{
  "id": "2302.03372",
  "version": "v1",
  "published": "2023-02-07T10:27:39.000Z",
  "updated": "2023-02-07T10:27:39.000Z",
  "title": "Wasserstein-$1$ distance between SDEs driven by Brownian motion and stable processes",
  "authors": [
    "Changsong Deng",
    "Rene L. Schilling",
    "Lihu Xu"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "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 + \\sigma\\,d L_t, \\quad X_0=x, %\\label{BM-SDE} d Y_t=b(Y_t)\\,dt + \\sigma\\,d B_t, \\quad Y_0=y, \\end{gather*} where $\\sigma$ is an invertible $d\\times d$ matrix, $L_t$ is a rotationally symmetric $\\alpha$-stable L\\'evy process, and $B_t$ is a $d$-dimensional standard Brownian motion. We show that for any $\\alpha_0 \\in (1,2)$ the Wasserstein-$1$ distance $W_1$ satisfies for $\\alpha \\in [\\alpha_0,2)$ \\begin{gather*} W_{1}\\left(X_{t}^x, Y_{t}^y\\right) \\leq C e^{-Ct}|x-y| +C_{\\alpha_0}d\\cdot\\log(1+d)(2-\\alpha)\\log\\frac{1}{2-\\alpha}, \\end{gather*} which implies, in particular, \\normal \\begin{equation} \\label{e:W1Rate} W_1(\\mu_\\alpha, \\mu) \\leq C_{\\alpha_0} d \\cdot \\log(1+d)(2-\\alpha) \\log \\frac{1}{2-\\alpha}, \\end{equation} where $\\mu_\\alpha$ and $\\mu$ are the ergodic measures of $(X^x_t)_{t \\ge 0}$ and $(Y^y_t)_{t \\ge 0}$ respectively. The term $d\\cdot\\log(1+d)$ appearing in this estimate seems to be optimal. For the special case of a $d$-dimensional Ornstein--Uhlenbeck system, we show that $W_1(\\mu_\\alpha, \\mu) \\geq C_{\\alpha_0,d} (2-\\alpha)$; this indicates that the convergence rate with respect to $\\alpha$ in \\eqref{e:W1Rate} is optimal up to a logarithmic correction. We conjecture that the sharp rate with respect to $\\alpha$ and $d$ is $d\\cdot\\log (1+d) (2-\\alpha)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-07T10:27:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "sdes driven",
      "stable processes",
      "dimensional standard brownian motion",
      "dimensional ornstein-uhlenbeck system",
      "valued stochastic differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}