Total variation distance between SDEs with stable noise and Brownian motion

TL;DR

This paper derives bounds on the total variation and Wasserstein-$p$ distances between the solutions of SDEs driven by stable noise and Brownian motion, revealing how these distances depend on the stability parameter and dimension.

Contribution

It provides explicit, optimal bounds for the total variation and Wasserstein-$p$ distances between SDE solutions driven by stable and Gaussian noises, extending previous work and using new interpolation techniques.

Findings

Bound on total variation distance depending on $eta$ and dimension

Bound on Wasserstein-$p$ distance for $0< p <1$

Optimality of the bounds with respect to $eta$

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.