How small changes in distributions affect privacy-preserving sampling?

Gradient-flow sampling generates samples from complex distributions by minimizing an energy functional. This work develops perturbation bounds for flows under spherical Hellinger-Kantorovich geometry, quantifying how errors in the target distribution propagate over time. The authors apply these bounds to differential privacy, deriving explicit privacy certificates (Pure-DP and Approximate-DP) for exponential-mechanism samplers and separating sampling error from inherent algorithmic loss.