<p>Domain Generalization (DG) aims to learn a well-generalized model for unseen target domains from multiple observed source domains. Most current approaches, e.g., domain-invariant representation, typically focus on learning a universal sample-to-label mapping (function) across domains and have achieved impressive performance. Nonetheless, they usually overlook semantically intrinsic domain-specific information, resulting in limited generalizability. For compensation, in this paper, we adopt a novel standpoint, that is, a domain can be regarded as a metasample sampling from a certain meta-distribution, namely an environment distribution. With this perspective, in DG, functions learned from individual domains can be seen as a collective set of functional samples. Consequently, we can establish a meta-function, mapping from the environment to functions, to induce specific functions for unseen domains from the above function set effectively. To achieve this metafunction, we propose a learning paradigm based on Gaussian process with theoretical guarantee, namely Generalization Process for Domain Generalization (GPDG). Specifically, analogous to traditional Gaussian process, we describe the inference process in DG as Gaussian process fed with samples and their corresponding domain distributions. Furthermore, we employ a domain augmentation strategy to refine its smoothness. Extensive experiments are constructed to demonstrate the effectiveness of GPDG.</p>