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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2302.03372 |
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| _version_ | 1866911789358252032 |
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| author | Deng, Changsong Schilling, Rene L. Xu, Lihu |
| author_facet | Deng, Changsong Schilling, Rene L. Xu, Lihu |
| contents | We are interested in the following two $\mathbb{R}^d$-valued stochastic differential equations (SDEs):
\begin{gather*}
d X_t=b(X_t)\,d t + σ\,d L_t, \quad X_0=x,
%\label{BM-SDE}
d Y_t=b(Y_t)\,d t + σ\,d B_t, \quad Y_0=y,
\end{gather*}
where $σ$ is an invertible $d\times d$ matrix, $L_t$ is a rotationally symmetric $α$-stable Lévy process, and $B_t$ is a $d$-dimensional standard Brownian motion (note that $B_t$ is a rotationally symmetric $α$-stable Lévy process with $α=2$). We show that for any $α_0 \in (1,2)$ the Wasserstein-$1$ distance $W_1$ satisfies for $α\in [α_0,2)$
\begin{gather*}
W_{1}\left(X_{t}^x, Y_{t}^y\right)
\leq C_1 e^{-C_2t}|x-y|
+\frac{C}{α_0-1}(2-α)d\log(1+d),
\end{gather*}
which implies, in particular,
\begin{equation} \label{e:W1Rate}
W_1(μ_α, μ_2)
\leq \frac{C}{α_0-1}(2-α)d\log(1+d),
\end{equation}
where $μ_α$ and $μ_2$ are the ergodic measures of $X_t$ and $Y_t$ respectively.
For the special case of a $d$-dimensional Ornstein--Uhlenbeck system, we show that $W_1(μ_α, μ_2) \geq C_{d} (2-α)$ for all $α\in(1,2)$; this indicates that the convergence rate with respect to $α$ in the second bound is optimal. The term $d\log(1+d)$ appearing in this bound seems to be optimal for the dimension $d$ as well. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_03372 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Optimal Wasserstein-$1$ distance between SDEs driven by Brownian motion and stable processes Deng, Changsong Schilling, Rene L. Xu, Lihu Probability We are interested in the following two $\mathbb{R}^d$-valued stochastic differential equations (SDEs): \begin{gather*} d X_t=b(X_t)\,d t + σ\,d L_t, \quad X_0=x, %\label{BM-SDE} d Y_t=b(Y_t)\,d t + σ\,d B_t, \quad Y_0=y, \end{gather*} where $σ$ is an invertible $d\times d$ matrix, $L_t$ is a rotationally symmetric $α$-stable Lévy process, and $B_t$ is a $d$-dimensional standard Brownian motion (note that $B_t$ is a rotationally symmetric $α$-stable Lévy process with $α=2$). We show that for any $α_0 \in (1,2)$ the Wasserstein-$1$ distance $W_1$ satisfies for $α\in [α_0,2)$ \begin{gather*} W_{1}\left(X_{t}^x, Y_{t}^y\right) \leq C_1 e^{-C_2t}|x-y| +\frac{C}{α_0-1}(2-α)d\log(1+d), \end{gather*} which implies, in particular, \begin{equation} \label{e:W1Rate} W_1(μ_α, μ_2) \leq \frac{C}{α_0-1}(2-α)d\log(1+d), \end{equation} where $μ_α$ and $μ_2$ are the ergodic measures of $X_t$ and $Y_t$ respectively. For the special case of a $d$-dimensional Ornstein--Uhlenbeck system, we show that $W_1(μ_α, μ_2) \geq C_{d} (2-α)$ for all $α\in(1,2)$; this indicates that the convergence rate with respect to $α$ in the second bound is optimal. The term $d\log(1+d)$ appearing in this bound seems to be optimal for the dimension $d$ as well. |
| title | Optimal Wasserstein-$1$ distance between SDEs driven by Brownian motion and stable processes |
| topic | Probability |
| url | https://arxiv.org/abs/2302.03372 |