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| Natura: | Preprint |
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2024
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| Accesso online: | https://arxiv.org/abs/2407.21306 |
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| _version_ | 1866918215064485888 |
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| author | Deng, Changsong Li, Xiang Schilling, Rene L. Xu, Lihu |
| author_facet | Deng, Changsong Li, Xiang Schilling, Rene L. Xu, Lihu |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_21306 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Total variation distance between SDEs with stable noise and Brownian motion Deng, Changsong Li, Xiang Schilling, Rene L. Xu, Lihu Probability 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. |
| title | Total variation distance between SDEs with stable noise and Brownian motion |
| topic | Probability |
| url | https://arxiv.org/abs/2407.21306 |