FRACTIONAL BROWNIAN VECTOR FIELDS 1. Introduction. A one ...

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8 P. D. TAFTI AND M. UNSER There exists a canonical regularization of the above singular integral, which is homogeneous and rotation- and shift-invariant. It can be shown that the canonical regularization, which is the one considered by Gel’fand and Shilov [17] and Hörmander [20], corresponds to a convolution with a homogeneous generalized function. Unfortunately, this regularization fails the third of our requirements, namely L 2 -boundedness. We shall therefore have to consider a different regularization of (2.11) in the sequel. In what follows, we shall always limit our consideration to values of γ such that 2γ − d ∉ . (2.12) 2 It will be seen later that this condition is equivalent to requiring that the Hurst exponent H ∉ in the definition of fBm (see the discussion following Theorem 3.2). To extend the definition of the (left) inverse from functions with vanishing moments to arbitrary test functions f ∈ d let us introduce the regularization operator R γ : f (·) → f (·) − ∑ T k [f ](·) k (2.13) |k|≤⌊2γ− d 2 ⌋ where T k [f ] denotes the (vector) coefficient of (·) k in the Taylor series expansion of f (·) around 0. Next, consider the operator ∫ −γ (− `∆) : f → −ξ (2π)−d j〈x,ω〉 e ˆΦ−γ d −ξ (ω)[Rγ ˆf ](ω) dω (2.14) (defined in the sense of the L 2 Fourier transform). This operator essentially removes sufficiently many terms from the Taylor expansion of ˆf (ω) at ω = 0 so as to make the singularity −γ of (ω) square-integrable. Of key importance is the fact that (− `∆) maps Schwartz ˆΦ −γ ξ test functions in d to square-integrable functions (assuming, as we already stated, that 2γ − d 2 ∉ ): −γ PROPOSITION 2.3. The operator (− `∆) maps d into (L 2 ) d on the condition that −ξ 2γ − d/2 ∉ . Proof. By Parseval’s identity, −γ ‖(− `∆) f −ξ ‖2 = (2π) −d ‖ˆΦ −γ −ξ [Rγ ˆf ](ω)‖ 2 ∫ = (2π) −d [R γ ˆf ] H (ω)ˆΦ −2γ d −2Re ξ (ω)[Rγ ˆf ](ω) dω ∫ ∑ = (2π) −d R γ ˆf m (ω) ˆΦ−2γ (ω) −2Re ξ mn Rγ ˆf n (ω) dω. d 1≤m,n≤d We may consider the behaviour of the integrand separately about ω = 0 and at infinity. First, note that R γ ˆf m (ω)R γ ˆf n (ω) has a zero of order at least 2⌊2γ − d/2⌋ + 2 at ω = 0 (cf. the definition of R γ in (2.13)). Since the singularity of ˆΦ−2γ (ω) at ω = 0 is of order −4γ −2Re ξ mn and 2⌊2γ − d/2⌋ + 2 − 4γ > −d for 2γ − d/2 ∉ , the integral converges about ω = 0. −ξ
FRACTIONAL BROWNIAN VECTOR FIELDS 9 At infinity, R γ ˆf m (ω)R γ ˆf n (ω) is dominated by the polynomial term and grows at most like ‖ω‖ 2⌊2γ−d/2⌋ , while ˆΦ−2γ −2Re ξ (ω) mn decays like ‖ω‖−4γ . We have 2⌊2γ − d/2⌋ − 4γ < −d, from where it follows that the integral also converges at infinity. The (L 2 ) d −γ norm of (− `∆) f −ξ is therefore bounded. The Hermitian adjoint 5 −γ of (− `∆) is the operator −ξ ∫ −γ (− ´∆) : f → −ξ (2π)−d ⎤ ⎣ ej〈x,ω〉 − ∑ j |k| x k ω k ⎥ k! ⎦ |k|≤⌊2γ− d 2 ⌋ ⎡ ⎢ d ˆΦ −γ −ξ (ω) ˆf (ω) dω (2.15) (see Appendix B). −γ −γ As was suggested, (− `∆) and (− ´∆) are respectively named the left and right inverses −ξ −ξ of (−∆) γ . They satisfy ξ −γ (− `∆) −ξ (−∆)γ = Id and ξ (−∆)γ −γ (− ´∆) = Id (2.16) ξ −ξ over d −γ . We may further extend the domain of (− ´∆) to a subset of generalized functions −ξ (distributions) 6 on d , using as definition the duality relation −γ 〈(− ´∆) −ξ g, f 〉 := 〈g,(− `∆) −γ −ξ f 〉 wherever the right-hand-side is meaningful and continuous for all f ∈ d . −γ It is easily verified that (− , and by duality (− ´∆) , are rotation-invariant and `∆) −γ −ξ homogeneous. This fact is captured in our next proposition, which we shall prove with the aid of the following lemma. LEMMA 2.4. R γ [f (M −1·)](x) = [R γ f (·)](M −1 x). Proof. By the uniqueness of the Taylor series expansion, r.h.s. = f (M −1 x) − ∑ T k [f ](M −1 x) k = l.h.s. |k|≤⌊2γ− d 2 ⌋ −γ PROPOSITION 2.5. The operators (− `∆) −ξ homogeneous in the sense of Eqns (2.3) and (2.4). Proof. For a non-singular real matrix M, −ξ −γ and (− ´∆) −ξ are rotation invariant and ∫ −γ (− `∆) f −ξ M (x) = (2π)−d j〈x,ω〉 |det M|e ˆΦ−γ d −ξ (ω)[Rγ M ˆf ](M T ω) dω ∫ = (2π) −d e j〈M−1 x,ρ〉 ˆΦ−γ d −ξ (M−T ρ)M[R γ ˆf ](ρ) dρ 5 With respect to the d scalar product ∫ 〈f , g〉 := f H (x)g(x) dx = ∑ 〈 f i , g i 〉. d 1≤i≤d 6 By these we mean members of the dual ( d ) ′ of d . As a matter of fact, ( d ) ′ can be identified with ( ′ ) d .
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FRACTIONAL BROWNIAN VECTOR FIELDS 1. Introduction. A one ...

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