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Abstract
Analysing and predicting dynamics of stochastic transport systems is paramount in fields including chemistry, physics and finance. A major component of these systems is diffusion; a transport process whereby particles move from a region of high concentration to a region of low concentration. Incorporating non-Markovian behaviour, as many real systems do, may lead to subdiffusion.
A widely used model of diffusion is a continuous time random walk, which involves separate jump and waiting time densities. A waiting time density with a finite mean leads to standard diffusion whilst a density where the mean diverges leads to subdiffusion. Additional processes such as reactions and forcing for single and multiple species can be readily incorporated.
In this thesis we review the rich literature of continuous time random walk models of subdiffusion with extensions to reactions and forcing and we put forward a model incorporating all three components. We analyse the behaviour of the continuous time random walk model both in and away from the diffusion limit. This limit corresponds to instantaneous local jumps in a spatial continuum. We then develop a new model of subdiffusion known as the discrete time random walk and highlight beneficial properties including it being an explicit numerical method. Historically continuous time random walks and subdiffusion have been analysed on lattice models of the continuum but we extend this to a generalized network model. We highlight interesting behaviour that arises due to the irregular nature of the topology of many random and real-world networks.
We conclude by considering the relationship of both continuous and discrete time random walks on regular lattices and networks as well as the diffusion limit. We have also included examples of both toy and real systems to guide the reader through technical discussions as well as highlighting the
importance of diffusion in many stochastic transport systems.