Tafti, P.D.
Unser, M.
Fractional Brownian Vector Fields
Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
8
5
fractional Brownian motion
random vector fields
self-similarity
invariance
Helmholtz decomposition
generalized random processes
Gel'fand-Vilenkin stochastic analysis
Fractal Processes
Self-Similarity
Wavelets
Motion
Representation
Decomposition
Divergence
Turbulence
Splines
Models
2010
2010
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.
SIAM
Multiscale Modeling & Simulation
Journal Articles
10.1137/090752638