Total variation distance between SDEs with stable noise and Brownian motion

We consider d -dimensional stochastic differential equations (SDEs) of the form dUₜ = b(Uₜ)\,dt + σ\,dZₜ . Let Xₜ denote the solution if the driving noise Zₜ is a d -dimensional rotationally symmetric α -stable process ( 1< α< 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(Xₜ)-law(Yₜ)\|TV for all t > 0 , and we show that the ergodic measures μ_α and μ₂ of Xₜ and Yₜ , respectively, satisfy \|μ_α-μ₂\|TV ≤ Cdlog(1+d)(2-α)/(α-1) . We show that this bound is optimal with respect to α 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 ): \|μ_α-μ₂\|Wₚ ≤ Cd(p+3)/2log(1+d)(2-α)/α-1 .