Non-homogeneous random walks

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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.