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Total variation distance between SDEs with stable noise and Brownian motion

Changsong Deng, Xiang Li, Rene L. Schilling, Lihu Xu

TL;DR

The paper analyzes how stable noise versus Brownian noise affect the long-time and finite-time behavior of SDEs with drift $b$ and diffusion matrix $σ$. It derives explicit total-variation bounds between the laws of the stable-driven and Brownian-driven systems for all times, and bounds between their ergodic measures, showing a tight $(2-α)$-dependent dependence that is optimal in α. A key technical contribution is a gradient estimate for the subordinated semigroup, obtained via a random time-change and Malliavin calculus, enabling precise control of the Poisson-type nonlocal terms. By combining these with a recent interpolation result, the authors also obtain Wasserstein-$p$ bounds for $0<p<1$, highlighting the quantitative impact of the noise type on ergodic behavior. The work provides a rigorous bridge between nonlocal and diffusion-driven dynamics and offers sharp, dimensionally robust metrics for comparing invariant measures of SDEs with different noise structures.

Abstract

We consider a $d$-dimensional stochastic differential equation (SDE) of the form $d U_t = b(U_t) dt + σ\,d Z_t$, let $X_t$ be the solution if the driving noise $Z_t$ is a $d$-dimensional rotationally symmetric $α$-stable process ($1<α<2$), 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 $μ_α$ and $μ_2$ of $X_t$ and $Y_t$, respectively, satisfy $$\|μ_α-μ_2\|_{\rm TV} \leq \frac{Cd\log(1+d)}{α-1}(2-α).$$ We shall show that this bound is optimal with respect to $α$ 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*} \|μ_α-μ_2\|_{W_p} \leq\frac{Cd^{(p+3)/2}\log(1+d)}{α-1} (2-α). \end{gather*} {\bf Key Words:} Total variation distance, Wasserstein-$p$ distance, stochastic differential equation, Poisson equation, stable process.

Total variation distance between SDEs with stable noise and Brownian motion

TL;DR

The paper analyzes how stable noise versus Brownian noise affect the long-time and finite-time behavior of SDEs with drift and diffusion matrix . It derives explicit total-variation bounds between the laws of the stable-driven and Brownian-driven systems for all times, and bounds between their ergodic measures, showing a tight -dependent dependence that is optimal in α. A key technical contribution is a gradient estimate for the subordinated semigroup, obtained via a random time-change and Malliavin calculus, enabling precise control of the Poisson-type nonlocal terms. By combining these with a recent interpolation result, the authors also obtain Wasserstein- bounds for , highlighting the quantitative impact of the noise type on ergodic behavior. The work provides a rigorous bridge between nonlocal and diffusion-driven dynamics and offers sharp, dimensionally robust metrics for comparing invariant measures of SDEs with different noise structures.

Abstract

We consider a -dimensional stochastic differential equation (SDE) of the form , let be the solution if the driving noise is a -dimensional rotationally symmetric -stable process (), and let be the solution if the driving noise is a -dimensional Brownian motion. Continuing the work of [Deng,Schilling, Xu, Bernoulli, 23], we derive an estimate of the total variation distance for all , and we show that the ergodic measures and of and , respectively, satisfy We shall show that this bound is optimal with respect to by an Ornstein--Uhlenbeck SDE. Combining this bound with a recent interpolation result from \cite{HRW23}, we can derive a bound in Wasserstein- distance (): \begin{gather*} \|μ_α-μ_2\|_{W_p} \leq\frac{Cd^{(p+3)/2}\log(1+d)}{α-1} (2-α). \end{gather*} {\bf Key Words:} Total variation distance, Wasserstein- distance, stochastic differential equation, Poisson equation, stable process.
Paper Structure (11 sections, 13 theorems, 123 equations)

This paper contains 11 sections, 13 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Main results
  4. A key gradient estimate for Theorem \ref{['main1']}
  5. A random time-change method
  6. Integration by parts via Malliavin Calculus
  7. Proof of Proposition \ref{['2ndgrad']}
  8. Gradient estimates of $Q_t$ and comparison of the generators
  9. Proofs of Theorem \ref{['main1']} of Corollary \ref{['corollary']}
  10. Optimality for the Ornstein--Uhlenbeck process on ${\mathds{R}}$
  11. Appendix

Key Result

Theorem 1.1

Assume (H1) and (H2). For any $\alpha \in (1,2)$, $x,y\in{\mathds{R}^d}$ and $t>0$, In particular,

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Proposition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • proof : Proof of Proposition \ref{['2ndgrad']}
  • Lemma 3.1
  • proof
  • ...and 16 more