The below is a non-exhaustive list of topics of personal interest to me. I am always happy to discuss these ideas in greater detail, so if you're interested please do contact me! As a general rule I greatly enjoy discussions on most areas of mathematics and physics so unrelated scientific enquiries are also most welcome.
Lévy walks are a type of random walk wherein longer excursions are allowed. Consequently, the motility of the walker no longer obeys standard diffusion theory, but is rather characterised by the movement illustrated above.
Requirements that the speed of the walker be finite makes the problem non-Markovian and thus introduces further complications.
Anomalous transport refers to processes wherein the well-known diffusion regime is no longer in effect. This includes effects such as trapping, memory-dependent movement and superdiffusion.
The versatility of these models (often using power-law distribtions) means they can be applied to study phenomena observed in fields as distinct as financial modelling, cellular biology or ecology.
Some walkers are unaffected by the environment (or surface) over which they move, whereas others are intensely influenced. I am intrigued by how surfaces can lead to anomalous behaviour of a walker. Furthermore, the effect of surface topogies on the dynamics of a system is very interesting to me (particularly in the case of fractals).
S. Fedotov and H. Stage. "Subdiffusive Transport in Heterogeneous Patchy Environments." In L. Olivares-Quiroz and O. Resendis-Antonio, Quantitative Models for Microscopic to Macroscopic Biological Macromolecules and Tissues, pp. 37-58 (2018). Springer, Cham.
H. Stage. "Aging in mortal superdiffusive Lévy walkers." Physical Review E, Vol. 96, No. 6 (2017), p. 062150.
H. Stage and S. Fedotov. "Non-Linear Continuous Time Random Walk Models." European Physical Journal B, Vol. 90, No. 11 (2017), p. 225.
S. Fedotov and H. Stage. "Anomalous Metapopulation Dynamics on Scale-Free Networks." Physical Review Letters, Vol. 118, No. 9 (2017), p. 098301.
H. Stage, S. Fedotov and V. Méndez. "Proliferating Lévy walkers and Front Propagation." Mathematical Modelling of Natural Phenomena, Vol. 11, No. 3 (2016), pp. 157-178.
A heuristic introduction to the applications of Wiener-Hopf factorisation in random processes, Isaac Newton Institute, University of Cambridge, October 2019.
Applications of Renewal Theory to Human Behaviour, University of Sheffield, March 2019.
Density-Induced Changes in Anomalous Transport, Technical University of Berlin, February 2018.
Persistence in the Search for Avalon, Collective Dynamics Seminar Series, University of Bristol, November 2017.
Living Lévy Walkers, Current status and future directions of Lévy walk research, Company of Biologists workshop held at Wiston House, September 2017.
Anomalous Diffusion of Synaptic Vesicles - how a neuron fires, Manchester SIAM-IMA Student Chapter Conference held at The University of Manchester, May 2017. Awarded Best Talk.
Proliferating Lévy Walkers and Front Propagation, META workshop held at The University of Birmingham, December 2015.