A and B are two constants of integration. Notice that, as expected in this approximation scheme, µ is the inverse of the characteristic length 1. In absence of the barrier, the solution is obviously A = B = 0 leading to the trivial solution Pc,(x)= 0, which is indeed the limit of the log-normal form (2) when t —) oo. In the presence of the barrier, there are two equivalent ways to deal with it. The most obvious one is to impose normalization fr. pogsr=i, ao (12) where x0 a log we. This leads to Poo(x) = µe "(_—_°) (13) Alternatively, we can express the condition that the barrier at x0 is reflective, namely that the flux j(x0) = 0. Let us stress that the correct boundary condition is indeed of this type (and not absorbing for instance) as the rule of the multiplicative process is that we put back we to wo when it becomes smaller than tvo, thus ensuring we ≥ wo. An absorbing boundary condition would correspond to kill the process when we ≤ wo. Substituting (10) in (8) with j(xo) = 0, we retrieve (13) which is automatically normalized. Reciprocally, (13) obtained from (12) satisfies the condition j(xo) a. 0. There is a faster way to get this result (13) using an analogy with a Brownian motion in equilibrium with a thermal bath. The bias (I) < 0 corresponds to the existence of a constant force —Iv' in the —x direction. This force derives from the linearly increasing potential V = In thermodynamic equilibrium, a Brownian particle is found at the position x with probability given by the Boltzmann factor e- Oluir. This is exactly (13) with D = 1/$ as it should from the definition of the random noise modelling the thermal fluctuations. Translating in the initial variable we = e, we get the Paretiau distribution pw 1+" PoD(wi) — °4, (14) we EFTA01147168 436 JOURNAL DE PHYSIQUE I N°3 with p given by (11): l(log A)I µ = ((log A)2) — (log A)2 (15) These two derivations should not give the impression that we have found the exact solution. As we show below, it turns out that the exponential form is correct but the value of µ given by (15) is only au approximation. As already stressed, the Fokker-Planck is valid iu the limit of narrow distributions of step lengths. The Boltzmann analogy assumes thermal equilibrium, i.e. that the noise is distributed according to a Gaussian distribution, corresponding to a log-normal distribution for the A's. These restrictive hypothesis are not obeyed in general for arbitrary 11(A). The power law distribution (14) is sensitive to large deviations not captured within the Fokker-Planck approximation. 2.2. Exatrr ANALYSIS. - In the general case where these approximations do not hold, we have to address the general problem defined by the equations (5, 6). Let us consider first the case where the barrier is absent. As already stated, the random walk eventually escapes to —0O with probability one. However, it will wander around its initial starting point, exploring maybe to the right and left sides for a while before escaping to —oo. For a given realization, we can thus measure the rightmost position xmaa it ever reached over all times. What is the distribution P..(141ax(0,x.))? The question has been answered in the mathematical literature using renewal theory ( [7], p. 402) and the answer is 9,,,a4(Max(0, •-••• , (16) with p given by r e° x(1)91d1 = +co 11(A)AedA = 1. (17) 00 0 The proof can be sketched in a few lines [7] and we summarize it because it will be useful in the sequel. Consider the probability distribution function M(x) r e. P.,„(x„,,,j(lx,„„„, that x„. ≤ x. Starting at the origin, this event r m.. ≤ x occurs if the first step of the random walk verifies xi. = y ≤ x together with the condition that the rightmost position of the random walk starting from —xt is less or equal to x — y. Summing over all possible y, we get the Wiener-Hopf integral equation Af (z) = LMfr- y)r(y)dy. (18) It is straightforward to check that AI(x) cox for large x with p given by (17). We refer to [7] for the questions of uniqueness and to [9,10] for classical methods for handling Wiener- Hopf integral equations. We shall encounter the same type of Wiener-Hopf integral equation in Section 2.5 below which addresses the general case. How is this result useful for our problem? Intuitively, the presence of the barrier, which prevents the escape of the random walk, amounts to reinjecting the random walker and enabling it to sample again and again the large positive deviations described by the distribution (16). Indeed, for such a large deviation, the presence of the barrier is not felt and the presence of the drift ensures the validity of (16) for large x. These intuitive arguments are shown to be exact in Section 2.5 for a broad class of processes. Let us briefly mention that there is another way to use this problem, on the rightmost position r m.., ever reached, to get an exponential distribution and therefore a power law dis- tribution in the ws variable. Suppose that we have a constant input of random walkers, say at EFTA01147169 N°3 CONSTRAINED CONVERGENT MULTIPLICATIVE PROCESSES 437 the origin. They establish a uniform flux directed towards —oo. The density (number per unit length) of these walkers to the right is obviously decaying as given by (16) with (17). This provides an alternative mechanism for generating power laws, based on the superposition of many convergent multiplicative processes. Let us now compare the two results (15, 17) for µ. It is straightforward to check that (15) is the solution of (17) when r(l) is a Gaussian i.e. 11(A) is a log-normal distribution. (15) can also be obtained perturbatively from (17): expanding e l as Oa = 1 + µl + + ... up to second order and re-exponentiating, we find that the solution of (17) is (15). This was expected from our previous discussion of the approximation involved in the use of the Fokker-Planck equation. 2.3. RELATION WITH KESTEN VARIABLES. — Consider the following mixture of multiplicative and additive process defining a random affine map: 5144 = be + AiSi, (19) with A and b being positive independent random variables. The stochastic dynamical process (19) has been introduced in various occasions, for instance in the physical modelling of 1D disordered systems [11] and the statistical representation of financial time series [12]. The variable S(t) is known in probability theory as a Kesten variable [13]. Consider as an example the number of fish Si in a lake in the t-th year. The population .98+1 in the (t + 1)st year is related to the population St through (19). The growth rate Ai depends on the rate of reproduction and the depletion rate due to fishing as well as environmental conditions, and is therefore a variable quantity. The quantity be describes the input due to restocking from an external source such as a fish hatchery in artificial cases, or from migration from adjoining reservoirs in natural cases. This model (19) can be applied to the problems of population dynamics, epidemics, investment portfolio growth, and immigration across national borders [8]. The justification of our interest in (19) relies on the fact that it is the simplest linear stochastic equation that can provide au alternative modelling strategy for describing complex time series to the nonlinear deterministic maps. Notice that the multiplicative process, with a At that can take values larger than 1, ensures an intermittent sensitive dependence on initial conditions. The restocking term be, or more generally the repulsion from the origin, corresponds to a reinjection of the dynamics. It is noteworthy that these two ingredients, of sensitive dependence on initial conditions and reinjection, are also the two fundamental properties of systems exhibiting chaotic behavior. b = 0 recovers (1) (without the barrier). For b 0 0, it is well-known that for (log A) < 0, S(t) is distributed according to a power law P(St) Si11+") (20) with µ determined by the condition (17) [13] already encountered above (A") = 1. In fact, the derivation of (20) with (17) uses the result (16) of the renewal theory of large positive excursions of a random walk biased towards —oo (12]. Figure 3 shows the reconstructed probability density of the Kesten variable Si for Ai and be uniformly sampled in the interval [0.48;1.48] and in [0,1] respectively. This corresponds to the theoretical value µ 1.47. We have also constructed the probability density function of the variations - Se of the Kesten variable for the same values. We observe again a power law tail for the positive and negative variations, with the same exponent. This is not by chance and we now show that the multiplicative process with the reflective barrier and the Kesten variable are deeply related. First, notice that for (log A) < 0 in absence EFTA01147170 I3S JOURNAL DE PHYSIQUE I Probability density for Kasten variable Fig. 3. — Reconstructed natural logarithm of the probability density of the Kesten variable Sr as a function of the logarithm of Si, for 0.48 ≤ A, ≤ 1.48 and 0 ≤ b, ≤ 1, uniformly sampled. The theoretical prediction p 1.47 from (17) is quantitatively verified. of b(t), Si would shrink to zero. The term b(t) can be thought of as an effective repulsion from zero and thus acts similarly to the previous barrier coo. To see this more quantitatively, we form Si+1 Sr bt — + AI — 1. (21) s "tt, , as 4, 1.1. It has the same We make the approximation of writing the finite difference — status as the one used to derive the Fokker-Planck equation and will lead to results correct up to the second cumulant. Introducing again the variable x a log S, expression (21) gives the overdamped Langevin equation: dr at = b(t)en — Iv' + q(t), (22) where we have written A(t) —1 as the sum of its mean and a purely fluctuating part. We thus get v = (A) —1 t•-• (log A) and D a (q2) = (A2) — (A)2 = (log(A)2) — (log A)2. Compared to (9), we see the additional term b(t)e—', corresponding to a repulsion from the x < 0 region. This repulsion replaces the reflective barrier, which can itself in turn be modelled by a concentrated force. The corresponding Fokker-Planck equation is OP(r,0 OP(z,t) 02P(t t) — b(t)e'P(x,t) — (v + b(t)e—r) +D . (23) at Oza EFTA01147171 N°3 CONSTRAINED CONVERGENT MULTIPLICATIVE PROCESSES 439 It also presents a well-defined stationary solution that we can easily obtain in the regions x —) +oo and x —) —oo. In the first case, the terms b(t)en can be neglected and we recover the previous results (13) with ro now determined from asymptotic matching with the solution at x a —co. For x —) —co, we can drop all the terms except those iu factor of the exponentials which diverge and get P(x) V. Back in the we variable, Poo(St) is a constant for St —) 0 and decays algebraically as given by (14) with the exponent (11, 15) for St —) +oc. Beyond these approximations, we can solve exactly expression (21) or equivalently (19) and we recover (17). This is presented in Section 2.5 below. Again, notice that (11, 15) is equal to the solution of (17) up to second order in the cumulant expansion of the distribution of log A. It is interesting to note that the Kesten process (19) is a generalization of branching processes [14]. Consider the simplest example of a branching process in which a branch can either die with probability pc, or give two branches with probability p2 = 1 —po. Suppose in addition that, at each time step, a new branch nucleates. Then, the number of branches St+1 at generation t + 1 is given by equation (19) with lit a 1 and At — 24—ad , where 51+1 is the number of branches out of the St which give two branches. The distribUtion 11(A) is simply deduced from the binomial distribution of ji+1, namely (le+i :sr )1:4'+'pos i _i145:41/Iiii+ . For large St , II(A) is approximately a Gaussian with a standard deviation equal to 4Pasi—P° , i.e. it goes to zero for large S. We thus pinpoint here the key difference between standard branching processes and the Kesten model: in branching models, large generations are self-averaging in the sense that the number of children at a given generation fluctuates less and less as the size of the generation increases, in contrast to equation (19) exhibiting the same relative fluctuation amplitude. This is the fundamental reason for the robustness of the existence of a power law distribution in contrast to branching models in which a power law is found only for the special critical case Po = (for po > p2, the population dies off, while for po < po the population prolifates exponentially). The same conclusion carries out directly for more general branching models. Note finally that it can be shown that the branching model previously defined becomes equivalent to a Kesten process if the number of branches formed from a single one is itself a random variable distributed according to a power law with the special exponent p = 1, ensuring the scaling of the fluctuations with the size of the generations. 2.4. GENERALIZATION TO A BROAD CLASS OF MULTIPLICATIVE PROCESS WITH REPULSION AT THE ORIGIN. - The above considerations lead us to propose the following generalization testi. = ef (4" {At iee"-})Atwe, (24) where f —> 0 for wt oo and f {Ai, bt, ...})) —> co for tilt O. The model (1) is the special case f (we, {At, be,...}) = 0 for we > wo and f (tot, {At, be, ...}) = log(er, ) for tot ≤ coo. The Kesten model (19) is the special case f(ve,{Ah be,...})= log(1 + irlD-) w, • More generally, we can consider a process in which at each time step t, after the variable At is generated, the new value At tot (or At tut + bt in the case of Kesten variables) is readjusted by a factor egvhs{Ane"•••}) reflecting the constraints imposed on the dynamical process. It is thus reasonable to consider the case where (wt, {Abbt, ...}) depends on t only through the dynamical variables At (and in special cases b1), a condition which already holds for the two examples above. In the following Fokker-Planck approximation, we shall consider the case where f(tvi,{Ah lgi,...}) is actually a function of the product Aim, which is the value generated by the process at step t and to which the constraint represented by f(Aim) is applied. We shall turn back to the general case (24) in Section 2.5. In the Fokker-Planck approximation, f(Alto) defines an effective repulsive stochastic force. 'lb illustrate the repulsive mechanism, it is enough to consider the restricted case where f(wt) EFTA01147172 440 r( )i UN A', DE PHYSIQUE I N°3 V V(x) -tvl x ••• 0 7C Fig. 4. — Generic form of the potential whose gradient gives the force felt by the random walker. This leads to a steady-state exponential profile of its density probability, corresponding to a power law distribution of the we -variable. is only a function of we. This corresponds to freezing the random part in the noise term A, leading to the definition of the diffusion coefficient. In the random walk analogy, we thus have the force F(xt ) = f (we ) acting on the random walker. The corresponding Fokker-Planck equation is 02(x,1) 0(v + F(x))P(x,t) DO2P(x,t) (25) Ot Ox Ox2 F(x) decays to zero at x co and establishes a repulsion of the diffusive process in the negative x region: this is the translation in the random walk analogy of the condition f(wt ) oo for wt a0. With these properties, the tail of P(x) for large x and large times is given by 2,,c(x) and as a consequence tot is distributed according to a power law, with exponent µ given again approximately by (11, 15). The shape of the potential defined by v t F(x) -4 1f1, showing the fundamental mechanism, is depicted in Figure 4. As we have already noted, the bound wo leading to a reflecting barrier is a special case of this general situation, corresponding to a concentrated repulsive force at so. The expression (24) for the general model can be "derived" from the overdamped Langevin equation equivalent to the Fokker-Planck equation (25): dx = F(x) - Iv' + 71(t). (26) dt Let us take the discrete version of (26) as x1+1 = zt +F(xt)— +tie, replace with x1 = log tot and exponeutiate to obtain 1.01+1 = eF(lostai )Aftet, (27) where A, a e— l°1+q8. Since F(x) 0 for large to,, we recover a pure multiplicative model w1+1 = Aiw, for the tail. The condition that F(x) becomes very large for negative x ensures that we cannot decrease to zero as it gets multiplied by a diverging number when it goes to zero. 2.5. EXACT DERIVATION OF THE TAIL OF THE POWER LAW DISTRIBUTION. — The existence of a limiting distribution for we obeying (24), for a large class of f (w, {A, b, ...}) decaying to zero for large w and going to infinity for w 0, is ensured by the competition between EFTA01147173 N°3 CONSTRAINED CONVERGENT MULTIPLICATIVE PROCESSES 441 the convergence of w to zero and the sharp repulsion from it. We shall also suppose in what follows that Of (w, b, ...))18x 0 for w oo, which is satisfied for a large class of smooth functions already satisfying the above conditions. It is an interesting mathematical problem to establish this result rigorously, for instance by the method used iu (1,10). Assuming the existence of the asymptotic distribution P(w), we can determine its shape, which must obey = w emfors{"4.•••}) '12' At; (28) where (A, 6, represents the set of stochastic variables used to define the random process. The expression (28) means that. the l.h.s. and r.h.s. have the same distribution. We can thus write += += I'D da P„(v) = I dA II(A) f dw P,,.(w)5(v — Aug) = f —A II(A)P„,(1). o o o A Introducing V = log; x a log w and 1 a log A, we get +go P(V) =J di II(1)P,(1/ — (29) Taking the logarithm of (28), we have V = x — f (x, {A, b,...}), showing that V x for large x > 0, since we have assumed that f(x, {A,b, ...}) —> 0 for large x. We can write PoldV = Px (x)dx leading to P(V) — rzt Atov..? P1(V) for x oo. We thus recover the Wiener-Hopf integral equation (18) yielding the announced results (16) with (17) and therefore the power law distribution (14) for we with p given by (17). This derivation explains the origin of the generality of these results to a large class of con- vergent multiplicative processes repelled from the origin. 3. Discussion 3.1. NATURE OF THE SOLUTION. — To sum up, convergent multiplicative processes repelled from the origin lead to power law distributions for the multiplicative variable we itself. Ideally, this holds true in the asymptotic regime, namely after an infinite number of stochastic products have been taken. This addresses a different question than that answered by the log-normal distribution for unconstrained processes which describes the convergence of the reduced variable *(log — (log tue)) to the Gaussian law. Notice that this reduced variable tends to zero for our problem and thus does not contain any useful information. We have presented an intuitive approximate derivation of the power law distribution and its exponent, using the Fokker-Planck formulation in a random walk analogy. Our main result is the explicit calculation of the exponent of the power law distribution, as a solution of a Wiener-Hopf integral equation, showing that it is controlled by extreme values of the process. We have also been able to extend the initial problem to a large class of systems where the common feature is the existence of a mechanism repelling the variable away from zero. We have in particular drawn a connection with the Kesteu process well-known to produce power law distributions. The results presented in this paper are of importance for the description of many systems in Nature showing complex intermittent self-similar dynamics. 3.2. THE EXPONENT p. — In the Fokker-Planck approximation of the random walk analogy, µ is the inverse of the size of the effective cavity trapping the random walk. In this approximation, µ is a function of, and only of, the first two cumulants of the distribution of log A. lu particular, if the drift Itil < 2D, µ < 2 corresponding to variables with no variance and even no mean EFTA01147174 442 JOURNAL DE PHYSIQUE I N°3 when p < 1 avl < D). It is rather intuitive; large fluctuations in A lead to a large diffusion coefficient D and thus to large fluctuations in wi quantified by a small µ. Recall that the smaller µ is, the wilder are the fluctuations. Within an exact formulation, we have shown that there is a rather subtle phenomenon which identifies µ as the inverse of the typical value of the largest excursion against the flow of a particle in random motion with drift. This holds true for a large class of models characterized by a negative drift and a sufficiently fast repulsion from the negative domain (in the z-variable), i.e. from the origin (in the w-variable). 3.3. ADDITIONAL CONSTRAINT FIXING µ. — We recover the relationship relating µ to the minimum value wo in the reflecting barrier problem by specifying [1) the value C of the average (tot). Calculating the average straightforwardly using (14), we get (tot ) = wo t, leading to 1 µ— (30) 1— (wo/C) Notice that this expression is a special case of (17) and should by no mean be interpreted as implying that A is controlled by wo in general. This is only true with an additional constraint, here of fixing the average. The general result is that p is given by (17), i.e. at a minimum by the two first cumulants of the distribution of log A. 3.4. POSITIVE DRIFT IN THE PRESENCE OF AN UPPER BOUND. - Consider a purely multi- plicative process where the drift is reversed (log A) > 0, corresponding to an average exponen- tial growth of to, in the presence of a barrier wo limiting to, to be smaller than it. The same reasoning holds and a parallel derivation yields Poo(wt) = w. tupt (31) o with A ≥ 0 again given by (17). This distribution describes the values 0 < we < too. Notice that, if p > 1, the distribution is increasing with wt. This is obviously no more a power law of the tail, rather a power law for the values close to zero. For A < 1, 1%O(m) decays as a power law, however bounded by wo and diverging at zero (while remaining safely normalized). This shows that, when speaking of general power law distribution for large values, this regime is not relevant. Only the regime with negative drift and lower bound is relevant. However, in the case of Kesteu variables (21), if S, is growing exponentially with an average rate (log A1) > 0, and if the input flow b, is also increasing with a larger rate r, we define bt = where th is a stochastic variable of order one. We also define At = Ate. If r > (log at), then (log Ai) < 0. The equation (1) thus transforms into = Ate, +61, with St = ertät, and where i t and b, obey exactly the conditions for our previous analysis to apply. The conclusion is that, due to input growing exponentially fast, the growth rate of tus becomes that of the input, its average (which exists for p > 1) grows exponentially as (SO e" and its value exhibits large fluctuations governed by the power law probability density function P(St ) p. with µ solution of (Ar) = ep, leading to p - Att,t; in the second order cumulant approximation. 3.5. TRANSIENT BEIIAVIOR. - For t large but finite, the exponential (16) with (17) is trun- cated and decays typically like a Gaussian for z > r/v. t. Translated in the to variable, the power law distribution (14) extends up to wi and transforms into an approximately log-normal law for large values. Refining these results for finite t using the theory of renewal processes is an interesting mathematical problem left for the future. EFTA01147175 N°3 CONSTRAINED CONVERGENT MULTIPLICATIVE PROCESSES 443 3.6. NON-STATIONARY PROCESSES. - When the multiplicative process (1) is not stationary in time, for instance if v(t),D(t) or xs(t) become function of time, then their characteristic timer of evolution must be compared with r(z) = x2 /D. For "small" x such that t" (x) C r, the distribution P(x,t) keeps an exponential tail with an exponent adiabatically following v(t),D(t) or zo(t). We thus predict a power law distribution for ws but with an exponent varying with v and D according to equations (11, 15). For "large" x such that t*(x) ≥ r, the diffusion process has not time to reach x and to bounce off the barrier that the parameters have already changed. It is important to stress again the physical phenomenon at the origin of the establishment of the exponential profile: the repeated encounters of the diffusing particle with the barrier. For large x, the repeated encounters take a large time, the time to diffuse from x to the barrier back and forth. In this regime r(x) ≥ r, the exponential profile for P(x) has not time to establish itself since the parameters of the diffusion evolve faster that the "scattering time" off the barrier. The analysis of the modification of the tail in the presence of non-stationarity effects is left to a separate work. In particular, we would like to understand what are the processes which lead to an exponential cut-off of the power law in the we variable, corresponding to an exponential of an exponential cut-off in the x-variable. 3.7. STATUS OF THE PROBLEM. — Levy and Solomon [1] propose that the power law (14) is to multiplicative processes what the Boltzmann distribution is to additive processes. In the latter case, the fluctuations can be described by a single parameter, the temperature (/3-1) defined from the factor in the Boltzmann distribution e'es. In a nutshell, recall that the exponential Boltzmann distribution stems from the fact that the number It of microstates constituting a macro-state in an equilibrium system is multiplicative in the number of degrees of freedom while the energy E is additive. This holds true when a system can be partitionned into weakly interactive sub-systems. The only solution of the resulting functional equation fl(E1 + E2) = MEL )1/(E2) is the exponential. No such principle applies in the multiplicative case. Furthermore, the Boltzmann reasoning that we have used in Section 2.1 is valid only under restrictive hypotheses and provides at best an approximation for the general case. We have shown that the correct exponent it is in fact controlled by extreme excursions of the drifting random walk against the main "flow" and not by its average behavior. This rules out the analogy proposed by Levy and Solomon. Acknowledgments D.S. acknowledges stimulating correspondence with S. Solomon and U. Frisch. This work is Publication no. 4642 of the Institute of Geophysics and Planetary• Physics, University of California, Los Angeles. References [1] Levy M. and Solomon S., Power laws are logarithmic Boltzmann laws, Int. J. Mod. Phys. C 7 (1996) 595-601; Spontaneous scaling emergence in generic stochastic systems, Int. J. Mod. Phys. C7 (1996) 745-751. [2] Gnedenko B.V. and Kolmogorov A.N, Limit distributions for sum of independent random variables (Addison Wesley, Reading MA, 1954). EFTA01147176 444 JOURNAL DE PHYSIQUE I N°3 [3] Aitcheson J. and Brown J.A.C., The log-normal distribution (Cambridge University Press, London, England, 1957). [4] Redner S., Random multiplicative processes: An elementary tutorial, Am. J. Phys. 58 (1990) 267-273. [5] Montroll E.W. and Shlesinger M.F., Proc. Nat. Acad. Set USA 79 (1982) 3380-3383. [6] Risken H.. The Fokker-Planck equation: methods of solution and applications, 2nd ed. (Berlin; Nev. York: Springer-Verlag, 1989). [7] Feller W., An introduction to probability theory and its applications, Vol II, second edition (John Wiley and sons, New York, 1971). [8] Sornette D. and Knopoff L., Linear stochastic dynamics with nonlinear fractal properties, submitted. [9] Morse P.M. and Feshbach H., Methods of theoretical physics (McGraw Hill, New York, 1953). [10] Frisch If., J. Quant. Spectrosc. Radial. Transfer 39 (1988) 149. [11] de Calan C., Luck J.-M., Nieuwenhuizen Th.M. and Petritis D., On the distribution of a random variable occurring in 1D disordered systems, J. Phys. A 18 (1985) 501-523. [12] de Haan L., Resnick S.I., Rootzen H. and de Vries C.G., Extremal behavior of solutions to a stochastic difference equation with applications to ARCH processes, Stoch. Proc. 32 (1989) 213-224. [13] Kesteu H., Random difference equations and renewal theory for products of random ma- trices, Acta Math. 131 (1973) 207-248. [14] Harris T.E., The theory of branching processes (Springer, Berlin, 1963). EFTA01147177