Stochastic Spatial Processes

我们研究了众所周知的正态分布的空间版本，它适用于通过空间分布的经历繁殖和迁移的相当一般的种群。例子包括经历随机替换和突变的特定遗传类型的种群，在种群中传播的疾病，争夺资源的稀有物种，或通过多孔介质渗透的液体。虽然进化的局部规则可能是复杂的和完全不同的，但在大的空间和时间尺度上分析这些过程可能会容易得多，因为这些过程可以表现出普遍的行为。我的大部分研究都是为了找到、证明和分析这种大规模进化的数学精确图像，它描述了上述所有现象。通过拟合这个极限的几个参数，人们就可以预测潜在种群在无数环境下的大规模行为。特别令人感兴趣的是在其范围边缘附近的人群或在疾病感染前沿附近的一组受感染地点的行为。然后我们可以分析这个随机锋面的分形行为。******极限随机过程本身被称为道森-渡边超过程（它的共同发现者唐·道森是加拿大数学界的领军人物），一个重要的特例是超布朗运动（SBM）。它表现出一些令人惊讶的规律行为，尽管其固有的随机性。我们研究了它的定性和定量性质。我们还开发了一些方法来构建和表征其他随机过程，这些随机过程的局部行为与这些基本对象相似。**

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. **