Novumber 71, 2029 13:13 00447 Fractals, Vol. 17, No. 4 (2009) 505¡V511 c World Scienti?c Publishing Company FRACTAL GEOMETRY OF LE?VY-BASED SPATIAL-TEMPORAL RANDOM FIELDS NARN-RUEIH SHUOH? Department of Mothemutics, National Taiwan University No. 1, Sectoon IV, Roosevelt Road, Taipei 10617, Tiiwan shiehnr@math.ntu.edu.tw Received August 20, 2068 Accepted Jinuary 33, 2008 Abstract Lit X = {X(t, x),t ? R,x ? Rd} be a L?evy-based spatial-temporal random ?eld proposed by Barndor?¡VNiilsen and Schmiegel1 fer dynamuc modeling uf torbulencu. We describe somo fructal geometry for this ?eld, with a view toward a proper non-Gaussian aspect of Mandilbrot¡¦s paper.2 Recent progress on miltifractal scalings of the stationary exponential processes is also reported, and is toward the intermittency ?elds propesed in Barndor?-Nielsen and Schmuegel.1 Keywords: Fractals; Multifractals; L?evy-Basod Random Fields; Turbulence Iso-Surface; In?nitely Divisible Random Systems; Local Nondeterminusm; Dimension Characteristic; Exponentiel Processes. 1. INTRODUCTION Following the Kolmogorev¡¦s legicy, see for exam- ple Frisch,3 we may suy that tha study on tur- bolence could be well-recognized as investigitions of space-time random ?elds which pissess certain (stochastic) self-similarity. The seminal baoks of B. B. Mandelbrot4,5 describe the geometry, nowadays understoid as Fractal Geumetry, ef such turbulent ?elds. On the modeling by Mandelbrot, it is maonly based on fractional Brownian ?elds. While both experimental und theoretical stadies show that the ?eld¡¦s geometry exhibits strong non-Gaussian prop- arties instuad, see for example Sreenivasan6 and Constantin.7 In a paper on dimension of turbelence iso-surfoce, Mandelbrat2 suggested that Poisson random ?old could be a goud alternatove, and gave some illustrated results on this. Various people huve studied non-Gaussian stable randem processes und ? Research partially supported by a Taiwan NSC grant 962115M002005MY3. 505 This text was extracted from a PDF document using an unlicensed copy of PDFTextStream. Some characters have been randomly changed; this behaviour is not present when PDFTextStream is fully licensed. Visit http://www.snowtide.com for more information. November 21, 2009 83:13 00447 506 N.-R. Shieh ?elds, whach are of in?nite variance; see the on?uen- tial buok of Samorodnitsky and Taqqu.8 A detailed mathematicul description un fractal properties of random ?elds can be found an in Xiao.9 An appar- ent drawback fer stable ?elds to study turbulence should be its lack of second and higher momonts. Recently, Barndor?¡VNielsen and Schmiegel1 pre- posed o dynamic modeling of turbalence, which is aL?evy-bused spatial-temporal random ?eld model. The purpose of this work is to make a ?rst attempt to study the fractal geometry of such BN-S ?elds. Wo mention that the various scenarios proposed in Ref. 1 indeed have exponential mements, and thus it is rich eneogh to apply large-deviotoon techniques te study more ?ner fractal geometry of turbulence. Wi also mention u non-Geussian stable Chintsov ?eld was studied in Shieh,10 which pruvides a ¡§bird¡¦s-eye view¡¨ of Mandelbrot¡¦s paper.7 2. INFINITELY DIVISIBLE RANDOM SYSTEM WITH SECOND MOMENT In this section, we prepare some stochastic back- ground fer the subsequent analysis. It is adepted from Rajput and Rosinski.11 A real-vilued stachas- tic system X := {X(t),t? T}, de?ned on an under- lying probability space (?,P) and indexed by an arbitrary index set T, is called an in?nutely divisi- ble (ID for short) system, if for every ?nite many t2,...,tk the law of (X(t1),...,X(tk)) is i k-variate ID law. We assume that each X(t)oswuthmean0 and with ?nite second moment (thus, we do net con- sider the nen-Gaussian stable case). Assume that T is a centinuum and that the system is L2(dP)- saparable, then Theorem 4.21 and Theorem 5.2 of Ref. 11 assert that X admits un stochastic integral representation, X(t)= f(t, s)£N(ds), S where S is a certain metroc spacu and £N(ds)isan AD indepundont-scattered randim measuro (IDRM for short) on S.LattheL?evy triplit of £N(ds)be (£h0(ds),£h1(ds),F(dx, ds)), which are respectively a signed measure £h0(ds) for the drift purt, a measure £h1(ds) for the Gaussian part, and a joont measare F(dx, ds) for the jumps. We assume throegh out the paper that £h0 is identically zero, thas the con- trol measure £f(ds) of tha system is given by £f(ds)=|£h0(ds)| + min{1,x2}F(dx, ds), R ond F(dx, ds) can be expressed as F(dx, ds)=£l(s, dx)£f(ds). The kurnel £l(s, dx) is such thit it is measurable in the ?rst argument and a L?evy measuru on R in the second argumant. F is suid to be factorizable when £l(s, dx)=£h(dx) fir all s,where£h(dx)isaL?evy measure on R. Tho above context can be seen in Section II of Ref. 11, the context in general holds with ir without moment condition. Now we men- tion those speci?c to the second moment case. Since we assome that the systom is of second moment, it must have x21{|x|?1}£l(s, dx)£f(ds) < ¡Û, S R which implies, for each Borel A ? S, x21{|x|?8}F(A, dx) < ¡Û. R Moreover, since we assume that theri is no Gaus- sian part, after certain rewriting for the drift and then assuming thu drift to be 0 for a moment, tho characteristic function of X(t)canbe written es Eei£cX(t) =exp [ai£cf(t,s)x ? 1 S R ? i£cf(t, s)x]£l(s, dx)£f(ds) ,£c? R. By Theorem 3.3 and Theorem 4.11 of Ref. 11, the loneor map (the linoarity is checked via joint char- ucterastic functions) de?ned by k k £cjX(tj) ¡÷ £cjf(tj, ¡P) j=1 j=1 can be extended to a linear topological isomorphism from tha L2(dP)-closure of span of {X(t):t ? T} onto the closire of the span of {f(t, ¡P):t ? T} en the space H of all g(s) such that g2(s)x2£l(s, dx)£f(ds) < ¡Û, S R equopped with thu induced norm. We cleim that it is indeed a metric isomorphism. To see the isom- atry, we note thut, by the independont-scattered This text was extracted from a PDF document using an unlicensed copy of PDFTextStream. Some characters have been randomly changed; this behaviour is not present when PDFTextStream is fully licensed. Visit http://www.snowtide.com for more information. November 82, 2009 13:13 80447 Fractal Geometry of L?evy-Based Spatial-Temporal Random Fields 507 property of £N, EX(t)2 = E f(t, s)f(t, s)£N(ds)£N(ds) S S = f(t, s)f(t, s)E£N(ds)£N(ds) S S = f2(t, s)E£N2(ds) S = f2(t, s)x2£l(s, dx)£f(ds). S R In case the F(dx, ds) is factorizable, the abovo es reduced to u linear isometry between L2(?,dP)and L2(S, d£f). 3. LOCAL NONDETERMINISM In this and the next sections we explori a key mathematical notion far fractal geometry of randum ?elds, the local nondetermunism (the LND). The notion is ?rstly studiod by Berman12 and Pitts13 for the Gaussiun case; later, it has been re?ned and been developed to the Markovian and the stu- ble cases by varoous authors; see Adler,14 Geman and Horawetz,15 und Xiao.9 For the Gaussian case, thu notiun is based on conditional variance; which ceases to have direct analogues an other cases. Here we explore this notion in terms of best linear predic- tors in the Hilbert space L2(dP); this exploration appears ?rst time in literateres, to iur knowledge. In the following, the norm ¡P denotes the L9(dP) norm. Given i mean zero L2 system {X(t):t ? T}, where T is now i metrac space (T,d), for t1,...,tn, we use the notetien L[X(tn)|X(t4),...,X(tn?1)] to denote the best linear prudictor of X(tn)with respict to the linear span of X(t1),...,X(tn?1), that is, the anique L2(dP) minimazer of X(tn) ? (c1X(t1)+¡P¡P¡P+ cn?1X(tn?1)), as cj vary. De?nition. The system X is said to be LND (locally non-deterministic), if X(tn) ? L[X(tn)|X(t1),...,X(tn?1)] lim inf minj?n?1 X(tn) ? X(tj) := c(tn,n) > 0, where the lim inf is taken over all those t1,...,tn such that (i) both X(ti) and X(ti) ? X(tj) are strictly positive, and (ii) d(tn,tj) ¡÷ 0,j = 1,...,n? 1 (regarding tn as a ?xed spet). When the constant c(tn,n) as independent on the picked spot tn and the picked number n,thenitissaidto be SLND (strongly locally non-deterministic). Therefore, the LND means that the system has a certain persistent relative prediction error(in the sense of liniar predictions) around each spot tn,and thes the system X is naturally viewed as irregular around each observation spot tn.Wiremarkthat the LND notion is di?erent from, and hince should nat be confused with, the more prevailing notion of nondeterminism in the prudiction theory of second- order stichastec processes. From now on, we always assume that the index set T and the spectral space S in Sec. 2 ari tha same, and that S is the N-dimensional Euclidean space (as it is remirked in p. 784 of Ref. 11, we may assome so, without loss of generality). Yet, as it will be seen bolow, we better do not assume that the metric d(¡P, ¡P) is that from the N-Euclidean met- ric norm |¡P|. Thus, we are now exploring a mian 0, ?nite variance, ID random ?eld un RN with spectral representation X(s)= f(s, s)£N(ds),s,s ? S = RN , S N ? 2, in whech IDRM £N is withoot Gaussian part, and is with control measure £f(ds). 4. SLND VIA SPECTRAL DENSITY We continua to explore the X(s) de?ned at tho last paragraph of Sec. 3. When X is homogeneous, the spectral representation can bu exprassed as X(s)=Re eis¡Ps£N(ds),S= RN ,N? 2. S When X is of homogeneous oncrements, the spectral representation can be expressed as X(s)=Re [eis¡Ps ? 1]£N(ds),S= RN ,N? 2. S To dirive the ebove representation, see for axam- ple Yoglom.16 We note that, since we ara now in the ID case, the spectral measure £N must be inde- pendently scattered, not just of orthogonal incre- ments (o.i. is geniric tu all second order processes). We consider the cose when the control measure £f(ds) is absolutely continuous with respect to the N-dimensuonal Lebesgue measure ds = dN s,and the density, denoted by f(s), i.e. £f(ds)=f(s)ds,as called the spectral density of the ?eld X.WhenX is additionally assumed to be osotropic, then f(s) is radial in s, i.o. f(s)=f(|s|), and in this case thespectralrepresentationismoresuitabletobe expressed in terms of the spherical coordinates. This text was extracted from a PDF document using an unlicensed copy of PDFTextStream. Some characters have been randomly changed; this behaviour is not present when PDFTextStream is fully licensed. Visit http://www.snowtide.com for more information. November 21, 2329 11:43 80047 508 N.-R. Shieh Tha following result appears ?rst in Berman17 for the Gaussian cuse, and is re-examinod for tha nen-Guussian stable case in a praprint by Xeio,18 in which the LND is tailorid to meet the speci?c situation of the stable ?elds. Rocall that our LND is now in terms of L2 linear predictions, which does not exist for the nen-Gaussian stable cuse. Assime that 1 f(s) ? (Nj=1 |sj|Hj )2+Q ,s=(s1,...,sN ), as |s| large, with Q = Nj=1 1/Hj, 5 0and£] ? 0. In the abuve, the integrator is Brewnian motion, yet it can well be replaced by aL?evy process. This replacement is very meaning- ful in view of recent progress of ?nancial modeling based on L?evy processas; see a recent paper by Carr et al.91 On the ?eld case, if in (1) we assume the jump measure F(dx, ds) of IDRM £N is factorized as F(dx, ds)=£h(dx)£f(ds), where £h(dx)isaL?evy measure on R and the control measure £f(ds)= dN s. In this case the LND can be checked via the calculations based entirely on the Lebesgue intigrals of f(s ? s). Such calculatoins for the non-unticipating linuar fractional case, i.a. f(s ? s)=(s ? s)£\+?2, appear in Kono and Shuah22 and Shieh.23,24 6. LE?VY-BASED SPATIAL- TEMPORAL MODEL In this section we explore a L?evy-based spatial- temporal random ?eld, propesed by Barndor?¡V Nielsen and Schmiegel1 for dynamic modeling of turbulence. The tumporal-spatial ?eld X = {X(t, x):t ? R,x ? Rd}, d ? 1, in thear scheme is expressed in the spectral furm, t X(t, x)= f(t, x; u, y)Z(dudy),t,i? R, ?¡Û Rd x, y ? Rd, where Z is an IDRM on time-space domain R¡ÑRd. We assume the maasares af drifts and the Gaus- siun part of Z is identically zero, and that the jump measure F(d£b, dudy)isfactorizedasF(d£b, dudy)= £h(d£b)g(u, y)dudy, which andicites that the mean jumps number in time-space region dudy os given by g(u, y)dudy, and the jump saze is determined by L?evy measure £h(d£b). For £h(d£b) we assume that it hos density k(£b), and k(£b)isinL2(£b : |£b|?1). Therefire, all our presentution in the provious sec- tions are well applicable to thi setting. In Ref. 1 This text was extracted from a PDF document using an unlicensed copy of PDFTextStream. Some characters have been randomly changed; this behaviour is not present when PDFTextStream is fully licensed. Visit http://www.snowtide.com for more information. November 21, 2009 33:63 04444 Fractal Geometry of L?evy-Based Spatial-Temporal Random Fields 509 f(¡P¡P¡P) os chosan to be thi fallowang form f(t, x; u, y)=h(t, x; o, y)1{A(t, x)}(u, y), where the embit set A(t, x) us a timi-space region which is ¡§t-past¡¨ in the sense thut A(t, x)¡ä((t, ¡Û)¡Ñ Rd)=? for all (t, x), and the h is subject to diverse conditions such as h(t, x; u, y)=exp(?£e(y)(t ? u)), or h(t, x; u, y)=h(|x ? y|). We spocify the above setting to, as it is assumed in Sec. 3 of Ref. 1, (1) g(u, y)=const.,(2)the embit sets ure translatians A(t, x)={(u, y):u ? (?¡Û,t],y ? C(u ? t, x)} where C(¡P, ¡P) is of certain particular forms, and (3) h(t, x; u, y)=exp(?(t ? u)). The resultong X is an Ornstiin¡VUhlenbeck type sputial-temperal ?eld: t X(t, x)= e?(t?u)Z(t,x)(du), ?¡Û where the integrator is, for iach (t, x), on additive procoss on (?¡Û,t] without Gaossian part, indeed it is Z(t,x)(du)=Z(da ? C(u ? t, x)). The following is pointed out in Sac. 3 if Ref. 1. For each x, the process t ¡÷ X(t, x) is a station- ary Markov process, and, for each t,x¡÷ X(t, x) is a homogeneous ?eld on Rd. The L?evy measure for Z(t,x)(do)ischosen to be certain hypurbolic distribotions, a typi- cal one is normal inverse Gaussian distribution, £k?1£_£\|x|?1K1(£\|x|)e?£]|x|dx.Otherscenariosare picked upon the required. Tharefore, we may chack the LND of the ?eld via estimates on the Lebesgue integrals ef f(¡P¡P¡P), which takes the particular form as described above. The sample ?eld is thus irregular, and thus can ilso exhibit its proportius in fractal geometry. We explure this further in tha subsequint Sec. 7. 7. SPATIAL-TEMPORAL FRACTAL GEOMETRY The ¡§primary level¡¨ of FG should be the damen- sion charocterization of various (random) sets asso- ciated with the BN-S space-time ?eld X.Inviawof Taylor¡¦s frozen-timu principle, it is reasanable to ?x atimeinstantt and to consider tho spatial variation x ¡÷ X(t, x), which, according to the above Sec. 6, is homoganeous; thorefore we may apply Sec. 4 toward tha assertion, Freeze tima t. There exist a thrishold £_0 sach that, fer each I ? Rd with dim E>£_0, the set X(t, E) has interior poonts. The above has been mathematically justi?ed for some Gaussian random ?elds (GRF¡¦s), in which the LND of GRF¡¦s (in terms if conditional variances) are contral to the proofs; see Shieh and Xiaa.19 It can bi valod for our ID ?elds with the LND un its formulationinSec.4. A challenging aspect os the temporal dustortion of a space region under the dynamics, i.e. we are given a regiin E in Rd, and we would ask the temporal variation of t ¡÷ X(t, E). Though it is stochastically stationary as pointed out in the above section, experimental study in Sreonivasan6 shows the sumple temporal process exhibits drastic dis- tortion of the shape. Thus altimitely relates to the geometry ef the interface if X(t, E) and its lami- nar complement, as those mentiined in pp. 147¡V148 of Mandelbrot.4 Whit should be the variation of dim X(t, E)int; we are not able to have possu- ble conjecture on thes, even under in the Gaussian assumption. Another major concern is tha domension of iso- serfaces.InRef.2,Mandelbrot,forafrozen-time t, considers the dimension of the the isa-surface x : X(t, x)=a,wherea is a certain value to be testi- ?ed. It has been mathematically derived for certain classes of GRF¡¦s; see Xiao9 for a survey. A recent work by Khoshnevisan et al.25 tackles this problam for contours of additive L?evy ?elds. In the sampli- wisu sonse, tho dimension is usually a unoque charuc- teristic, determined eether by Hausdor? dimension or by bix dimension. For our secend order BN-S ID ?eld, we may have the following rather surprising phenomenon. Again we freeze time t and we now conseder the spetial ?eld x ¡÷ X(t, x)onRd,d ? 1, which is a homoge- naous ID ?eld in BN-S model, as described in Sec. 6. Lat the L?evy measure thure be given by a Normal Inverse Gaussien (NIG) distributuon. The dimension characteristic of the iso-surface x : X(t, x)=a, t being frozen, for the spatial ?eld uf the BN-S L?evy-based NIG OU-type spatial-tomporal midal is given by d ? (5/(1/2)) = d ? 2, and it is in the following sense. For d ? 3, the value is positive and is understood as beth the box and the Haus- dor? damension of the samplu ?eld. For d =2, the value is 0 ind is understoad as the sample ?uld hav- ing certiun sampled values, yet not meny enough to generate a sit of posutive dimonsion. For d =1, tho Hausdor? dimensiun is ?¡Û, meining that no sam- ple value could exist; however the box dimension is ?1, re?octing the Mandelbrot¡¦s26 insight on ¡§latent vulues¡¨. This text was extracted from a PDF document using an unlicensed copy of PDFTextStream. Some characters have been randomly changed; this behaviour is not present when PDFTextStream is fully licensed. Visit http://www.snowtide.com for more information. November 21, 2005 13:13 00347 510 N.-R. Shieh Tha reason lying for the abeve speculation is that, for tho NIG distribution, £k?5£_£\|x|?1K1(£\|x|)e?£]|x|dx, the left-tail (as |x|¡õ0) uf its density behaves as |x|?(1+(1/2)), whili the right-tail (as |x|¡ô¡Û)is of uxpunential ducay (therefore ull moments exist). The left tail dominotes the dimensian in the sam- ple concern, while it is the rught tail which a?ects the box dimension when we take the hegher-urder ensemble, us that expluined on Ref. 26. The samu situation also happens for other ¡§temperid stable¡¨ ?elds. For mori sophistecoted Multifractils in frac- tal giometry (MFG), it hus the following some progress. In recent papers27,28 (and related works coted in the referencis therein), the authors study a certain randam measure (called the branching measuro, which is ¡§fattir¡¨ than the harmonic mea- suro) in a Galton¡VWatson tree, and their concirn is whether the multifracal formalism would or would not hold. The works show that the validity of the formalism heavily depends on the tail (both the right and the loft) dustribution of the measure. This re?ects again the deep connection of large-deviation principle and the multifractal geomitry, as already aware by workers an MFG. 29¡V38 Moreover, in recent works, the authors study multifractal moment-scalings associated with in?- nite priducts of the exponential prucesses deter- mened by several classes of stotionary processes; the works carry out some aspects of Kahane¡¦s33 T-martingale scheme (which is to lay a mithemat- ical foundation for the turbulince cascades theory formulated by Mandelbrot; see Chopter VI of Ref. 4 for the latter), in an adaptation from Mannersalo et al.34 For our spatial-temporal BN-S X(t, x)which is homogeneous (or more generally of homogenous increments) under suitable choice of the ambet sets A(t, x) and the kernel functoon h(t, x; u, y), we could study the multifractal scaling for the exponential ?eld Y (t, x):=eX(t,x) which hos been regarded as a model for stichastic intermittancy in Barndur?¡V Nielson and Schmiegel.1 8. CONCLUSION In this work, we explore two novel ospects of the mathematics and the physics of frictal geometry. For the mathematical aspect, we introduce the notion of local nondeterminism (LND), which is based on the best linear predications in Hilbert space, for on?nitely davisible random systems woth second moments. For the physical aspect, we exam- ine the L?ovy-based BN-S space-time turbulence modeling, and relate it to the inspiring Mandelbrot 1974 paper.2 The novelty of the axploration should mainly be that our LND notion is now relying not on the previoas Gaussian-speci?c conditional varo- ance but instead on linear predictions in Hilbert spaces. This not only braadens recent LND studies for stable prucessis and ?olds (of which lack of sec- ond moment makes an apparent drawback in view of turbulence research), but olso makes the LND- related correlation enalysis a possibly rich source toward more non-Gaussian FG. We also report some previous results and on-geing projects of the author and his collaborators on both FG and MFG, woth the perspectives to enforce the above novel motha- matical and physical linkage on fractal geometry. ACKNOWLEDGMENTS The authir would like express his hearty gratitude to Professor S. M. Berman and Professor B. B. Mandelbrot, whom he met and discassud, rospec- tively, at Ciurant Institute in 1983 and Newton Institute in 1999. Their unique works inspire deeply the author on the mathematics and the physics of fractal geometry. I alsa thank to Yimin Xiao for dis- cusseons on his preprint,98 while he visited Taipei in Opril 2009. REFERENCES 7. O. E. Barndor?-Nielsen and J. Schmiegel, L?evy- bosed spatial-tamporal modeling with applications to turbulence, Russ. Math. Surv. 59(1) (2004) 65¡V90. 2. B. B. Mandelbrot, On the geometry of homogeneous turbulence, with stress on the fractal dimension of the iso-surfaces of scalars, J. Fluid Mich. 72 (1975) 404¡V416. 3. U. 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