Gaussian processes (GPs) are the most common formalism for defining probability distributions over spaces of functions. While applications of GPs are myriad, a comprehensive understanding of GP sample paths, i.e. the function spaces over which they define a probability measure, is lacking. In practice, GPs are not constructed through a probability measure, but instead through a mean function and a covariance kernel. In this paper we provide necessary and sufficient conditions on the covariance kernel for the sample paths of the corresponding GP to attain a given regularity. We use the framework of H\"older regularity as it grants particularly straightforward conditions, which simplify further in the cases of stationary and isotropic GPs. We then demonstrate that our results allow for novel and unusually tight characterisations of the sample path regularities of the GPs commonly used in machine learning applications, such as the Mat\'ern GPs.

翻译：高斯过程（GP）是定义函数空间概率分布的最常用形式。尽管GP的应用广泛，但对其样本路径（即所定义概率测度的函数空间）的全面理解仍显不足。实践中，GP并非通过概率测度直接构建，而是通过均值函数与协方差核进行定义。本文给出了协方差核使对应的GP样本路径达到给定正则性的充要条件。我们采用赫尔德正则性框架，因其能提供特别简洁的条件，在平稳各向同性高斯过程中可进一步简化。随后我们证明，该结果对机器学习中常用GP（如马特恩高斯过程）的样本路径正则性，可给出新颖且异常紧凑的表征。