  Diffusion models have rapidly become a vital part of deep generative
architectures, given today's increasing demands. Obtaining large,
high-performance diffusion models demands significant resources, highlighting
their importance as intellectual property worth protecting. However, existing
watermarking techniques for ownership verification are insufficient when
applied to diffusion models. Very recent research in watermarking diffusion
models either exposes watermarks during task generation, which harms the
imperceptibility, or is developed for conditional diffusion models that require
prompts to trigger the watermark. This paper introduces WDM, a novel
watermarking solution for diffusion models without imprinting the watermark
during task generation. It involves training a model to concurrently learn a
Watermark Diffusion Process (WDP) for embedding watermarks alongside the
standard diffusion process for task generation. We provide a detailed
theoretical analysis of WDP training and sampling, relating it to a shifted
Gaussian diffusion process via the same reverse noise. Extensive experiments
are conducted to validate the effectiveness and robustness of our approach in
various trigger and watermark data configurations.
