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

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4 P. D. TAFTI AND M. UNSER equation by requiring that the right inverse of U interact in a particular way with rotations and scalings of the coordinate system. 2 The “invariance” properties the operator U is required to satisfy are the following. Uf Ω = (Uf ) Ω ; (rotation invariance) (2.3) Uf σ = σ 2γ (Uf ) σ (degree 2γ homogeneity) (2.4) (γ relates to one of the main parameters of the family of the random solutions, namely the Hurst exponent, by the relation H = 2γ − d/2). Note that we shall assume invariance with respect to improper rotations (with detΩ = −1) as well as proper rotations (with detΩ = 1). The above properties translate, respectively, to the following conditions on the Fourier expression of the operator U: Û(Ωω) = ΩÛγ (ω)Ω T ; (rotation invariance) (2.5) Û(σω) = σ 2γ Û γ (ω). (homogeneity) (2.6) The following theorem was proved by Arigovindan for d = 2,3 [1]. It can be shown more generally to hold in any number of dimensions. THEOREM 2.1 (Arigovindan [1]). A vector convolution operator satisfying properties (2.3) and (2.4) has a Fourier expression of the form ˆΦ e γ (ω) = ξ ‖ω‖2γ ξ ωω T irr ‖ω‖ + 2 eξ sol I − ωωT , (2.7) ‖ω‖ 2 with ξ = (ξ irr ,ξ sol ) ∈ 2 . It is easy to verify that the converse of the theorem is also true for arbitrary dimension d. Since we shall be considering real operators, in what follows we shall implicitly assume e ξ irr ,e ξ sol ∈ 2 without further mention. The operators introduced in Theorem 2.1 generalize vector Laplacians in two senses (fractional orders and re-weighting of solenoidal and irrotational components). We shall therefore call them fractional (vector) Laplacians, and use the symbol (−∆) γ to denote ξ the operator with Fourier expression ˆΦ γ . To gain a better understanding of the action of ξ fractional vector Laplacians, it is instructive at this point to recall the Fourier expressions (in 2 We shall have to consider a right inverse of U that—unlike U—is not shift invariant, and does not correspond to a convolution; hence the solution B H will not be stationary (more on this later).
FRACTIONAL BROWNIAN VECTOR FIELDS 5 standard Cartesian coordinates) of some related vector differential operators. 3 grad div curl graddiv curlcurl ∆ E (−∆) γ ξ ←→ jω; ←→ (jω) T ; ←→ 0 −jω3 jω 2 jω 3 0 −jω 1 −jω 2 jω 1 0 ←→ −ωω T ; ←→ ‖ω‖ 2 I − ωω T ; ←→ −‖ω‖ 2 I; ; ←→ ωω T /‖ω‖ 2 ; ←→ ˆΦ γ (ω) = ξ ‖ω‖2γ e ξ irr ωωT + e ξ ‖ω‖ 2 sol I − ωωT ‖ω‖ 2 . (2.8) The penultimate operator (E) and its complement (Id − E) project a vector field respectively onto its curl-free and divergence-free components. In other words, together they afford a Helmholtz decomposition of the vector field on which they act (these operators appear prominently in fluid dynamics literature [8–10, 46]). This is because In addition, one has div (Id − E) = 0 and curl E = 0. E grad = grad and E curl = 0. (Id − E) is known as the Leray projector in turbulence literature. Our notation for the fractional vector Laplacian (−∆) γ is motivated by the observation ξ that it can be factorized as (−∆) γ ξ = (−∆)γ 0 e ξ irrE + e ξ sol (Id − E) . In view of the properties of the operator E, this factorization means that the operator (−∆) γ combines a coordinate-wise fractional Laplacian with a re-weighting of the curl- and ξ divergence-free components of the operand. 2.2. Some properties of ˆΦ γ . Let us now take a closer look at the family of matrixvalued functions ˆΦ γ , γ ∈ , ξ eξ irr ,e ξ ξ sol ∈ . They, of course, satisfy the required invariances: ˆΦ γ ξ (Ωω) = ΩˆΦ γ ξ (ω)ΩT ; ˆΦ γ ξ (σω) = σ 2γ ˆΦγ ξ (ω). But they exhibit, in addition, the following properties. (ˆΦ1) Closedness under multiplication: We have ˆΦ γ 1 ξ 1 (ω)ˆΦ γ 2 ξ 2 (ω) = ˆΦ γ 1 +γ 2 ξ 1 +ξ 2 (ω), 3 Note that, while the curl operator is classically defined in three dimensions, the equivalents of graddiv, curlcurl, and ∆ = graddiv−curlcurl can be defined in any number of dimensions, for instance by their Fourier symbols. More precisely, for arbitrary d, the equivalents of −curlcurl and −graddiv that appear in the definition of the vector Laplacian correspond, respectively, to the product of d-dimensional curl and divergence with their adjoints.
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