Optimal transport for stationary point processes

本项目的目标是在平稳随机测度的情况下，发展一个与丰富的概率测度之间最优传输理论相对应的理论，特别关注平稳点过程，即平稳离散无限测度。首先，我们的目标是在静止点过程的空间上构造测地线距离，这将引出用最短曲线在点过程之间插值的自然概念。这个结构将为项目的后续目标提供基础。一方面，我们将研究沿插值的点过程泛函的凸性，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，我们希望利用静止点过程的距离来分析无限相互作用粒子系统的动力学，将它们视为新发展的几何中的梯度流。最后，我们的目标是将开发的技术应用于具体的感兴趣的挑战性点过程模型。

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.