Non-homogeneous random walks

随机游动是随机过程理论中的基本模型，它与纯数学和应用数学的重要领域有着深刻的联系，并在科学和其他领域得到了广泛的应用。一般来说，随机游动是描述粒子(或随机游走者)在空间中运动的随机过程。粒子的轨迹由时间上离散时刻的一系列随机跳跃表示。这些模型的基本问题涉及游走者的长期渐近行为。随机游动涉及几个学科，有着丰富的历史。经典的一维随机游动最早是在几百年前作为机会游戏的模型被研究的，例如所谓的赌徒破产问题。在他1900年的论文中，路易斯·巴切里耶将类似的推理运用到他的股价模型中。大约在同一时间，人们首次研究了多维随机行走，这源于科学先驱在不同应用领域的工作，如声学(瑞利勋爵的声音理论大约从1880年发展起来)、生物学(卡尔·皮尔逊1906年的物种随机迁徙理论)和统计物理学(爱因斯坦的布朗运动理论在1905-08年间发展)。在20世纪20年代波利亚的工作之后，随机游走问题的数学重要性变得清晰起来，在过去60年左右的时间里，出现了将随机游走理论与调和分析、位势理论、组合学和谱理论等有影响力的数学领域联系起来的美丽联系。随机游走模型在现代科学的许多高度活跃的领域中继续得到新的和重要的应用；最近的具体发展包括微生物学中的微生物运动模型、分子化学中的聚合物构象和经济学中的金融系统。空间均匀随机行走是大量文献的主题，其中跳跃的概率性质是相同的，而与步行者当前的空间位置无关。在许多建模应用中，经典的空间同质性假设是不现实的：随机游走者的行为可能取决于空间中当前的位置。因此，应用促进了对非均匀随机游动的研究。此外，数学动机自然产生于通过严格的数学证明加深我们对基础研究问题的理解的观点：具体地说，非齐次随机游走是探索近临界行为并更好地理解经典随机游走模型中存在的相变的自然背景。拟议的研究是分析近临界随机系统的广泛研究方案的一部分。非均匀随机行走通常不能用均匀随机行走通常使用的技术来研究：需要新的方法(同样重要的是，新的直觉)。自然，对近临界系统的分析比对远离临界的系统的分析更具挑战性和微妙。这种方法论是基于鞅思想的。这些方法是健壮和强大的，预计在项目期间开发的方法将适用于许多其他近临界模型，包括那些跨现代概率理论和其他领域的应用，如排队理论、相互作用的粒子系统和随机介质。

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.