This is a brief account of the current work by Dzhaparidze, van Zanten and Zareba, delivered as a lecture note at the conference Small deviations and related topics II held in St. Petersburg, September 12-19, 200

Dzhaparidze, K.O. (Kacha)

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Dzhaparidze, Kacha

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Representations of isotropic random fields with homogeneous increments, with applications to spacial fractional Brownian motion

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C e n t r u m  v o o r  W i s k u n d e  e n  I n f o r m a t i c aPNAProbability, Networks and Algorithms Probability, Networks and AlgorithmsRepresentations of isotropic random fields with homogeneous increments, with applications to spacial fractional Brownian motion K.O. DzhaparidzeREPORT PNA-E0510 OCTOBER 2005CWI is the National Research Institute for Mathematics and Computer Science. It is sponsored by the Netherlands Organization for Scientific Research (NWO).CWI is a founding member of ERCIM, the European Research Consortium for Informatics and Mathematics.CWI's research has a theme-oriented structure and is grouped into four clusters. Listed below are the names of the clusters and in parentheses their acronyms.Probability, Networks and Algorithms (PNA)Software Engineering (SEN)Modelling, Analysis and Simulation (MAS)Information Systems (INS)Copyright © 2005, Stichting Centrum voor Wiskunde en InformaticaP.O. Box 94079, 1090 GB Amsterdam (NL)Kruislaan 413, 1098 SJ Amsterdam (NL)Telephone +31 20 592 9333Telefax +31 20 592 4199ISSN 1386-3711Representations of isotropic random fields withhomogeneous increments, with applications tospacial fractional Brownian motionABSTRACTThis is a brief account of the current work by Dzhaparidze, van Zanten and Zareba, delivered asa lecture note at the conference "Small deviations and related topics II" held in St. Petersburg,September 12-19, 2005.2000 Mathematics Subject Classification:  60G60 60G15Keywords and Phrases: isotropic random field, homogeneous increments, fractional Brownian motionNote: This research is carried out in collaboration with Harry van Zanten and Pawel Zareba under the NWO projectP1303: Spectral analysis of Gaussian processesRepresentations of isotropic random fields withhomogeneous increments,with applications to spacial fractional BrownianmotionKacha DzhaparidzeCenter for Mathematics and Computer ScienceKruislaan 413, 1098 SJ Amsterdam, The Netherlandskacha@cwi.nlSeptember 14, 2005Isotropic random field with homogeneous increments. The results pre-sented here are taken over from the current work by Dzhaparidze, van Zantenand Zareba [9] in which some of the results of the previous papers [5]–[8] areextended to isotropic Gaussian random fields with homogeneous increments.Our work, inspired in large extend by Malyarenko [13], does improve upon hisgeneral result a` la our representation (8) and somewhat simplify applications tothe spacial fractional Brownian motion.The departure point in the aforementioned papers is the spectral representa-tion of the covariance function of a mean zero random process Xt, t ∈ R1, withstationary increments:EXsXt =∫R1(eiλt − 1)(e−iλs − 1)d%(λ) s, t ∈ R1 (1)where % is a spectral function. In the special case of fBm(H) withEXsXt =12(|t|2H + |s|2H − |t− s|2H), (2)the spectral function is known to be given byd%(λ) = C2Hdλ|λ|1+2Hwith a certain positive constant C2H . Malyarenko [13] treats at once a multi-dimensional case, a random field with homogeneous increments characterized1by the spectral representationEXtXs =∫RN(ei(v,t) − 1)(e−i(v,s) − 1)d%(v) s, t ∈ RN (3)under the additional isotropy requirement in the sense that XΩtd= Xt for anyorthogonal matrix Ω. The spectral function %(v) then depends only on thelength |v| of the vector v ∈ RN (for simplicity we use the same symbol % alsofor the resulting scalar function). The main object of study in [13] is a spacialversion of fBm(H) that is characterized by the covariance of the same form (2)with s, t ∈ RN , whose spectral representation (3) holds withd%(v) = CN 2Hdv|v|N+2H ,CN 2H a certain positive constant. The intention is to extend results of [5] to themulti-dimensional case. The first step is finding the multi-dimensional analogueto the following simple reformulation of (1): for s, t ∈ R1EXsXt =∫ ∞0(cosλs− 1λcosλt− 1λ+sinλsλsinλtλ)µ(dλ) (4)with µ(dλ) = 2λ2%(dλ). However, this step is not elementary. It turns out (see[13] for details or the books [19] or [12] where the basic methodology can befound) that (3) can be rewritten in the following form: with s, t ∈ RNEXsXt =piNΓ2(N2 )∞∑`=0h(`,N)∑m=1∫ ∞0gm` (s, λ)− gm` (0, λ)λgm` (t, λ)− gm` (0, λ)λµ(dλ)(5)withµ(dλ) =2piN |λ|N+1%(dλ)Γ2(N/2).It will be seen in the concluding section how the system of functions gm` isdefined and how the numbersh(`,N) =(2`+N − 2)(`+N − 3)!(N − 2)!`! (6)occur (in case N = 1 only two terms will remain in the series (5) and it turnsinto (4), of course). Meanwhile in the next section we present an interestingconsequence of the representation (5).Series expansion. In this section we restrict our attention to the unit ball|t| < 1. As is demonstrated in [8], in the scalar case one can make use ofKrein’s spectral theory of vibrating strings (see [11] or [4]) that allows us toswitch over to the discrete spectrum. The discrete counterpart of the spectralfunction µ in (4) is defined in terms of eigenvalues and eigenfunctions of the2corresponding string equation, and we have the representation of the followingform: for s, t ∈ R1EXsXt =∞∑n=1(cosλns− 1λncosλnt− 1λn+sinλnsλnsinλntλn)σ2n.Krein has developed procedures for finding the numbers λn and σ2n in manypractically important cases (see the list of rules in [4], section 6.9), except inthe fBm(H) case. The latter is discussed in [8] where one can see that λn’s inthat case are the positive zero’s of the Bessel function J−H and thatσ2n =2C2Hλ2Hn J21−H(λn).In the paper [9] under preparation we show how the same method of vi-brating strings does extend to the multi-dimensional case and does lead to therepresentation of the following form: for s, t ∈ RNEXsXt =piNΓ2(N2 )∞∑`=0h(`,N)∑m=1∞∑n=1gm` (s, λn)−gm` (0, λn)λngm` (t, λn)−gm` (0, λn)λnσN 2n(7)with the same spectrum {λn, n = 1, 2, . . .} as before and the jumps {σN 2n , n =1, 2, . . .} that are determined in the similar way as before. For the spacialfBm(H), for instance, λn’s are again the positive zero’s of the Bessel functionJ−H , whileσN 2n =2CN 2Hλ2Hn J21−H(λn).Precisely as in the scalar case, it follows from the representation (7) that thefollowing series expansion holds a.s. and uniformly in |t| ≤ 1:Xt =piNΓ2(N2 )∞∑`=0h(`,N)∑m=1∞∑n=1gm` (t, λn)−gm` (0, λn)λnηm`n (8)with independent Gaussian N(0, σN 2n ) random variables ηm`n. We have to skipthe details on the necessary arguments and conclude this note by characterizingthe system of functions gm` .Wave propagation in space These functions naturally occur in the spectraltheory of random fields (as developed in [19], [12], etc.) Actually, this stemsfrom the basic roˆle they play in harmonic analysis in L2(RN ) (see in particular[17], section 10.2.5; see also [3] or [10]). Another context to be mentioned brieflyin the sequel, is from mathematical physics (see e.g. [1], [2], [15], [14], [18]).The wave equation that describes the propagation of sound in a media withdensity ρ(x), x ∈ RN takes the form of a hyperbolic equation for the inducedpressure p:ρ∂2p∂t2= ∆p x ∈ RN3where ∆ =∑Ni=1 ∂2/∂x2j is the Laplace operator. To separate the time andspace variables t and x, substitute p(t, x) = Θ(t) Ξ(x) and use the separationconstant λ2 to get two equations Θ′′ = λ2Θ and ∆Ξ = λ2ρΞ. The generalsolution of the first equation in the time component t is expressed in terms ofthe linear combination of trigonometric functions sinλt and cosλt. Thereforewe focus our attention to the second equation in the space component x ∈ RN .It is handy to rewrite the Laplace operator in terms of the spherical coordinates(r, θ1, . . . , θN−2, φ) related to the vector x = (x1, . . . , xN ) byx1 = r cos θ1x2 = r sin θ1 cos θ2... ...xN−1 = r sin θ1 sin θ2 · · · sin θN−2 cosφxN = r sin θ1 sin θ2 · · · sin θN−2 sinφwhere r = |x|. In these coordinates∆ =1rN−1∂∂r(rN−1∂∂r)+1r2∆0where ∆0 is the Laplace–Beltrami operator on the unit sphere SN−1, given by∆0 =1sin2 θN−2∂∂θN−2(sinN−2 θn−2∂∂θN−2)+1sin2 θN−2 sin2 θN−3∂∂θN−3(sinN−3 θN−3∂∂θN−3)+ · · ·+ 1sin2 θN−2 · · · sin2 θ1∂∂φ,that is a symmetric operator possessing the complete orthonormal set of squireintegrable eigenfunctions on the unit sphere, the so-called spherical harmonics,corresponding to the eigenvalues −`(`+N − 2), ` = 0, 1, . . ., with multiplicities(6). Denoting these eigenfunctions by {Y m` ,m = 1, . . . , h(`,N)}, we thus have∆0Y m` + `(`+N − 2)Y m` = 0.Let us turn back to our problem of solving the equation∆Ξ = λ2ρΞ. (9)By treating only the case of a radial density ρ(r), we can separate the radialand angular coordinates r and (θ, φ) by substituting Ξ(x) = u(r)v(θ, φ) andusing a separation constant k2. We get then two equations: ∆0v + k2v = 0 forthe angular coordinates and r3−N(rN−1u′)′+ (λ2r2ρ− k2)u = 0 for the radius.As was already said, the former equation is integrated for k2 = `(`+N − 2) interms of the spherical harmonics. Prescribe therefore the same value to k2 alsoin the radial equation. We getr3−N(rN−1u′)′ + [λ2r2ρ− `(`+N − 2)]u = 0. (10)4Equation (10) is the subject of hard study in the literature, but the explicitsolutions are known only in several particular cases of the density ρ(r) (see,for instance, the books on mathematical physics and quantum mechanics, citedabove). The simplest case is ρ ≡ 1, of course. The solution in this case iswell-known, since (10) is reducible to the Bessel equation. Subject to the initialconditions u`(0) = δ` 0, we haveu`(r) = 2N−22 Γ(N2)J`+N−22 (λr)(λr)N−22(see e.g. [18], p. 351, or [15], p. 231). Thus the productgm` (x, λ) = 2N−22 Γ(N2)J`+N−22 (λ|x|)(λ|x|)N−22Y m`(x|x|)satisfy equation (9) for ρ ≡ 1, that is in fact the characteristic equation for theLaplace operator (the so-called Helmholtz equation; see e.g. [18], section 31, or[16], section 2.6).Thus the explicit expression is given for functions gm` in the representations(5), (7) and (8).As was already mentioned, in the scalar case N = 1 the representation (5)turns into (4), because the unit sphere in this degenerate case is interpreted asthe two point set {−1, 1} and we have only two spherical harmonics Y 10 (x) = 1and Y 11 (x) = x. Besides, in this case u0(r) = cosλr and u1(r) = sinλr.In the planar case N = 2 we have for each ` only h(`, 2) = 2 sphericalharmonics Y 1` (φ) = cos `φ/√2pi and Y 2` (φ) = sin `φ/√2pi, while u`(r) = J`(λr).Finally, in the case N = 3, most important for physical applications, foreach ` we haveu`(r) =√pi2λrJ`+ 12 (λr)and h(`, 3) = 2`+ 1 spherical harmonicsY m` (θ, φ) = (−1)m√(1 + 2`)(`−m)!4pi (`+m)!Pm` (cos θ) eimφ |m| ≤ `with the associated Legendre functions Pm` (cos θ) (see e.g. [14], Appendix B, or[16], section 2.4).References[1] Berezin, F. A. and Shubin, M. A. (1991) The Schro¨dinger Equation.Kluwer.[2] Chadan, K. and Sabatier P. C. (1977) Inverse Problems in Quantum Scat-tering Theory. Springer.5[3] Dym, H. and McKean, H. P. (1972) Fourier Series and Integrals. AcademicPress.[4] Dym, H. and McKean, H. P. (1976) Gaussian Processes, Function Theory,and the Inverse Spectral Problem. Academic Press.[5] Dzhaparidze, K. and van Zanten, H. (2004) A series expansion of fractionalBrownian motion. Prob. Theory Relat. Fields 130, 39–55.[6] Dzhaparidze, K. and van Zanten, H. (2005) Krein’s spectral theory andthe Paley-Wiener expansion for fractional Brownian motion. Ann. Probab.33(2), 620–644.[7] Dzhaparidze, K. and van Zanten, H. (2005) Optimality of an explicit seriesexpansion of fractional Brownian sheet. Statist. Prob. Lett. 71, 295–301.[8] Dzhaparidze, K., van Zanten, H. and Zareba, P. (2005) Representations offractional Brownian motion using vibrating strings. To appear in StochasticProcesses and their Applications.[9] Dzhaparidze, K., van Zanten, H. and Zareba, P. (2005) On a series expan-sion and moving average representation of homogeneous isotropic randomfield with application to multi-dimensional fractional Brownian motion.[10] Helgason, S. (1981) Topics in Harmonic Analysis on Homogeneous Spaces.Birkha¨user.[11] Kac, S. and Krein, M. G. (1974) On the spectral functions of the sting.Amer. Math. Soc. Transl. (2) Vol. 103. 19–102.[12] Leonenko, N. (1999) Limit Theorems for Random Fields with Singular Spec-trum. Kluwer.[13] Malyarenko, A. (2005) A series expansion of a certain classof isotropic gaussian random fields with homogeneous increments.arXiv:math.ST/0411539 v3 17 Jun 2005.[14] Messiah, A. (1961) Quantum Mechanics. North-Holland Publishing Com-pany.[15] Nikiforov, A. F. and Uvarov V. B. (1961) Special Functions of MathematicalPhysics. Birkha¨user.[16] Ne´de´lec, A. (2001) Acoustic and Electromagnetic Equations. Springer.[17] Vilenkin, N. Ja. and Klimyk, A. U. (1993) Representation of Lie Groupsand Special Functions. Kluwer.[18] Vladimirov, V. S. (1984) Equations of Mathematical Physics. Mir Publish-ers, Moscow.[19] Yadrenko, M. I. (1980) Spectral Theory of Random Fields. OptimizationSoftwere, Inc.6[16] Ne´de´lec, A. (2001) Acoustic and Electromagnetic Equations. Springer.[17] Vilenkin, N. Ja. and Klimyk, A. U. (1993) Representation of Lie Groupsand Special Functions. Kluwer.[18] Vladimirov, V. S. (1984) Equations of Mathematical Physics. Mir Publish-ers, Moscow.[19] Yadrenko, M. I. (1980) Spectral Theory of Random Fields. OptimizationSoftwere, Inc.7

Representations of isotropic random fields with homogeneous increments, with applications to spacial fractional Brownian motion

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