Stochastic Spatial Models

我们研究分布在空间上随时间演化的随机过程。例子包括随机繁殖和迁徙的种群，在种群中传播的疾病，争夺资源的物种，或通过多孔介质（渗透）流动的石油。虽然进化的局部规则可能看起来很复杂，但在大空间和时间尺度上分析这些过程往往更容易，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精确数学图景。某些流行病模型在数学上等同于某些类型的渗透。这两个模型都有一个临界参数（感染概率或渗透概率），超过这个参数，流行病就可以长期存活，否则石油就会渗透。一个目标是找出这个关键参数在特定状态下的表现，对应于一种可以进行长期感染的疾病。对于竞争物种，我们的结果有时可以告诉我们它们什么时候可以平衡共存，或者什么时候一种类型可以驱逐另一种。用来处理这个问题的一般数学方法也告诉我们，合作行为是如何从简单的达尔文选择进化而来的——这是现代进化论的一个核心问题。从数学的角度来看，这些问题在二维环境中（这是最相关的）比在高维环境中更难，我们将尝试在这个具有挑战性和关键的环境中解决这个问题。对于合作的进化，我们将考虑多类型模式，其中多种类型（明显存在于自然界）的共存应该是可能的。我们还研究了在大尺度下观察这些随机空间模型所产生的极限模型。一个核心问题是，这些过程是否捕获了所有的潜在信息，或者是否存在需要更丰富结构的隐藏变量。对于连续体流行病模型，我们相信疾病将在随机波中存活，我们将研究这些波的定性和定量性质。最后的目标将是加强许多原始空间模型与其尺度连续体极限之间的联系。对于某些模型，我们知道，如果我们在固定时间拍摄快照，那么重新缩放的空间模型的概率将收敛于某个众所周知的称为超级布朗运动的普遍极限的概率。我们想要证明，整个时间演化的重新缩放模型收敛于这个无处不在的极限。

We study random processes distributed over space which evolve in time. Examples include populations undergoing random reproduction and migration, diseases spreading through a population, species competing for resources, or oil flowing through a porous medium (percolation). Although the local rules of evolution can appear complicated, often it is 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. Certain epidemic models turn out to be mathematically equivalent to certain kinds of percolation. Both models have a critical parameter (infection probability or percolation probability) above which the epidemic can survive in the long term, or oil will percolate. One objective is to find how this critical parameter can behave in a certain regime, corresponding say to a disease which can perform long range infection. For competing species our results can sometimes tell us when can they co-exist in equilibrium or when one type can drive the other out. The general mathematical methods designed to handle this question also tell us how cooperative behaviour can evolve out of simple Darwinian selection--a central problem in modern evolutionary theory. From a mathematical perspective it turns out that these questions can be harder in a 2-dimensional environment (which is most relevant) than in higher dimensions and we will try to address this question in this challenging and crucial setting. For the evolution of cooperation we will consider multi-type models where the coexistence of many types (clearly present in nature) should be possible. We also study the limiting models that arise from viewing these stochastic spatial models in the large scale. A central question is whether or not these processes capture all the underlying information or if there are hidden variables which require a richer structure. For the continuum epidemic models, we believe the disease will survive in stochastic waves and we will study the qualitative and quantitative nature of these waves. A final objective will be to tighten the connection between many of the original spatial models and their scaling continuum limits. For some models we know that if we take snapshots at fixed times then the probabilities of the rescaled spatial models will converge to those of a certain well-known universal limit called super-Brownian motion. We want to show that the entire time evolution of the rescaled models converge to that of this ubiquitous limit.