Longterm behaviour of interacting stochastic (partial) differential equations and combinatorial stochastic processes, with a focus on the method of duality

对偶方法是一种数学形式主义，它允许找到两个不一定相关的随机过程的某些性质之间的密切联系，例如矩或长期行为。通常，通过分析简单的、通常是离散的或组合的过程的性质，可以研究复杂的随机过程的重要性质，例如由随机偏微分方程式描述的过程。该方法在相互作用粒子系统理论、随机偏微分方程论、数学遗传学和随机种群生物学等领域都取得了巨大的成功。在过去的几年里，已经取得了许多重要的突破，特别是在方法层面上。然而，当试图寻找双重进程时，通常被限制在特别方法上。这个项目有三个主要目标。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双重进程的问题上来(I)。其次，我们对对偶过程本身的长期性质感兴趣(II)。最后，我们旨在系统地分析对偶方法的适用性，扩展它的范围和在不同类型的对偶下保留的性质(III)。

The method of duality is a mathematical formalism that allows to find close connections between certain properties, such as moments or the long-term behaviour, of two not necessarily related stochastic processes. Often it is possible to study important properties of a complicated stochastic process, e.g. described by a stochastic partical differential equation, by analysing the properties of a simpler, typically discrete or combinatorial, process. This method has been used with great success in the theory of interacting particle systems, stochastic partial differential equations, mathematical genetics and stochastic population biology. In the last years, many important break throughs have been achieved, especially on a methodological level. However, often when trying to find dual processes one is restricited to ad-hoc methods. This project has three main objectives. First of all, we would like to transfer several concrete questions about certain SPDEs to questions about their dual processes (I). Secondly, we are interested in the longterm properties of the dual processes itself (II). Finally, we aim at a systematic analysis of the method of duality focussing on applicability, extending its scope and properties that are preserved under different types of duality (III).