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Total variation distance between SDEs with stable noise and Brownian motion

Part of: Stochastic analysis Stochastic processes

Published online by Cambridge University Press:  23 February 2026

Chang-Song Deng ,
Xiang Li ,
René Schilling  and
Lihu Xu
Chang-Song Deng*
Affiliation:
Wuhan University
Xiang Li*
Affiliation:
Southern University of Science and Technology
René Schilling*
Affiliation:
TU Dresden
Lihu Xu*
Affiliation:
Michigan State University and University of Macau
*
*Postal address: School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China. Email: dengcs@whu.edu.cn
**Postal address: Department of Statistics and Data Science, Southern University of Science and Technology, Shenzhen 518055, China. Email: lixiang3@sustech.edu.cn
***Postal address: TU Dresden, Fakultät Mathematik, Institut für Mathematische Stochastik, 01062 Dresden, Germany. Email: rene.schilling@tu-dresden.de
****Postal address: Department of Statistics and Probability, Wells Hall, Michigan State University, East Lansing, MI 48824, USA. Email: lihuxu@msu.edu
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Abstract

We consider d-dimensional stochastic differential equations (SDEs) of the form $\textrm{d}U_t = b(U_t)\,\textrm{d}t + \sigma\,\textrm{d}Z_t$. Let $X_t$ denote the solution if the driving noise $Z_t$ is a d-dimensional rotationally symmetric $\alpha$-stable process ($1\lt \alpha\lt 2$), and let $Y_t$ 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 $\|\textrm{law}(X_{t})-\textrm{law}(Y_{t})\|_\textrm{TV}$ for all $t \gt 0$, and we show that the ergodic measures $\mu_\alpha$ and $\mu_2$ of $X_t$ and $Y_t$, respectively, satisfy $\|\mu_\alpha-\mu_2\|_\textrm{TV} \leq {Cd\log(1+d)}(2-\alpha)/({\alpha-1})$. We show that this bound is optimal with respect to $\alpha$ 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 ($0 \lt p \lt 1$): $\|\mu_\alpha-\mu_2\|_{W_p} \leq {Cd^{(p+3)/2}\log(1+d)}(2-\alpha)/{\alpha-1}$.

Keywords

Total variation distanceWasserstein-p distancestochastic differential equationstable processBrownian motion

MSC classification

Primary: 60H10: Stochastic ordinary differential equations
Secondary: 60G52: Stable processes

Information

Type
Original Article
Information
Journal of Applied Probability , First View , pp. 1 - 19
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Copyright
© The Author(s), 2026. Published by Cambridge University Press on behalf of Applied Probability Trust

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