Non-homogeneous random walks

Random walks are fundamental models in stochastic process theory that exhibit deep connections to important areas of pure and applied mathematics and enjoy broad applications across the sciences and beyond. Generally, a random walk is a stochastic process describing the motion of a particle (or random walker) in space. The particle's trajectory is represented by a series of random jumps at discrete instants in time. Fundamental questions for these models involve the long-time asymptotic behaviour of the walker.

Random walks have a rich history involving several disciplines. Classical one-dimensional random walks were first studied several hundred years ago as models for games of chance, such as the so-called gambler's ruin problem. In his 1900 thesis, Louis Bachelier applied similar reasoning to his model of stock prices. Many-dimensional random walks were first studied at around the same time, arising from work of pioneers of science in diverse applications such as acoustics (Lord Rayleigh's theory of sound developed from about 1880), biology (Karl Pearson's 1906 theory of random migration of species), and statistical physics (Einstein's theory of Brownian motion developed during 1905-08). The mathematical importance of the random walk problem became clear after Polya's work in the 1920s, and over the last 60 years or so beautiful connections have emerged linking random walk theory to influential areas of mathematics such as harmonic analysis, potential theory, combinatorics, and spectral theory. Random walk models have continued to find new and important applications in many highly active domains of modern science; specific recent developments include for example modelling of microbe locomotion in microbiology, polymer conformation in molecular chemistry, and financial systems in economics.

Spatially homogeneous random walks, in which the probabilistic nature of the jumps is the same regardless of the present spatial location of the walker, are the subject of a substantial literature. In many modelling applications, the classical assumption of spatial homogeneity is unrealistic: the behaviour of the random walker may depend on the present location in space. Applications thus motivate the study of non-homogeneous random walks. Moreover, mathematical motivation arises naturally from the point of view of deepening our understanding, via rigorous mathematical proofs, of fundamental research problems: concretely, non-homogeneous random walks are the natural setting in which to probe near-critical behaviour and obtain a finer understanding of phase transitions present in the classical random walk models.

The proposed research is part of a broad research programme to analyse near critical stochastic systems. Non-homogeneous random walks can typically not be studied by the techniques generally used for homogeneous random walks: new methods (and, just as importantly, new intuitions) are required. Naturally, the analysis of near-critical systems is more challenging and delicate than that for systems that are far from criticality. The methodology is based on martingale ideas. The methods are robust and powerful, and it is to be expected that methods developed during the project will be applicable to many other near-critical models, including those with applications across modern probability theory and beyond, to areas such as queueing theory, interacting particle systems, and random media.

Fundamental mathematical research has an impressive but unpredictable history of impact. A well-known example is provided by classical results in number theory developed by Fermat and others in the 17th century that are cornerstones of today's electronic encryption mechanisms central to one of the most widespread and important technologies on the planet, the internet. There are many examples where developments in probability theory in particular have come to play vital but unforeseen (and largely unforeseeable) roles in knowledge, society and economics: to name just two, Ito's stochastic calculus spawned the vast mathematical modelling industry in finance, and Google's PageRank algorithm is based on a random walk idea.

Applied probability has been identified as a key area of UK mathematics by the International Review of Mathematical Sciences and has been singled out by the EPSRC as a high profile area worthy of support. As described elsewhere in this proposal, random walks have already found numerous applications across the sciences, and it is likely that new ones will be found. The proposed research will improve our understanding of near-critical random walks, potentially enabling new insights in these modelling applications. Moreover, the mathematical techniques that the project develops are likely to be applicable to other near-critical stochastic systems, with potentially broad impact across many areas of applications. For example, previous work on Lyapunov function methods has enabled the analysis of complex systems arising in queueing theory, interacting particle systems, stochastic growth models, and so on. These models have applications in network optimization, operations research, statistical physics, and biology, to name a few.

The proposed research will open new directions for scientists who apply random walks in their work but who have previously been restricted to using homogeneous models due to our limited understanding of the more general situation in which the random walk is non-homogeneous. Since random walks are applied in vibrant and increasingly relevant fields throughout science (including economics, statistical physics, microbiology, amongst many others), it is to be anticipated that, over time, such developments will impact upon these important applications.

The project will produce a well-trained PDRA, which will be a valuable resource for research and development in the UK. The PDRA will become part of an active research environment in Strathclyde, including the active research seminar programme, and will receive detailed training in mathematical techniques appropriate to the project; the PDRA will also bring expertise to the project. The PDRA will benefit from opportunities to collaborate with researchers in Strathclyde and in the wider international community, to take part in national and international conferences and workshops, and to publish work in high quality academic journals. The PDRA will have the opportunity to engage in teaching activities, including undergraduate tutorials and graduate-level courses as part of the Scottish Mathematical Sciences Training Centre of which Strathclyde is a member and contributor. Thus the PDRA will receive significant academic benefit. Strathclyde provides excellent training in the development of generic skills and competencies that will help the PDRA's long-term employability prospects.