Fractional random fields on domains with fractal boundary

Abstract

For random fields with fractional regularity order (respectively, fractional singularity order), an orthogonal decomposition of the associated reproducing kernel Hilbert space with respect to domains with fractal boundary is derived. The approach presented is based on the theory of generalized random fields on fractional Sobolev spaces. The orthogonal decomposition derived is equivalent to the weak-sense Markov condition, and based on the concept of splitting Hilbert spaces. A mean-square fractional order differential representation on bounded domains with fractal boundary is also obtained. In the Gaussian case, the random fields studied have fractal sample paths. Examples of fractional-order differential models in the class considered are provided.