Anomalous diffusion via self-interaction and reflection

Broadly interpreted, a diffusion is a continuous-time stochastic process with continuous trajectories, whose stochastic evolution is driven by a drift and a diffusion coefficient. Diffusions, and their discrete-time cousins, random walks, are ubiquitous in stochastic modelling (e.g., conformation of polymer molecules and roaming of animals) as well as in stochastic optimization algorithms of computational statistics and machine learning. Moreover, diffusions and random walks are prototypical stochastic systems, exhibiting phase transitions in their behaviour depending on values of the underlying parameters of the model. The most well-studied diffusions and random walks have two simplifying features: (i) they are Markov, meaning that the future evolution of the process depends only on the current state, and not its previous history, and (ii) the evolution is homogeneous in space.

The proposed research programme will extend the state-of-the-art to non-Markovian and reflecting processes exhibiting anomalous diffusion, in which processes explore space more rapidly than the classical case under assumptions (i) and (ii). A rich and deep classical theory of diffusions and random walks is available. For example, a cornerstone is the result that, in the continuum scaling limit, homogeneous random walks converge to Brownian motion, a universal mathematical object of central importance. This theory extends, in a highly non-trivial way, to broader classes of walks satisfying (i) but a regularity condition weaker than (ii), as shown in recent work of the research team, in which the limit object, Brownian motion, is replaced by a certain class of spatially non-homogeneous diffusion process. While not Brownian motion, these diffusions were nevertheless diffusive, meaning that the rate at which they explore space is the same as in the classical case.

Applications motivate more complex models. In one direction, the evolution may depend on the entire history of the walk: to access fresh resources, roaming animals do not retrace their steps, while the excluded volume effect in polymers ensures that no two monomers can occupy the same physical space. In another direction, certain processes are constrained by natural boundaries at which they are forced to reflect or otherwise deviate from the bulk behaviour: queue-length processes in operations research are usually constrained to be non-negative, for example. Both self-interactions and reflections lead to considerably more challenging mathematical models.

This proposal sets out an ambitious project to develop novel robust probabilistic techniques for the analysis of certain multidimensional processes exhibiting non-Markovian and/or reflecting behaviour and possessing universal features. The analysis is facilitated by a common underlying structure of these seemingly disparate models, which we identify and then exploit. This structure is easier to identify in the context of continuum models, so that is our focus in this proposal. Once the structure has been identified, we expect these ideas to pave the way for further developments also in discrete versions of the models.