Path Properties of Subdiffusion—A Martingale Approach

In statistical physics, subdiffusion processes constitute one of the most relevant subclasses of the family of anomalous diffusion models. These processes are characterized by certain power-law deviations from the classical Brownian linear time dependence of the mean-squared displacement. In this article we study sample path properties of subdiffusion. We propose a martingale approach to the stochastic analysis of subdiffusion models. We verify the martingale property, Hölder continuity of the trajectories, and derive the law of large numbers. The precise asymptotic behavior of subdiffusion is obtained in the law of the iterated logarithm. The presented results may be applied to identify the type of subdiffusive dynamics in experimental data. Keywords: Inverse subordinatorLaw of the iterated logarithmLaw of large numbersMartingaleSubdiffusionMathematics Subject Classification: 6G176G52 ACKNOWLEDGMENTS The author thanks the anonymous referees for their constructive comments leading to improvement of the presentation. The author was partially supported by the Foundation for Polish Science through the Domestic Grant for Young Scientists (2009).