Cross-domain recommendation (CDR) has been proven as a promising way to tackle the user cold-start problem, which aims to make recommendations for users in the target domain by transferring the user preference derived from the source domain. Traditional CDR studies follow the embedding and mapping (EMCDR) paradigm, which transfers user representations from the source to target domain by learning a user-shared mapping function, neglecting the user-specific preference. Recent CDR studies attempt to learn user-specific mapping functions in meta-learning paradigm, which regards each user's CDR as an individual task, but neglects the preference correlations among users, limiting the beneficial information for user representations. Moreover, both of the paradigms neglect the explicit user-item interactions from both domains during the mapping process. To address the above issues, this paper proposes a novel CDR framework with neural process (NP), termed as CDRNP. Particularly, it develops the meta-learning paradigm to leverage user-specific preference, and further introduces a stochastic process by NP to capture the preference correlations among the overlapping and cold-start users, thus generating more powerful mapping functions by mapping the user-specific preference and common preference correlations to a predictive probability distribution. In addition, we also introduce a preference remainer to enhance the common preference from the overlapping users, and finally devises an adaptive conditional decoder with preference modulation to make prediction for cold-start users with items in the target domain. Experimental results demonstrate that CDRNP outperforms previous SOTA methods in three real-world CDR scenarios.