{
  "id": "2312.11145",
  "version": "v2",
  "published": "2023-12-18T12:35:46.000Z",
  "updated": "2024-01-15T08:54:39.000Z",
  "title": "SDEs with supercritical distributional drifts",
  "authors": [
    "Zimo Hao",
    "Xicheng Zhang"
  ],
  "comment": "40pages",
  "categories": [
    "math.PR",
    "math.AP"
  ],
  "abstract": "Let $d\\geq 2$. In this paper, we investigate the following stochastic differential equation (SDE) in ${\\mathbb R}^d$ driven by Brownian motion $$ {\\rm d} X_t=b(t,X_t){\\rm d} t+\\sqrt{2}{\\rm d} W_t, $$ where $b$ belongs to the space ${\\mathbb L}_T^q \\mathbf{H}_p^\\alpha$ with $\\alpha \\in [-1, 0]$ and $p,q\\in[2, \\infty]$, which is a distribution-valued and divergence-free vector field. In the subcritical case $\\frac dp+\\frac 2q<1+\\alpha$, we establish the existence and uniqueness of a weak solution to the integral equation: $$ X_t=X_0+\\lim_{n\\to\\infty}\\int^t_0b_n(s,X_s){\\rm d} s+\\sqrt{2} W_t. $$ Here, $b_n:=b*\\phi_n$ represents the mollifying approximation, and the limit is taken in the $L^2$-sense. In the critical and supercritical case $1+\\alpha\\leq\\frac dp+\\frac 2q<2+\\alpha$, assuming the initial distribution has an $L^2$-density, we show the existence of weak solutions and associated almost sure Markov processes. Moreover, under the additional assumption that $b=b_1+b_2+{\\rm div} a$, where $b_1\\in {\\mathbb L}^\\infty_T{\\mathbf B}^{-1}_{\\infty,2}$, $b_2\\in {\\mathbb L}^2_TL^2$, and $a$ is a bounded antisymmetric matrix-valued function, we establish the convergence of mollifying approximation solutions without the need to subtract a subsequence. To illustrate our results, we provide examples of Gaussian random fields and singular interacting particle systems, including the two-dimensional vortex models.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2024-01-15T08:54:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10"
    ],
    "keywords": [
      "supercritical distributional drifts",
      "weak solution",
      "mollifying approximation",
      "gaussian random fields",
      "sure markov processes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}