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，它允许人们在关于一类“对偶函数”的两个随机马尔可夫过程之间建立紧密的联系。如果建立了形式对偶，通常可以通过分析更简单的、典型的离散组合对偶过程的性质来研究“复杂”空间随机系统的重要性质，如长期行为或其谱系的性质。该方法已成功地应用于相互作用粒子系统理论和相互作用随机DEs (P)模型中的许多过程，例如垫脚石模型或Wright-Fisher模型。在过去几年中，取得了重要进展。然而，对偶仍然没有系统的理论（“发现对偶过程是一种黑暗的艺术”，a . Etheridge [Eth06] p.519），许多理论和实践重要性的系统等待着进一步的分析。这个项目有三个主要目标。首先，我们想把关于某些特殊经济平台的几个具体问题转移到关于它们的双重过程(1)的问题上。其次，我们对双过程本身的长期性质感兴趣（II）。最后，我们对对偶方法进行了系统的分析

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