We analyse how the sampling dynamics of distributions evolve in score-based diffusion models using \emph{cross-fluctuations}, a centered-moment statistic from statistical physics. Specifically, we show that starting from an unbiased isotropic normal distribution, samples undergo sharp, discrete transitions, eventually forming distinct events of a desired distribution while progressively revealing finer structure. As this process is reversible, these transitions also occur in reverse, where intermediate states progressively merge, tracing a path back to the initial distribution. We demonstrate that these transitions can be detected as discontinuities in $n^{\text{th}}$-order cross-fluctuations. For variance-preserving SDEs, we derive a closed-form for these cross-fluctuations that is efficiently computable for the reverse trajectory. %, thus tying cross-fluctuation dynamics to event formation within the desired distribution. We find that detecting these transitions directly boosts sampling efficiency, accelerates class-conditional and rare-class generation, and improves two zero-shot tasks--image classification and style transfer--without expensive grid search or retraining. We also show that this viewpoint unifies classical coupling and mixing from finite Markov chains with continuous dynamics while extending to stochastic SDEs and non Markovian samplers. Our framework therefore bridges discrete Markov chain theory, phase analysis, and modern generative modeling.

This work tackles the important task of understanding out-of-distribution behavior in two prominent types of generative models, i.e., GANs and Diffusion models. Understanding this behavior is crucial in understanding their broader utility and risks as these systems are increasingly deployed in our daily lives. Our first contribution is demonstrating that diffusion spaces outperform GANs' latent spaces in inverting high-quality OOD images. We also provide a theoretical analysis attributing this to the lack of prior holes in diffusion spaces. Our second significant contribution is to provide a theoretical hypothesis that diffusion spaces can be projected onto a bounded hypersphere, enabling image manipulation through geodesic traversal between inverted images. Our analysis shows that different geodesics share common attributes for the same manipulation, which we leverage to perform various image manipulations. We conduct thorough empirical evaluations to support and validate our claims. Finally, our third and final contribution introduces a novel approach to the few-shot sampling for out-of-distribution data by inverting a few images to sample from the cluster formed by the inverted latents. The proposed technique achieves state-of-the-art results for the few-shot generation task in terms of image quality. Our research underscores the promise of diffusion spaces in out-of-distribution imaging and offers avenues for further exploration. Please find more details about our project at \url{http://cvit.iiit.ac.in/research/projects/cvit-projects/diffusionOOD}