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，它允许关于一类“对偶函数”在两个随机马尔可夫过程之间建立密切联系。如果建立了形式上的对偶性，通常可以通过分析一个更简单的、通常是离散的或组合的对偶过程的性质，来研究“复杂的”空间随机系统的重要性质，例如其谱系的长期行为或性质。这种方法已经在相互作用粒子系统理论和相互作用随机(P)DES模拟种群演化的许多过程中获得了巨大的成功(例如，踏脚石或Wright-Fisher模型)。在过去的几年里，已经取得了重要进展。然而，目前还没有系统的对偶理论(A.Etheridge[eth06]p.519)，许多具有理论和实践意义的系统还有待进一步的分析。本项目有三个主要目的。首先，我们想把关于某些SPDEs的几个具体问题转移到关于它们的对偶过程的问题上(I)。其次，我们对对偶过程本身的长期性质感兴趣(Ii)。最后，我们致力于系统地分析对偶方法

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