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.

随机步行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具有深厚的联系，并在整个科学及其他地区享受广泛的应用。通常，随机步行是一个随机过程，描述了空间中粒子（或随机步行者）的运动。粒子的轨迹在及时离散的瞬间以一系列随机跳动表示。这些模型的基本问题涉及步行者的长期渐近行为。几百年前，首次将古典的一维随机步行作为机会游戏，例如所谓的赌徒的毁灭性问题。在1900年的论文中，路易斯·巴切尔（Louis Bachelier）采用了与他的股票价格模式相似的推理。大约在同一时间首次研究了多维随机步行，这是由科学先驱的作品（例如，雷利勋爵（Lord Rayleigh）从1880年发达的声音理论），生物学（卡尔·皮尔森（Karl Pearson）的1906年1906年的1906年，《物种随机迁移理论》）和统计物理学（Einstein of Brownian of Brownian of Brownian of Brownian of Brownian Motion of Brownistic of Brownian Motion of the Prownian of Brownian Motion of Stregien of the 1905 of 19055-05-08-08-08-08-08-08-08-08-08-08-08-08-08-08-08-08。在Polya在1920年代的工作之后，随机行走问题的数学重要性变得很明显，在过去的60年左右的时间里，已经出现了将随机行走理论与诸如谐波分析，潜在理论，组合学和光谱理论等数学领域的有影响力领域联系起来。随机步行模型继续在现代科学的许多高度活跃领域中找到新的重要应用。最新的具体发展包括在微生物学中进行微生物运动的建模，分子化学中的聚合物构象和经济学中的金融系统。在空间均匀的随机步行中，无论步行者的当前空间位置如何，跳跃的概率性质都是相同的，它是大量文献的主题。在许多建模应用中，空间同质性的经典假设是不现实的：随机助行器的行为可能取决于空间中的当前位置。因此，应用激发了对非均匀随机步行的研究。此外，数学动机自然而然地源于通过严格的数学证据，基本研究问题加深我们的理解的观点：具体，非均匀的随机步行是探测近乎临界行为并获得对经典随机步行模型中存在的相位过渡的预识的自然环境。 拟议的研究是一项广泛的研究计划的一部分，该计划旨在分析关键随机系统附近。通常不用于均匀随机步行的技术通常不研究非均匀的随机步行：新方法（以及重要的是，需要新的直觉）。自然，对近乎临界系统的分析比远非关键的系统更具挑战性和精致。该方法基于Martingale的想法。这些方法是强大而强大的，可以预见，在项目期间开发的方法将适用于许多其他近临界模型，包括那些在现代概率理论及其他方面具有应用的模型，用于排队理论，相互作用的粒子系统和随机媒体等领域。

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