Abstract We consider d -dimensional stochastic differential equations (SDEs) of the form dUₜ = b (Uₜ) \, dt + \, dZₜ. Let Xₜ denote the solution if the driving noise Zₜ is a d -dimensional rotationally symmetric -stable process (1 2), and let Yₜ be the solution if the driving noise is a d -dimensional Brownian motion. Continuing the work started in Deng et al. (2025), we derive an estimate of the total variation distance \|law (Xₓ) -law (Yₓ) \|TV for all t 0, and we show that the ergodic measures _ and ₂ of Xₜ and Yₜ, respectively, satisfy \|_-₂\|TV Cd (1+d) (2-) / (-1). We show that this bound is optimal with respect to by an Ornstein–Uhlenbeck SDE. Combining this bound with a recent interpolation result from Huang et al. (2023), we can derive a bound in the Wasserstein- p distance (0 p 1): \|_-₂\|ₖ䂹 Cd^{ (p+3) /2 (1+d) } (2-) /-1.
Deng et al. (Mon,) studied this question.