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 focus primarily on Hölder 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 Matérn GPs.

翻译：高斯过程是定义函数空间上概率分布最常用的形式化方法。尽管高斯过程应用广泛，但对其样本路径（即它们定义概率测度的函数空间）的全面理解仍显不足。实践中，高斯过程并非通过概率测度构建，而是通过均值函数和协方差核构建。本文针对协方差核提出了充分必要条件，使得对应高斯过程的样本路径能够达到给定的正则性。我们主要关注Hölder正则性，因为它能导出特别简洁的条件，这些条件在平稳和各向同性高斯过程的情形下可进一步简化。随后我们证明，本文结论能够对机器学习应用中常用高斯过程（如Matérn高斯过程）的样本路径正则性给出新颖且异常精确的表征。