{
  "id": "2407.09222",
  "version": "v1",
  "published": "2024-07-12T12:38:17.000Z",
  "updated": "2024-07-12T12:38:17.000Z",
  "title": "Energy solutions to SDEs with supercritical distributional drift: A stopping argument",
  "authors": [
    "Lukas Gräfner"
  ],
  "comment": "11 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "In this note we consider the SDE \\begin{equation*} \\text{d}X_t = b (t, X_t) \\text{d} t + \\sqrt{2} \\text{d} B_t, \\label{mainSDE} \\end{equation*} in dimension $d \\geqslant 2$, where $B$ is a Brownian motion and $b : \\mathbb{R}_+ \\rightarrow \\mathcal{S}' (\\mathbb{R}^d ; \\mathbb{R}^d)$ is distributional, scaling super-critical and satisfies $\\nabla \\cdot b \\equiv 0$. We partially extend the super-critical weak well-posedness result for energy solutions from [GP24] by allowing a mixture of the regularity regimes treated therein: Outside of environments of a small (and compared to [GP24] ''time-dependent'') set $K \\subset \\mathbb{R}_+ \\times \\mathbb{R}^d$, the antisymmetric matrix field $A$ with $b^i = \\nabla\\cdot A_i$ is assumed to be in a certain supercritical $L^q_T H^{s, p}$-type class that allows a direct link between the PDE and the SDE from a-priori estimates up to the stopping time of visiting $K$. To establish this correspondence, and thus uniqueness, globally in time we then show that $K$ is actually never visited which requires us to impose a relation between the dimension of $K$ and the H\\\"older regularity of $X$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-12T12:38:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "supercritical distributional drift",
      "energy solutions",
      "stopping argument",
      "antisymmetric matrix field",
      "super-critical weak well-posedness result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}