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= CH1 a(t) BH1 (u)
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(628)
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A proof of this result can be found in [BEN 98a], regarding the multifractional processes 633 Elliptic Gaussian random elds (EGRP) Another manner of generalizing FBM, which constitutes the approach adopted in [BEN 97b], consists of starting from the reproducing kernel Hilbert space By returning to De nition 63, we can already represent the reproducing kernel Hilbert space norm by means of the operator J and the formula: J ( 1 ), J ( 2 )
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L2 (
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However, this formula can be presented differently For every function f, g of space D0 of zero functions in 0, C with compact support: f, g
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(629)
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2 where AH is a pseudo-differential operator of symbol CH constant de ned in (68)), ie: 2 eix CH 1+2H
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(630)
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A demonstration of (629) is found in Lemma 11 of [BEN 97b] This equation is also equivalent to: AH = J 2 (631)
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2 It should be noted that the symbol of the operator AH , ( ) = CH 1+2H is homogenous in and does not depend on x, which respectively corresponds to the self-similarity property and increment stationarity of the process Consequently, it is natural to consider Gaussian processes which are associated with the symbols (x, ) which also depend on the position The property of stationarity of the increments is lost However, if we impose that (x, ) is elliptic of order H, ie, controlled by 1+2H when + , in the precise sense that there exists C > 0 such that, for all x, d :
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C(1 + )2H+1
(x, )
1 (1 + )2H+1 C
(632)
we then obtain processes called elliptic Gaussian random processes (EGRP), which locally preserve many properties of the FBM In this chapter, we will de ne the elliptic Gaussian random processes in a less general manner than in [BEN 97b] DEFINITION 66 Let AX be the pseudo-differential operator de ned by: f D0 , AX (f ) = eit (t, )f ( ) d (2 )1/2 (633)
of the symbol (t, ) verifying for 0 < H < 1, the following hypothesis: HYPOTHESIS 63 (H) There exists R > 0: for every t and for i = 0 to 3: i (t, ) i > such that: | (s, ) (t, )| it is elliptic of order H (see (632)) We will then call elliptic Gaussian random processes of order H the Gaussian processes of reproducing kernel Hilbert space given by adherence of D0 for the norm ( AX f, f L2 ( ) )1/2 and provided with the Hermitian product: f, g
Ci (1 + | |)2H+1 i
for | | > R
(1 + | |)2 +1+ |s t|
= AX f, g
L2 ( )
Locally Self-similar Fields
Let us make some comments to clarify De nition 66 First, we restrict ourselves to one dimension mainly for the same reasons of simplicity as for the ltered white noises In addition, let us note that if does not depend on t, then AX still veri es:
2 AX = JX
(634)
for the isometry: JX ( )(y) =
d e iy 1 ( ) ( ) (2 )1/2 ( )
L2 ( )
This is the same, in fact, as saying that we have a harmonizable representation of X: X(t) = e it 1 W (d ) ( )
It is therefore enough to have an asymptotic expansion of (e it 1)/ ( ) of the type (626) so that X is a ltered white noise On the other hand, the FBMs are not elliptic Gaussian random processes of the type de ned earlier, since the lower inequality of ellipticity (632) is not veri ed Moreover, if the symbol depends on t, relation (634) is no longer true and the elliptic Gaussian random processes are no longer ltered white noises Let us reconsider Hypothesis 63 The rst two points are necessary so that the symbol AX behaves asymptotically at high frequency as if it does not depend on t If we want to distinguish the two models roughly, we can consider that elliptic Gaussian random processes have more regular trajectories, whereas ltered white noises lend themselves better to identi cation Consequently, let us reconsider the manner of determining the local regularity of an elliptic Gaussian random process and very brie y summarize the reasoning of [BEN 97b] The starting point for the study of the regularity in the elliptic Gaussian random processes is, as for FBM, a Karhunen-Loeve expansion of the elliptic Gaussian random processes in adapted bases The selected orthonormal base is built starting from the base (613) of L2 ( ) by supposing: = (AX ) 2 ( ) where the fractional power of AX is de ned by means of a symbolic calculation on the operators This leads to: X=
as and in L2 ( )
(635)
The regularity of X is the consequence of wavelet type estimates which relate to and its rst derivative and which resemble to (616) A precise statement is