This article provides a novel approach to nonstationarity by considering a bridge between differential equations and spatial fields. We consider the dynamical transformation of a given spatial process undergoing the action of a temporal flow of space diffeomorphisms. Such dynamical deformations are shown to be connected to certain classes of ordinary and partial differential equations. The natural question arises of how such dynamical diffeomorphisms convert the original spatial covariance function, specifically if the original covariance is spatially stationary or isotropic. We first challenge this question from a general perspective, and then turn into the special cases of both d-dimensional Euclidean spaces, and hyperspheres. Several examples of dynamical diffeomorphisms defined in these spaces are given and some emphasis has been put on the stationary reducibility problem. We provide a simple illustration to show the performance of the maximum likelihood estimation of the parameters of a family of dynamically deformed covariance functions.

中文翻译：

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由动力系统引起的非平稳时空协方差函数

本文通过考虑微分方程和空间场之间的桥梁，提供了一种新的非平稳性方法。我们考虑经历空间微分同胚的时间流作用的给定空间过程的动态变换。这种动态变形被证明与某些类别的常微分方程和偏微分方程有关。自然的问题是这种动态微分同胚如何转换原始空间协方差函数，特别是如果原始协方差在空间上是静止的或各向同性的。我们先从一般的角度来挑战这个问题，然后转入两者的特例维欧几里得空间和超球面。给出了在这些空间中定义的几个动态微分同胚的例子，并强调了静止可约性问题。我们提供了一个简单的说明来展示动态变形协方差函数族参数的最大似然估计的性能。