See [MEY 90] for traditional results on operators in .NET

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3 See [MEY 90] for traditional results on operators
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THEOREM 54 ([BEN 97]) Let X be a self-similar Gaussian (r, H) self-similar process, with r-stationary increments, with 0 < H < 1: X admits the following harmonic representation: X(x) = eix
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Rd r 1 (ix )k k=0 k! d | | 2 +H S | |
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X is the unique solution of the following stochastic elliptic differential equation: LX = W Q(D)X(0) = 0 for any Q such that d Q < r As a particular case, we can mention the harmonic representation [MAN 68] of fractional Brownian motion of parameter H: r = 1 and S 1: BH (t) = eix 1
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| | 2 +H
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523 Hyperbolic processes DEFINITION 53 The operator L de ned in (108) is called hyperbolic if its symbol 1 d is of the form ( ) = i=1 | i |Hi + 2 THEOREM 55 (FRACTIONAL B ROWNIAN SHEET [LEG 99]) Fractional Brownian sheet, de ned as:
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X(x) =
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satis es the following equality: X( 1 x1 , , d xd ) =
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COROLLARY 51 When 1 = = d and H = H1 + + Hd , the hyperbolic process X obtained is self-similar with parameter H, with H between 0 and d In contrast with the elliptic case, H > 1 is hence allowed, though the Brownian fractional sheet is non-derivable
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524 Parabolic processes Let A be a pseudo-differential operator of dimension n 1, ie, its symbol is a function of Rn 1 in R Let L be the pseudo-differential operator of dimension n, whose symbol is a function of R Rn 1 in R and de ned by L = t A Let us consider the stochastic differential equation LX = W By analogy with the classi cation of operators, X is said to be parabolic The most prominent example is the Ornstein-Uehlenbeck process:
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The operator t is renormalized with a factor 1 ; the operator A can be 2 renormalized with an arbitrary factor Generally, a parabolic process is not self-similar 525 Wavelet decomposition In this section, we expand Gaussian self-similar processes on a wavelet basis, which hence also constitutes a basis for the self-reproducing Hilbert space of the process4 5251 Gaussian elliptic processes Let u , with u Ed , be a Lemari -Meyer generating system [MEY 90] Let , with d , be the generated orthonormal basis of wavelets Let us assume that X veri es hypotheses and notations, as in Theorem 54 Let us de ne u , with u Ed , with the harmonic representation: (x) =
eix
r 1 (ix )k k=0 k! d 2 +H S | | | |
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Let us then de ne the associated family of wavelets , with d THEOREM 56 ([BEN 97]) There is a sequence of normalized Gaussian normal random 2D variables such that: X(x) =
d
2 j(r 1+H) (x)
If X is self-similar in the usual sense, then this decomposition is a renormalized distribution of mass, as de ned in section 513
4 See [NEV 68] for self-reproducing Hilbert spaces
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5252 Gaussian hyperbolic process THEOREM 57 With the same notations as those of Theorem 56, we obtain the decomposition: X(x) =
( 1 ,, d ) ( 1 )d
2 (n1 H1 + +nd Hd ) 1 (x1 ) d (xd ) 1 ,, d
where the sequence 1 ,, d consists of normalized Gaussian random 2D variables Hyperbolic processes enable us to model multiscale random structures, with preferred directions 526 Renormalization of sums of correlated random variable Let BH denote fractional Brownian motion of parameter H Let us consider the increments of size h > 0: Xk = h H (BH ((k + 1)h) BH (kh)) The following properties are well-known: X0 is a normalized Gaussian random variable; the sequence (Xk ) is stationary; a law of large numbers can be written as: 1 n + n + 1 lim