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Video editing and generation methods often rely on pre-trained image-based diffusion models. During the diffusion process, however, the reliance on rudimentary noise sampling techniques that do not preserve correlations present in subsequent frames of a video is detrimental to the quality of the results. This either produces high-frequency flickering, or texture-sticking artifacts that are not amenable to post-processing. With this in mind, we propose a novel method for preserving temporal correlations in a sequence of noise samples. This approach is materialized by a novel noise representation, dubbed $\int$-noise (integral noise), that reinterprets individual noise samples as a continuously integrated noise field: pixel values do not represent discrete values, but are rather the integral of an underlying infinite-resolution noise over the pixel area. Additionally, we propose a carefully tailored transport method that uses $\int$-noise to accurately advect noise samples over a sequence of frames, maximizing the correlation between different frames while also preserving the noise properties. Our results demonstrate that the proposed $\int$-noise can be used for a variety of tasks, such as video restoration, surrogate rendering, and conditional video generation.