Sharp Wasserstein-1 Stability for Diffusions under Radial Drift Bounds

We study how drift and initialization errors propagate in diffusions that may expand locally but contract at large distances. Under a radial constraint on the reference drift, we derive a finite-time Wasserstein-1 bound for diffusions with the same additive isotropic noise coefficient bounded away from zero. The bound separates initialization error from expected drift error along the perturbed process. Each term is optimal when the other error is absent. Reflection coupling reduces the analysis to a one-dimensional diffusion absorbed at zero. For fixed radial parameters allowing expansion, the amplification coefficient has a polynomial peak as the noise lower bound decreases. Its integral over all time grows exponentially; accumulating a fixed fraction takes an exponentially long time. A reverse Gaussian mixture shows why a sharp uniform coefficient can still overestimate a specified error: the optimal coefficient grows quadratically with mode separation, while adding a fixed positive constant to the drift gives bounded error per unit amplitude.