Random walks in irregular media

不规则介质中的随机游走

• 批准号: RGPIN-2016-03703
• 负责人: Barlow, Martin
• 金额: $ 2.91万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635954
• 关键词: Random; walks; irregular; media

The study of random walks in mathematics goes back as least as far as the work of Polya in 1921; he showed that a simple random walk in one or two dimensions will return to its starting point infinitely often, while in 3 or more dimensions it will ultimately leave its initial neighbourhood, never to return. At large length scales random walks can be approximated by Brownian motion. The connection between diffusion and partial differential equations (PDE) is due to Einstein; the transition probabilities of the diffusion process are solutions to the heat equation. These connections, which hold in great generality, were explored very thoroughly in the second part of the 20th century by mathematicians.

数学中对随机游走的研究至少可以追溯到1921年Polya的工作；他证明了一个简单的一维或二维随机漫步会无限次地回到起点，而在三维或多维中，它最终会离开它的初始邻域，永远不会回来。在大长度尺度上，随机游走可以用布朗运动来近似。扩散和偏微分方程（PDE）之间的联系是由于爱因斯坦；扩散过程的跃迁概率是热方程的解。数学家们在20世纪下半叶对这些普遍存在的联系进行了非常彻底的探索。