Fractional Generalized Random Fields of Variable Order

Abstract

We study the class of random fields having their reproducing kernel Hilbert space isomorphic to a fractional Sobolev space of variable order on n . Prototypes of this class include multifractional Brownian motion, multifractional free Markov fields, and multifractional Riesz-Bessel motion. The study is carried out using the theory of generalized random fields defined on fractional Sobolev spaces of variable order. Specifically, we consider the class of generalized random fields satisfying a pseudoduality condition of variable order. The factorization of the covariance operator of the pseudodual allows the definition of a white-noise linear filter representation of variable order. In the ordinary case, the Hölder continuity, in the mean-square sense, of the class of random fields introduced is proved, and its mean-square Hölder spectrum is defined in terms of the variable regularity order of the functions in the associated reproducing kernel Hilbert space. The pseudodifferential representation of variable order of the resulting class of multifractal random fields is also defined. Some examples of pseudodifferential models of variable order are then given.

Fractional Brownian Vector Fields

“…As far as the spaces of solutions are concerned, another possibility would be to use fractional Sobolev spaces, as proposed by Ruiz-Medina, Anh, and Angulo [45] and Kelbert, Leonenko, and Ruiz-Medina [23]. It appears that this approach would work especially well when considering Gaussian self-similar vector fields.…”

Multifractality in space–time statistical models

“…In particular, generalized random field models are useful to incorporate, in terms of test functions, sample information effects such as spatial deformation from turbulence or optical devices, and blurring due to object movement during image registration (Goitía et al 2004). In the case where test functions have uniform or variable local singularity exponents, and appropriate moment conditions (fast decay at infinity), the corresponding space-time random field model, defined in the weak sense, can be fractal or multifractal, and can also display long-range dependence (see Anh et al 1999;Ruiz-Medina et al 2001, 2004a, b, 2002, for the Gaussian case). Interesting applications of this approach such as orthogonal expansions in terms of Riesz bases, wavelet expansions, and, in general, atomic decompositions allow the implementation of prediction and filtering techniques for the class of space-time random fields described (see, for example, Ruiz-Medina et al 2003a, b).…”

“…The first one is based on the consideration of generalized random fields on test function spaces with variable local singularity orders on multifractal domains (see Ruiz-Medina et al 2004a), and is related to the generation of Feller semigroups. From this approach, space-time models with multifractal spatial variograms at each time cross-section can be introduced.…”

Numerical simulation for the variable-order Galilei invariant advection diffusion equation with a nonlinear source term

“…Several authors (e.g. [4,5,10,11,14,17,18,[24][25][26][27][28]) have investigated several classes of variable-order fractional differential equations. To date, only a few authors studied numerical methods and numerical analysis of variable-order fractional differential equations.…”