Sticky Jump Diffusions: A Unifying Framework for Discrete, Continuous, and Hybrid Diffusion

We introduce Sticky Jump Diffusions (SJDs), continuous-time Markov processes on ${\mathbb{R}}^d$ with a discrete anchor set identified with token embeddings. In forward time, anchors release their mass at a hazard rate and the released mass diffuses in the continuous ambient space; time reversal couples a score-driven SDE with a sticky jump kernel whose rate and destination are fixed by flux balance with the forward law. We estimate the score and the per-anchor reverse hazards from a single denoising classifier via Denoising Hazard Matching, the hazard analogue of denoising score matching, with simulation-free cross-entropy training. SJD generalizes the classical sticky-boundary diffusions of Feller and Itô to vocabulary-sized anchor sets, and recovers masked diffusion, continuous diffusion, and hybrid diffusion as limits. Beyond these limits, the framework exposes two design axes that previous work hold fixed: the un-sticking kernel, which encodes both per-anchor geometry and cross-position structure of the corruption, and the per-anchor forward hazard, which encodes the commitment schedule. We evaluate SJD on CIFAR-10, ImageNet $64\times 64$, Text8, and Sudoku, where it is competitive with discrete, continuous, and hybrid baselines.