Possible long-range dependence in fractional random fields

Abstract

Abstract Many existing stochastic models have been developed for description and analysis of Markov diffusion. This paper outlines the new concept of α -duality which lays the foundation for an extension of Markov diffusion to fractional diffusion. The theory of Riesz and Bessel potentials and the corresponding potential spaces play a key role in this new approach. We establish the existence of an important subclass of fractional random fields, namely that of fractional Riesz–Bessel motions, which extends the class of fractional Brownian motions. As a result, the scope of Markov diffusion is widened to cover random fields with long-range dependence.