Random fields of variable order on multifractal domains

Abstract

We introduce a class of random …eld models with variable regularity/singularity order on multifractal domains. The derivation of results on atomic decomposition of functions in fractional Sobolev spaces of variable order, and traces of these spaces on multifractal domains plays a key role. The non-triviality of the multifractal spectrum of the measure de…ning the multifractal domain ensures the non-triviality of the multifractal spectrum of the functions in the trace Sobolev spaces of variable order de…ned. The generalized reproducing kernel Hilbert space theory is considered to introduce a class of random …elds with non-trivial mean-square multifractal spectrum. In the Gaussian case, the random …eld models studied have multifractal sample paths