{
  "id": "1001.3007",
  "version": "v1",
  "published": "2010-01-18T10:18:00.000Z",
  "updated": "2010-01-18T10:18:00.000Z",
  "title": "Stochastic differential equations with coefficients in Sobolev spaces",
  "authors": [
    "Shizan Fang",
    "Dejun Luo",
    "Anto Thalmaier"
  ],
  "comment": "31 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We consider It\\^o SDE $\\d X_t=\\sum_{j=1}^m A_j(X_t) \\d w_t^j + A_0(X_t) \\d t$ on $\\R^d$. The diffusion coefficients $A_1,..., A_m$ are supposed to be in the Sobolev space $W_\\text{loc}^{1,p} (\\R^d)$ with $p>d$, and to have linear growth; for the drift coefficient $A_0$, we consider two cases: (i) $A_0$ is continuous whose distributional divergence $\\delta(A_0)$ w.r.t. the Gaussian measure $\\gamma_d$ exists, (ii) $A_0$ has the Sobolev regularity $W_\\text{loc}^{1,p'}$ for some $p'>1$. Assume $\\int_{\\R^d} \\exp\\big[\\lambda_0\\bigl(|\\delta(A_0)| + \\sum_{j=1}^m (|\\delta(A_j)|^2 +|\\nabla A_j|^2)\\bigr)\\big] \\d\\gamma_d<+\\infty$ for some $\\lambda_0>0$, in the case (i), if the pathwise uniqueness of solutions holds, then the push-forward $(X_t)_# \\gamma_d$ admits a density with respect to $\\gamma_d$. In particular, if the coefficients are bounded Lipschitz continuous, then $X_t$ leaves the Lebesgue measure $\\Leb_d$ quasi-invariant. In the case (ii), we develop a method used by G. Crippa and C. De Lellis for ODE and implemented by X. Zhang for SDE, to establish the existence and uniqueness of stochastic flow of maps.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-01-18T10:18:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60H10",
      "34F05",
      "60J60",
      "37C10"
    ],
    "keywords": [
      "stochastic differential equations",
      "sobolev space",
      "diffusion coefficients",
      "stochastic flow",
      "distributional divergence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}