{
  "titles": [
    {
      "title": "Elliptic stochastic quantization"
    }
  ],
  "creators": [
    {
      "name": "Albeverio, Sergio",
      "nameType": "Personal",
      "familyName": "Albeverio",
      "givenName": "Sergio"
    },
    {
      "name": "De Vecchi, Francesco C.",
      "nameType": "Personal",
      "familyName": "De Vecchi",
      "givenName": "Francesco C."
    },
    {
      "name": "Gubinelli, Massimiliano",
      "nameType": "Personal",
      "familyName": "Gubinelli",
      "givenName": "Massimiliano"
    }
  ],
  "publisher": {
    "name": "Papers of Massimiliano Gubinelli"
  },
  "publicationYear": 2018,
  "types": {
    "resourceTypeGeneral": "Preprint"
  },
  "version": "2",
  "url": "https://arxiv.org/abs/1812.04422v2",
  "dates": [
    {
      "date": "2018-12-11",
      "dateType": "Issued"
    },
    {
      "date": "2019-08-05",
      "dateType": "Available"
    }
  ],
  "language": "en",
  "formats": [
    "application/pdf",
    "application/x-tex",
    "application/x-texmacs",
    "application/x-bibtex"
  ],
  "subjects": [
    {
      "subject": "math.PR",
      "subjectScheme": "arXiv"
    },
    {
      "subject": "math-ph",
      "subjectScheme": "arXiv"
    },
    {
      "subject": "math.MP",
      "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 prove an explicit formula for the law in zero of the solution of a class of elliptic SPDE in $\\mathbb{R}^2$. This formula is the simplest instance of dimensional reduction, discovered in the physics literature by Parisi and Sourlas (1979), which links the law of an elliptic SPDE in $d + 2$ dimension with a Gibbs measure in $d$ dimensions. This phenomenon is similar to the relation between an $\\mathbb{R}^{d + 1}$ dimensional parabolic SPDE and its $\\mathbb{R}^d$ dimensional invariant measure. As such, dimensional reduction of elliptic SPDEs can be considered a sort of elliptic stochastic quantisation procedure in the sense of Nelson (1966) and Parisi and Wu (1981). Our proof uses in a fundamental way the representation of the law of the SPDE as a supersymmetric quantum field theory. Dimensional reduction for the supersymmetric theory was already established by Klein et al. (1984). We fix a subtle gap in their proof and also complete the dimensional reduction picture by providing the link between the elliptic SPDE and the supersymmetric model. Even in our $d = 0$ context the arguments are non-trivial and a non-supersymmetric, elementary proof seems only to be available in the Gaussian case.",
      "descriptionType": "Abstract"
    },
    {
      "description": "Journal reference: Ann. Probab. Volume 48, Number 4 (2020), 1693-1741",
      "descriptionType": "SeriesInformation"
    }
  ],
  "relatedIdentifiers": [
    {
      "relatedIdentifier": "arXiv:1812.04422v2",
      "relatedIdentifierType": "arXiv",
      "relationType": "IsIdenticalTo"
    },
    {
      "relatedIdentifier": "10.48550/arXiv.1812.04422",
      "relatedIdentifierType": "DOI",
      "relationType": "IsVersionOf"
    },
    {
      "relatedIdentifier": "10.1214/19-AOP1404",
      "relatedIdentifierType": "DOI",
      "relationType": "IsPreprintOf"
    }
  ]
}