Interacting stochastic (partial) differential equations, combinatorial stochastic processes and duality in spatial population dynamics

对偶方法是一种数学形式主义，它允许人们在两个随机马尔可夫过程之间建立关于一类“对偶函数”的密切联系。如果一个正式的对偶建立，它往往是可能的研究一个“复杂的”空间随机系统的重要属性，如长时间的行为或其谱系的属性，通过分析一个简单的，典型的discrete或组合，对偶过程的属性。这种方法已被用于相互作用的粒子系统和相互作用的随机（P）DE模拟人口的演变（例如垫脚石或赖特-费舍尔模型）的理论中的许多过程取得了巨大的成功。在过去几年中，取得了重要进展。然而，仍然没有系统的二元性理论（“寻找二元过程是一种黑色的艺术”，A。Etheridge [Eth06] p.519），许多具有理论和实践重要性的系统有待进一步分析。该项目有三个主要目标。首先，我们想把关于某些SPDE的几个具体问题转移到关于它们的双重过程的问题上（I）。其次，我们感兴趣的是对偶过程本身的长期性质（II）。最后，我们的目标是对对偶方法进行系统的分析

The method of duality is a mathematical formalism that allows one to establish close connections between two stochastic Markov processes with respect to a class of `duality functions'. If a formal duality is established, it is often possible to study important properties of a `complicated' spatial stochastic system, such as longtime-behaviour or properties of its genealogy, by analysing the properties of a simpler, typically discreteor combinatorial, dual process. This method has been used with great success for many processes in the theory of interacting particle systems and interacting stochastic (P)DEs modeling the evolution of populations (e.g. the stepping stone or the Wright-Fisher model). In the last years, important progress has been achieved. However, there is still no systematic theory of duality (“finding dual processes is something of a black art", A. Etheridge [Eth06] p.519), and many systems of theoretical and practical importance await further analysis. This project has three main objectives. Firstly, we would like to transfer several concrete questions about certain SPDEs to questions about their dual processes (I). Secondly, we are interested in the long-term properties of the dual processes themselves (II). Finally, we aim towards a systematic analysis of the method of duality