Terminal Localization of Empirical Reverse Diffusions under Singular Score Errors

Source abstract

For a finite empirical target, the exact reverse diffusion is driven toward training points by an attraction whose strength diverges as the forward noising time decreases to zero. We study whether this terminal localization persists under progressively measurable score errors that may themselves be singular. For reverse Ornstein--Uhlenbeck diffusions, from an arbitrary initial law and under a deterministic error envelope $E$ imposed along the perturbed trajectory satisfying $t\int_t^T E(σ)/σ\,dσ\to0$, we prove almost-sure convergence to a training point, together with explicit uniform bounds on a terminal window with probability arbitrarily close to one. Every training point has positive terminal probability, although the terminal selection probabilities need not match the empirical weights. The results cover power-law errors $E(t)=O(t^{-γ})$, $0<γ<1$, including infinite squared-error energy. For suitable errors in this range, the exact and perturbed full path laws can be mutually singular while their terminal laws share the same finite support. For suitable switched perturbations at the critical scale $t^{-1}$, arbitrarily small positive amplitudes can produce a nonzero terminal displacement with arbitrarily high probability. These results quantify stochastic robustness of empirical terminal attraction without requiring absolute continuity of the full path laws.