Stochastic Spatial Processes

随机空间过程

• 批准号: RGPIN-2019-03928
• 负责人: Perkins, Edwin
• 金额: $ 3.06万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=758558
• 关键词: Stochastic; Spatial; Processes

We study a spatial version of the well-known normal distribution which applies to quite general populations distributed through space undergoing reproduction and migration. Examples include populations of a particular genetic type undergoing random replacement and mutation, diseases spreading through a population, a rare species competing for resources, or a liquid percolating through a porous medium.  Although the local rules of evolution may be complex and quite different, it can be much easier to analyze the process at large space and time scales where these processes can exhibit universal behaviour.  Much of my research is to find, justify and analyze a mathematically precise picture of this large scale evolution which describes all of the above phenomena in the large. By fitting a few parameters of this limit one can then predict large-scale behaviour of the underlying population in myriad settings. Of particular interest is the behaviour of the population near the edge of its range or the set of infected sites near the infection front of the disease. We can then analyze the fractal behaviour of this random front. The limiting random process itself is called a Dawson-Watanabe superprocess (its co-discover, Don Dawson, is a leading figure in Canadian mathematics), an important special case being super-Brownian motion (SBM). It exhibits some surprising regular behaviour, in spite of its inherent randomness.  We study its qualitative and quantitative properties. We also develop methods to construct and characterize other random processes which behave locally like these fundamental objects.

我们研究了众所周知的正态分布的空间版本，该版本适用于通过经过繁殖和迁移的空间分布的相当一般的种群。例子包括经过随机替代和突变的特定遗传类型的种群，通过人群传播的疾病，竞争资源竞争的稀有物种或通过多孔培养基渗透的液体。尽管局部进化规则可能很复杂且完全不同，但是在大时空和时间尺度上分析这些过程可以表现出普遍行为的过程可能会更容易。我的大部分研究是找到，证明和分析这种大规模演化的数学精确图片，该图表描述了以上所有现象。通过拟合此限制的一些参数，可以预测在无数环境中基础人群的大规模行为。特别令人感兴趣的是人口在其范围边缘附近的人口或疾病感染前附近受感染部位的集合。然后，我们可以分析此随机阵线的分形行为。限制随机过程本身称为Dawson-Watanabe Superprocess（其共同发现，Don Dawson，是加拿大数学的领导人物），这是一个重要的特殊情况，是超棕色运动（SBM）。尽管其继承了随机性，但它表现出一些令人惊讶的规则行为。我们研究其定性和定量特性。我们还开发了构建和表征其他像这些基本对象的本地行为的其他随机过程的方法。