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

点击翻译按钮获取中文摘要

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).