Non-homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems
Non-homogeneous Random Walks: Lyapunov Function Methods for Near-Critical Stochastic Systems
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DOI：
10.1017/9781139208468
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发表时间：
2016-12
期刊：
影响因子：
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通讯作者：
M. Menshikov;S. Popov;A. Wade
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文献类型：
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作者：
M. Menshikov;S. Popov;A. Wade
随机系统为各种重要的现实应用提供了强大的抽象模型：例如，电力供应、交通流量、数据传输。它们(以及它们所模拟的真实系统)往往会发生相变，当某个参数低于某个临界值时，它们会以一种方式表现出来，一旦达到临界值，就会立即切换行为。在真实的系统中，我们不一定能够控制所有参数值，因此知道如何找到临界点并了解这些点附近的系统行为是很重要的。本书以非齐次随机游动为例，介绍了应用于近临界随机系统的“半鞅”或“李雅普诺夫函数”方法。应用程序处理近临界随机系统，涵盖从随机台球模型到相互作用的粒子系统的现代概率理论。深入探讨了空间非齐次随机游动，因为它们提供了典型的近临界系统。
Stochastic systems provide powerful abstract models for a variety of important real-life applications: for example, power supply, traffic flow, data transmission. They (and the real systems they model) are often subject to phase transitions, behaving in one way when a parameter is below a certain critical value, then switching behaviour as soon as that critical value is reached. In a real system, we do not necessarily have control over all the parameter values, so it is important to know how to find critical points and to understand system behaviour near these points. This book is a modern presentation of the'semimartingale'or'Lyapunov function'method applied to near-critical stochastic systems, exemplified by non-homogeneous random walks. Applications treat near-critical stochastic systems and range across modern probability theory from stochastic billiards models to interacting particle systems. Spatially non-homogeneous random walks are explored in depth, as they provide prototypical near-critical systems.