Stochastic Spatial Processes

我们研究了众所周知的正态分布的空间版本，它适用于在经历繁殖和迁移的空间中分布的相当一般的人口。例如，经历随机替换和突变的特定遗传类型的种群，通过种群传播的疾病，争夺资源的稀有物种，或通过多孔介质渗流的液体。尽管局部的进化规则可能很复杂，而且差异很大，但在大的空间和时间尺度上分析过程可能会容易得多，在那里这些过程可以表现出普遍的行为。我的大部分研究是为了找到、证明和分析这种大规模进化的数学精确图景，它描述了所有上述现象的大体。通过对这一极限的几个参数进行拟合，人们就可以预测在各种环境下潜在人群的大规模行为。特别令人感兴趣的是人口在其范围边缘附近的行为，或疾病感染前沿附近的一组受感染地点。然后我们可以分析这个随机前沿的分形行为。极限随机过程本身被称为Dawson-Watanabe超过程(它的共同发现Don Dawson是加拿大数学的领军人物)，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，尽管它具有内在的随机性。我们研究了它的定性和定量性质。我们还发展了构造和刻画其他随机过程的方法，这些随机过程的局部行为类似于这些基本对象。**

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. **