论文标题
时间依赖性球体随机场的水平曲线波动
Fluctuations of Level Curves for Time-Dependent Spherical Random Fields
论文作者
论文摘要
最近引起了人们的关注，对随机场的几何函数的行为进行了研究。在本文中，我们通过考虑一般各向同性高斯球体随机场的水平曲线的波动来扩展此框架。我们专注于长期和短暂的记忆假设；在前一种情况下，我们表明$ u $级曲线的波动由单个组件主导，对应于在随机场的多极组件的子集上评估的二阶混乱。我们证明存在差异渐近的取消点的存在；这些点不包括节点案例$ u = 0 $，这与最新的对淋巴结线的高频行为形成了鲜明对比，而无需时间依赖性的随机本征函数。在简短的内存案例中，我们表明所有混沌都在极限上贡献，没有发生取消，并且可以通过第四阶定理和Breuer-Major参数确定中心限制定理。
The investigation of the behaviour for geometric functionals of random fields on manifolds has drawn recently considerable attention. In this paper, we extend this framework by considering fluctuations over time for the level curves of general isotropic Gaussian spherical random fields. We focus on both long and short memory assumptions; in the former case, we show that the fluctuations of $u$-level curves are dominated by a single component, corresponding to a second-order chaos evaluated on a subset of the multipole components for the random field. We prove the existence of cancellation points where the variance is asymptotically of smaller order; these points do not include the nodal case $u = 0$, in marked contrast with recent results on the high-frequency behaviour of nodal lines for random eigenfunctions with no temporal dependence. In the short memory case, we show that all chaoses contribute in the limit, no cancellation occurs and a Central Limit Theorem can be established by Fourth-Moment Theorems and a Breuer-Major argument.
