arXiv:cond-mat/0102346v1 [cond-mat.stat-mech] 20 Feb 2001EPJ manuscript No. (will be inserted by the editor)Stochastic equations generating continuous multiplicative cascadesFran¸ cois Schmitt1and David Marsan21Vrije Universiteit Brussel, Dept. of Fluid Mechanics, Pleinlaan 2, 1050 Brussels, Belgium, e-mail:

[email protected] 2LGIT, Universit´ e de Savoie Campus scientifique 73376 Le Bourget du Lac, Cedex, FranceReceived: date / Revised version: dateAbstract. Discrete multiplicative turbulent cascades are described using a formalism involving infinitely divisible random measures. This permits to consider the continuous limit of a cascade developed on acontinuum of scales, and to provide the stochastic equations defining such processes, involving infinitelydivisible stochastic integrals. Causal evolution laws are also given. This gives the first general stochastic equations which generate continuous multifractal measures or processes.PACS. 02.50.Ey Stochastic processes – 05.40.Fb Random walks and L´ evy flights1 IntroductionMultiplicative cascades were first introduced in turbulenceto model the energy flux in the inertial range. The cas- cade formalism was originally introduced as a discrete (inscale) procedure, with a fixed (often 2) scale ratio be- tween the scale of a structure and that of the daughter structures (see [1,2,3,4]). Discrete cascade models lead to discrete scale invariance, characterized by log-periodic modulations [5,6,7,8]. On the other hand, a continuous symmetry leading to scaling for any scale ratio has beenproposed, and corresponds to a scale densification of cas-cade models [9,10,11,12,13,14]. Scale densification im-plies the use of infinitely divisible (ID) random variables,defining cascade models that can be called log-ID [10,11, 13], and have been compared to experimental data in var- ious studies [15,16,17,18,19,20]. This then leads to an interrogation: discrete cascade models are built using a simple recursive multiplicative procedure (see below), but what is the continuous limit of this procedure? What is the stochastic equation generating continuous multifrac- tals? Up to now, the process generated by such contin- uous multiplicative cascades has not been explicitely de- scribed in the general case; only its statistical momentsare given [13], or some general relation verified by the pdfat different scales [21]. This paper aims at detailing how ID random measures can be introduced for discrete cas- cades; their continuous limit is then given in the form of ID stochastic integrals. The stochastic evolution laws that generate causal continuous multifractal processes will also be provided. This is of direct importance for providing estimators of the future state of the process.2 Discrete multiplicative cascadesSoon after Kolmogorov and Obukhov published their log- normal proposal for the statistics of the small-scale dis-sipation field [22,23], experimental studies showed thatthe dissipation field had long-range power-law correlations [24,25]. This lead Yaglom to propose a random cascade model with long-range correlations and small-scale lognor- mal statistics [1]. Yaglom’s multiplicative cascade model is at the basis of most cascade models introduced later to ac- count for turbulent intermittency. It is a discrete (in scale) model, but most of its properties are shared by continuous models. A lognormal pdf is assumed, but this is an unnec- essary hypothesis, as is now well recognised. This model is multiplicative, nested in a recursive manner. The multi-plicative hypothesis generates large fluctuations, and the stacking generates long-range correlations, giving spatiallyto these large fluctuations their intermittent character.As is classically done (see e.g. Frisch [26]), we definethe cascade yielding a dissipation field ǫ(x) at the smallest scale ℓ0, as the productǫ(x) =nYi=1Wi,x(1)of n independent realisations Wi,xof a common, positive law. The cascade is developed from the largest scale L down to ℓ0= L/Λ where Λ = λn1is the total scale ra- tio and λ1> 1 is the constant scale ratio between two consecutive scales. Generally one assume for convenience λ1= 2, but we will later on consider the λ1→ 1 limit cor- responding to a continuous cascade [27]. Since all random variables are independent, one has the moments of order2Fran¸ cois Schmitt, David Marsan: Stochastic equations generating continuous multiplicative cascadesq > 0 of ǫ:=nYi=1=n= ΛK(q)(2)where K(q) = logλ1. For discrete cascades, the analytical expression taken by K(q) is only loosely con- strained a priori: by conservation K(0) = 0, K(1) = 0, and since K(q) is – up to a logλ1factor – the second Laplace characteristic function of the random variable logW, it is a convex function (see [28]). To densify the cascade described above, we keep thetotal scale ratio Λ large but fixed; the continuous limit can be obtained by increasing the total step number n, hence λ1= Λ1/n→ 1+(see [27,13,19,29]). Equation (1) then shows that, in this limit, logǫ is an ID random variable (see [28] for ID random variables): continuous cascade models are log-ID [10,11,13,14]. This restricts the eligible cascademodels, since ID laws define a specific family of probability distributions. Let us also mention one of the main properties of multi- plicative cascades: long-range power-law correlations. Fol- lowing the development given by Yaglom [1], one can con- sider two points separated by a distance r as having com- mon ancestors from steps 1 to p, and separated (hence the corresponding random variables are independent) paths for steps p to n, where λp1≈ r. Direct calculations then provides the classical result for two-points correlations ofmultifractal fieds [30,31]:≈ ΛK(p+q)rK(p)+K(q)−K(p+q)(3)for p > 0 and q > 0. Since K(q) is non-linear for multi- fractal distributions, the exponent K(p)+K(q)−K(p+q)quantifies the long-range power law correlations of multi- fractal measures. For p = q = 1, this yields the µ = K(2) exponent originally given for usual correlations by Yaglom [1]. The scaling law for the moments (1) and the power-law correlations (3) are the two signatures of multifractality, that are to be recovered by multifractal stochastic models,as we define them below.3 ID random measures and stochastic integralsThe densification of a multiplicative cascade implies that logǫ is an ID random variable; we now express this den-sification in the form of ID stochastic integrals. Since we need below to consider moments of order q > 0 of the ID random variable Γ = logǫ, we consider ID laws for which the second Laplace characteristic function ΨX(q) = log converges for a given domain Θ. For an ID random variable X, one has the general result that ∀n integer, ΨX(q) nis still a second characteristic function. This showsthat a family of ID laws can be defined: two ID laws belong to the same family if their second characteristic function is proportional. Then each ID family can be character- ized by a reference function Ψ0(q). We choose a referencefunction such that Ψ0(1) = 1, and for an ID random vari-able X, we define its scale S(X) as the proportionality factor, giving the general identity ΨX(q) = S(X) Ψ0(q). The scale is a positive real number; it is an additive func- tion since for two independent ID random variables X and Y of the same family, it is easily checked that we have S(X +Y ) = S(X)+S(Y ). An ID random variable is then uniquely characterized by its reference second char- acteristic function and its scale (relative to this reference function). As examples, Ψ0(q) = q2for a Gaussian law, and Ψ0(q) = log2(1 + q) for a Gamma law.We then define ID random measures as set functions M(A), such that ∀A, M(A) is an ID random variable, with a scale given by S(M(A)) = m(A), m(A) being the control measure of M(A). We can easily check that M(A) possesses the basic additive property of random measures: for two sets A and B with A ∩ B = ∅, let usnote C = A ∪ B. By definition, M(C) is a ID law, andits scale verifies: S(M(C)) = m(C) = m(A) + m(B) = S(M(A))+S(M(B)), hence that M(C) = M(A)+M(B). This corresponds to the following second characteristic function for M(A):ΨM(A)(q) = m(A) Ψ0(q)(4)This expression takes a more familiar form in the 1D case when A = [0,t] is an interval, and taking M(A) = Y (t) − Y (0) = Y (t) where Y is a process with independent and stationary increments. Then one has the classical result ΨY (t)(q) = t ΨZ(q) where Z is the stationary process given by Z(t) = Y (t) − Y (t − 1). In the following we keep the random measure notation, which is more general. Having introduced an ID random measure M, a stochas- tic integral can be built (see e.g. [33]), as a Stiltjes integral:Zbaf(t)M(dt) = limn→∞n−1Xi=0f(a + ib − an)M([a + ib − an,a + (i + 1)b − an]) (5)As we will see below, the densification of the cascade leads to a stochastic integral with f(t) = 1 ∀t. In this case, the second characteristic function of the integral has a simple expression. Let us note I =RAM(dx). By additive prop-erty, I is still an ID law of the same family as M, and its scale is given by S(I) = S?RAM(dx)?=RAdx = m(A), such that we have still Eq. (4) with M(A) =RAM(dx) and m(A) =RAdx. Let us note that when f is not identi-cally 1, the result is not so simple, since an addition of ID random variables belongs to the same family, but not a lin- ear combination. One has is this case (if the integral con- verges, see [33]): ΨI(q) =RAΨ0(qf(x))dx, showing thatin general, ΨI(q) is not proportional to Ψ0(q).4 Densification of the cascade and stochastic equationsLet us introduce λ a variable scale ratio, verifying 1 ≤λ ≤ Λ, where Λ is the fixed largest scale ratio. We alsoFran¸ cois Schmitt, David Marsan: Stochastic equations generating continuous multiplicative cascades3introduce R = logΛ and r = logλ. The elementary scale ratio introduced above writes now λ1= λ1/n= eR/n. The discrete cascade corresponds to introducing a stochastic kernel M and intervals Apand Bpsuch thatΓ(x) = logǫ(x) =n−1Xp=0M (Ap,Bp(x))(6)where here M(A,B) is a random variable depending only on m(A), giving ΨM(A,B)(q) = m(A)Ψ0(q). The intervals Apand Bp, responsible for the cascading parent/children structure, are built in the following way: the width of Apis linear in r, giving Ap= [pRn,(p+1)R n]. The intervals Bp(x) are centered in x and of width proportional to λp1= epR/n, giving Bp(x) = [x −τ 2epR/n,x +τ 2epR/n] where τ = L/Λis the resolution. The densification, corresponding to n → ∞, transforms then Eq. (6) into a stochastic integral andusing Eq. (5) (with f=1), we obtain finally that:ǫΛ(x) = Λ−cexpZΛ1M?cdλλ,DλI0(x)? (7)where c > 0 is a parameter, I0(x) is the interval of length τ centered in x, and Dλis the dilatation operator of factor λ. At a given position x, the stochastic integral corresponds to a kernel visiting a conical structure, as represented in Fig. 2. This expression can be generalized to d-dimensional domains, and also to anisotropic scaling symmetries.It can be easily verified that this stochastic equationgenerates a multifractal field. Indeed, the moments are scaling as = ΛK(q)withK(q) = c(Ψ0(q) − q)(8)Moreover the two-points statistics can also be recovered: as was done above for Eq. (3), the correlation involves two integrals which have no inter- section (and thus are independent random variables) for λ 0, where 0 ≤ α ≤ 2 is the L´ evy index; α = 2 for the Gaussian case [33,34]. We have here Ψ0(q) = qα; we note that when α x) ≈ x−α), whereas positive fluctuations have an exponential decay [9,32]. Then, by splitting Eq. (6) into two inte- grals, corresponding to backward and forward domains, and introducing the change of variables u = x −τ 2λ andv = x+τ 2λ respectively, one obtains (introducing the L´ evymeasure Lα(du) = M(du,[u,x])) a stable stochastic inte- gral:ǫΛ(x) = Λ−cexpZA(x)|u − x|−1/αdLα(cu)(10)where A(x) = [x−X/2,x−τ/2]∪[x+τ/2,x+X/2]. This equation corresponds to the exponential of a fractional integration (over a limited domain) of order 1 − 1/α of a L´ evy-stable noise. This expression was already given [9, 31] by phenomenological arguments, and is here derivedas a direct consequence of the densification. When the position is time, we obtain the following causal stochastic evolution equation for a logstable multifractal generatedwith a fixed scale ratio Λ = T/τ:ǫΛ(t) = Λ−cexpZt−τt−T(t − u)−1/αdLα(cu)(11)This can be directly used in numerical simulations. For lognormal multifractals Lαis replaced by the Wiener mea- sure W.5 ConclusionWe now discuss the new results provided by our approach, compared to related papers. She and Waymire [13] have given a general expression for K(q) for continuous cas- cades, using the canonical L´ evy-Khinchine representation for the second characteristic function of ID laws. This could of course also be provided here. On the other hand, we have pushed further the analysis, since the process it- self was not studied in [13]. Castaing and collaborators have studied the convolution properties of the probability density of velocity increments under a change of scale, for continuous cascades [21,14,18]. This property is recovered here (it can be obtained from Eq. (7)), and we also pro- vide the stochastic equation for the process itself. Other studies have provided Fokker-Planck [35] or Langevin [36] equations for the cascade process in the scale-ratio space. These equations apply only to lognormal cascades. Thesestudies consider a fixed spatial (or temporal) position and provide stochastic equations for the cascade process devel- oping at this particular position. This framework is thendifferent from ours by the fact that (i) the position is not taken into account, whereas we included the position in our equations, and (ii) we provided equations for the gen- eral log-ID case, and not only for the lognormal case.4Fran¸ cois Schmitt, David Marsan: Stochastic equations generating continuous multiplicative cascadesWe have obtained continuous multifractals as the ex- ponential of a stochastic integral. This formalisation is of great interest for theoretical studies since it provides thefirst general stochastic equations for continuous multifrac- tals. These equations can be generalized to anisotropic sit- uations, and their dynamical properties (i.e. error growth and the correspondingpredictability limit, or return times) are now open to theoretical studies, whereas before such studies were possible only through numerical simulations. We have also provided the general evolution equation for causal continuous multifractal processes. We have shown how these equations simplify for log-stable and lognormal multifractals. These equations can be used for numerical simulations of continuous multifractals. In the general log-ID case, Eq. (7) can be used as follows: a 2D ID noise must be gener- ated, in the r (r = logλ is the logarithm of the scale ratio) - position plane. Then for each position, a numeri- cal path-integration is performed in this plane, a