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

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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