Anomalous diffusion processes: Stochastic models and their properties

Abstract

Anomalous diffusion processes are those whose variances deviate from the usual linear scaling with respect to time exhibited by regular Brownian motion. There are several physical mechanisms that can induce anomalous dynamics in a given test particle including sporadic trapping phenomena, long-range jumps and correlations between increments that can either inhibit or exacerbate their spread. Modeling such phenomena by means of stochastic processes and governing partial (integro)-differential equations has been an active field of research within the statistical physics community for many years. This thesis is a collection of four papers contributing novel research into this field, developing numerical methods for simulation and density estimation, analyzing population behaviours arising from stochastic models, examining information contained in trajectories to determine benchmarks for optimal statistical inference, and developing new models to describe anomalous diffusion and related behaviours.