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 on, 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 us 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（GPs）是定义函数空间上概率分布最常用的形式体系。尽管GPs的应用广泛，但对其样本路径（即其定义概率测度的函数空间）的全面理解仍显不足。实践中，GPs并非通过概率测度直接构建，而是通过均值函数和协方差核来定义。本文给出了协方差核的必要且充分条件，使得对应GPs的样本路径具有特定正则性。我们采用Hölder正则性框架，因其提供了尤为简洁的条件，且在平稳和各向同性GPs情形下可进一步简化。随后证明，我们的结果能够对机器学习应用中常用GPs（如Matérn GPs）的样本路径正则性给出新颖且异常紧凑的刻画。