Temporal Resolution Limits for Score-Based Diffusion Models

Score-based diffusion models for trajectories involve two time variables. Physical time indexes the system represented by the data, while generative time indexes the noising and reverse sampling procedure. We ask what exact recovery of the joint distribution at one observation spacing implies about the underlying continuous process. A Brownian bridge construction gives continuous processes with identical joint distributions at the recorded times and different quadratic variation and crossing probabilities. A two-dimensional Ornstein--Uhlenbeck (OU) example shows the same ambiguity within smooth, time-homogeneous Itô diffusions. The population denoising score matching (DSM) objective therefore depends only on the sampled distribution. In a finite-sample OU experiment, two estimates have equal training DSM losses and differ by 8.28 nats in midpoint conditional negative log score. We evaluate such models at several observation spacings using intermediate observations excluded from training.