Optimal transport for stationary point processes

该项目的目标是开发一个对应的丰富的理论之间的最佳运输的概率措施的设置平稳随机措施，特别侧重于平稳点过程，即平稳离散无限措施。首先，我们的目标是在空间上的平稳点过程，这将导致自然的概念点过程之间的插值最短曲线的测地距离。这一结构将为项目的后续目标提供基础。一方面，我们将研究点过程的泛函沿着插值的凸性性质，以发展一种系统的方法来推导点过程的泛函不等式。另一方面，我们想利用驻点过程的距离来分析无限相互作用粒子系统的动力学，将它们视为新开发的几何中的梯度流。最后，我们的目标是将开发的技术应用到具体的挑战点过程模型的兴趣。

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.