Total variation distance between SDEs with stable noise and Brownian motion

Abstract

We consider a $d$-dimensional stochastic differential equation (SDE) of the form $d U_t = b(U_t) dt + \sigma\,d Z_t$, let $X_t$ be the solution if the driving noise $Z_t$ is a $d$-dimensional rotationally symmetric $\alpha$-stable process ($1<\alpha<2$), and let $Y_t$ be the solution if the driving noise is a $d$-dimensional Brownian motion. Continuing the work of [Deng,Schilling, Xu, Bernoulli, 23], we derive an estimate of the total variation distance $\|{\rm L} (X_{t})-{\rm law}(Y_{t})\|_{\rm TV}$ for all $t>0$, and we show that the ergodic measures $\mu_\alpha$ and $\mu_2$ of $X_t$ and $Y_t$, respectively, satisfy $$\|\mu_\alpha-\mu_2\|_{\rm TV} \leq \frac{Cd\log(1+d)}{\alpha-1}(2-\alpha).$$ We shall show that this bound is optimal with respect to $\alpha$ by an Ornstein--Uhlenbeck SDE. Combining this bound with a recent interpolation result from \cite{HRW23}, we can derive a bound in Wasserstein-$p$ distance ($0< p <1$): \begin{gather*} \|\mu_\alpha-\mu_2\|_{W_p} \leq\frac{Cd^{(p+3)/2}\log(1+d)}{\alpha-1} (2-\alpha). \end{gather*} {\bf Key Words:} Total variation distance, Wasserstein-$p$ distance, stochastic differential equation, Poisson equation, stable process.