Optimal transport for stationary point processes

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The goal of this project is to develop a counterpart to the rich theory of optimal transport between probability measures in the setting of stationary random measure with a particular focus on stationary point processes, i.e. stationary discrete infinite measures. First we aim at constructing geodesic distances on the space of stationary point processes that will induce natural notions of interpolation between point processes by shortest curves. This structure will provide the basis for subsequent goals of the project. On the one hand we will investigate convexity properties of functionals of point processes along interpolations in order to develop a systematic approach to derive functional inequalities for point processes. On the other hand, we want to leverage the distance on stationary point processes to analyse the dynamics of infinite interacting particle systems viewing them as gradient flows in the newly developed geometry. Finally, we aim at applying the developed techniques to concrete challenging point process models of interest.