Fractional Brownian Vector Fields

Abstract

Abstract. This work puts forward an extended definition of vector fractional Brownian motion (fBm) using a distribution theoretic formulation in the spirit of Gel'fand and Vilenkin's stochastic analysis. We introduce random vector fields that share the statistical invariances of standard vector fBm (self-similarity and rotation invariance) but which, in contrast, have dependent vector components in the general case. These random vector fields result from the transformation of white noise by a special operator whose invariance properties the random field inherits. The said operator combines an inverse fractional Laplacian with a Helmholtz-like decomposition and weighted recombination. Classical fBm's can be obtained by balancing the weights of the Helmholtz components. The introduced random fields exhibit several important properties that are discussed in this paper. In addition, the proposed scheme yields a natural extension of the definition to Hurst exponents greater than one.

On Regularized Reconstruction of Vector Fields

“…The vector regularization functionals that we have derived consist of combinations of (possibly fractional) curl-and divergence-like operators and their adjoints, wrapped in scalar, vector, and/or matrix norms (also introduced in this paper). The generalized vector Laplacians of [20] also fall within this framework. Finally, we have presented an application of the proposed framework to the problem of vector field denoising in 2-and 3-D, where we gave a natural generalization of (quadratic), as well as (TV-type) regularization for vector fields.…”

“…For , the operator defined in Theorem 2 has the Fourier expression and therefore corresponds, up to normalization, to the fractional vector Laplacian , that is, the scalar Laplacian applied coordinatewise. For this reason, we shall refer to the family of operators identified by (19) as generalized vector Laplacians, with notation [20]. To better understand the behavior of the operator when either or is zero, note that can be decomposed as Id (20) where operator is defined by its Fourier multiplier .…”

“…An illustrative example is Id curl which incorporates independent regularization of the curl-and divergence-free subspaces [see the definition of after (20)], as well as the fractional curl and divergence terms. Note that some of 's may be zero.…”

Left-inverses of fractional Laplacian and sparse stochastic processes

“…which is the natural L p extension of the fractional integral operator that was introduced in [1,12,13] for p = 2 and γ ∈ Z/2.…”

“…We One of the primary application of the p-integrable Riesz potentials is the construction of generalized random processes by suitable functional integration of white noise [12][13][14]. These processes are defined by the stochastic partial differential equation (1.3), the motivation being that the solution should essentially display the same invariance properties as the defining operator (fractional Laplacian).…”