Non-homogeneous random walks

非齐次随机游走

• 批准号: EP/J021784/1
• 负责人: Andrew Wade
• 金额: $ 11.71万
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2013
• 资助国家: 英国
• 起止时间: 2013 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FJ021784%2F1
• 关键词: 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.

随机游走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表现出深厚的联系，并在科学内外享有广泛的应用。一般来说，随机游走是描述空间中粒子（或随机游走者）运动的随机过程。粒子的轨迹由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步行者的长期渐近行为。随机游走有着丰富的历史，涉及多个学科。经典的一维随机游走在数百年前首次作为机会游戏的模型进行研究，例如所谓的赌徒破产问题。在 Louis Bachelier 1900 年的论文中，他将类似的推理应用于他的股票价格模型。大约在同一时间，多维随机游走首次被研究，源于科学先驱在不同应用领域的工作，例如声学（瑞利勋爵的声音理论于 1880 年左右发展）、生物学（卡尔·皮尔逊 1906 年的物种随机迁移理论）和统计物理学（爱因斯坦的布朗运动理论于 1905-08 年发展）。在 Polya 在 1920 年代的工作之后，随机游走问题的数学重要性变得清晰起来，并且在过去 60 年左右的时间里，出现了将随机游走理论与调和分析、势论、组合学和谱理论等有影响力的数学领域联系起来的美妙联系。随机游走模型在现代科学的许多高度活跃的领域中不断发现新的重要应用；最近的具体发展包括例如微生物学中的微生物运动建模、分子化学中的聚合物构象以及经济学中的金融系统。空间同质随机游走是大量文献的主题，其中无论步行者当前的空间位置如何，跳跃的概率性质都是相同的。在许多建模应用中，空间同质性的经典假设是不现实的：随机游走者的行为可能取决于空间中的当前位置。因此，应用激发了非齐次随机游走的研究。此外，通过严格的数学证明，加深我们对基础研究问题的理解，数学动机自然而然地产生：具体来说，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环境。 拟议的研究是分析近临界随机系统的广泛研究计划的一部分。非齐次随机游走通常不能通过通常用于齐次随机游走的技术来研究：需要新的方法（同样重要的是，新的直觉）。当然，近临界系统的分析比远临界系统的分析更具挑战性和微妙性。该方法基于鞅思想。这些方法稳健而强大，预计该项目期间开发的方法将适用于许多其他近临界模型，包括那些在现代概率论及其他领域应用的模型，例如排队论、相互作用粒子系统和随机介质等领域。

项目成果

Non-homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems

• DOI: 10.1017/9781139208468
• 发表时间: 2016-12
• 作者: M. Menshikov;S. Popov;A. Wade

New Constructions and Bounds for Winkler's Hat Game

温克勒帽子游戏的新结构和界限

• DOI: 10.1137/130944680
• 发表时间: 2015
• 期刊: SIAM Journal on Discrete Mathematics
• 影响因子: 0.8
• 作者: Gadouleau M

Non-homogeneous random walks on a semi-infinite strip

半无限带上的非齐次随机游走

• DOI: 10.1016/j.spa.2014.05.005
• 发表时间: 2014
• 期刊: Stochastic Processes and their Applications
• 影响因子: 1.4
• 作者: Georgiou N

A radial invariance principle for non-homogeneous random walks

非齐次随机游走的径向不变性原理

• DOI: 10.1214/18-ecp159
• 发表时间: 2018
• 期刊: Electronic Communications in Probability
• 影响因子: 0.5
• 作者: Georgiou N

Anomalous recurrence properties of many-dimensional zero-drift random walks

多维零漂移随机游走的反常递推性质

• DOI: 10.1017/apr.2016.44
• 发表时间: 2016
• 期刊: Advances in Applied Probability
• 影响因子: 1.2
• 作者: Georgiou N