Random walks in irregular media

不规则介质中的随机游走

• 批准号: 138460-2011
• 负责人: Barlow, Martin
• 金额: $ 3.35万
• 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: 2015
• 资助国家: 加拿大
• 起止时间: 2015-01-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=569354
• 关键词: Random; walks; irregular; media

My proposed research is on random walks in random environments, particularly on environments which arise in statistical physics. I will study the random conductance model, a model for a random walk in a random environment. The environment is made by giving the edges in Z^d non-negative random weights. The walk then jumps across an edge at a rate proportional to the edge weight. The main problems are to prove that the rescaled random walk converges to a multiple of Brownian motion, and to obtain Gaussian bounds on the transition density of the walk. If both properties hold, then combining them gives very good information concerning the process - one obtains Harnack inequalities, a local limit theorem, and good bounds on Green's functions. The case of independent weights has recently been solved, but many hard problems remain if one considers stationary ergodic edge weights, or edge weights which vary in time. A second question is to consider analogous problems in the continuum context - a particular problem here is the motion of a Brownian particle with reflection from a random collection of obstacles. The final topic is to study random walks on various random graphs, at or near criticality. Unlike the cases described above, one does not expect a Brownian limit, since these graphs have fractal properties. One particular problem is the uniform spanning tree in high dimensions.

我提议的研究是随机环境中的随机漫步，特别是统计物理中出现的环境。