{
  "titles": [
    {
      "title": "Averaging along irregular curves and regularisation of ODEs"
    }
  ],
  "creators": [
    {
      "name": "Catellier, R.",
      "nameType": "Personal",
      "familyName": "Catellier",
      "givenName": "R."
    },
    {
      "name": "Gubinelli, M.",
      "nameType": "Personal",
      "familyName": "Gubinelli",
      "givenName": "M."
    }
  ],
  "publisher": {
    "name": "Papers of Massimiliano Gubinelli"
  },
  "publicationYear": 2012,
  "types": {
    "resourceTypeGeneral": "Preprint"
  },
  "version": "4",
  "url": "https://arxiv.org/abs/1205.1735v4",
  "dates": [
    {
      "date": "2012-05-08",
      "dateType": "Issued"
    },
    {
      "date": "2016-02-04",
      "dateType": "Available"
    },
    {
      "date": "2026-11-14",
      "dateType": "Updated"
    }
  ],
  "language": "en",
  "formats": [
    "application/pdf",
    "application/x-tex"
  ],
  "subjects": [
    {
      "subject": "math.PR",
      "subjectScheme": "arXiv"
    },
    {
      "subject": "math.FA",
      "subjectScheme": "arXiv"
    }
  ],
  "rightsList": [
    {
      "rights": "arXiv.org perpetual, non-exclusive license to distribute; all other rights reserved to the authors",
      "rightsUri": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/"
    }
  ],
  "descriptions": [
    {
      "description": "We consider the ordinary differential equation (ODE) $dx_{t} =b(t,x_{t} ) dt+ dw_{t}$ where $w$ is a continuous driving function and $b$ is a time-dependent vector field which possibly is only a distribution in the space variable. We quantify the regularising properties of an arbitrary continuous path $w$ on the existence and uniqueness of solutions to this equation. In this context we introduce the notion of $\\rho$-irregularity and show that it plays a key role in some instances of the regularisation by noise phenomenon. In the particular case of a function $w$ sampled according to the law of the fractional Brownian motion of Hurst index $H \\in (0,1)$, we prove that almost surely the ODE admits a solution for all $b$ in the Besov–Hölder space $B^{\\alpha+1}_{\\infty , \\infty}$ with $\\alpha >-1/2H$. If $\\alpha >1-1/2H$ then the solution is unique among a natural set of continuous solutions. If $H>1/3$ and $\\alpha >3/2-1/2H$ or if $\\alpha >2-1/2H$ then the equation admits a unique Lipschitz flow. Note that when $\\alpha <0$ the vector field $b$ is only a distribution, nonetheless there exists a natural notion of solution for which the above results apply.",
      "descriptionType": "Abstract"
    }
  ],
  "relatedIdentifiers": [
    {
      "relatedIdentifier": "arXiv:1205.1735v4",
      "relatedIdentifierType": "arXiv",
      "relationType": "IsIdenticalTo"
    },
    {
      "relatedIdentifier": "10.48550/arXiv.1205.1735",
      "relatedIdentifierType": "DOI",
      "relationType": "IsVersionOf"
    }
  ]
}