非局部连续方程周期测度解的存在性 via 最优运输
Existence of periodic measure-valued solutions to the nonlocal continuity equation via optimal transport
AI总结：
本文利用最优运输构造的CAT(0)空间，通过Schauder不动点定理证明了非局部连续方程周期测度解的存在性，并推导出周期图测度解的存在性。
摘要：
我们研究了一类非局部连续方程的周期解的存在性，这些方程包括由耦合振子系统导出的均场方程。虽然粒子层面的周期解通过在不变集的一个截面构造Poincaré映射已得到研究，但将其扩展到连续方程层面存在非平凡挑战。特别是，在无限维空间中设定适当的拓扑以显示不变性并应用不动点论证并不容易。为克服这一困难，我们使用由最优运输构造的测地凸空间中的不动点定理。具体而言，从关于 stationary 变量的分解出发，我们定义一个利用Wasserstein-2距离在一线性空间上的度量，从而得到一个CAT(0)空间。在此拓扑中，我们构造了概率测度的不变集，并通过Schauder定理证明了周期测度解的存在性。作为推论，我们的方法直接给出了周期图测度解的存在性。
英文摘要：
We investigate the existence of periodic solutions for a class of nonlocal continuity equations, which include mean-field equations derived from systems of coupled oscillators. While periodic solutions at the particle level have been studied through the construction of a Poincaré map on a section of an invariant set, extending this analysis to the level of continuity equations presents nontrivial challenges. In particular, setting an appropriate topology for the infinite-dimensional space to show invariance and apply the fixed point argument is not easy. To overcome this difficulty, we use fixed point theorem for geodesically convex spaces constructed by optimal transportation. Specifically, from the disintegration with respect to stationary variable, we define a metric using the Wasserstein-$2$ distance over one-dimensional space, which yields a $CAT(0)$ space. In this topology, we construct an invariance set of probability measures and prove the existence of the periodic measure-valued solution from Schauder's fixed point theorem on geodesically convex spaces. As a corollary, our method directly gives an existence of periodic graph measure solution.
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