On uniformly subelliptic operators and stochastic area View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2007-11-27

AUTHORS

Peter Friz, Nicolas Victoir

ABSTRACT

Let X a be a Markov process with generator ∑i,j∂i( aij∂j· ) where a is a uniformly elliptic symmetric matrix. Thanks to the fundamental works of T. Lyons, stochastic differential equations driven by X a can be solved in the “rough path sense”; that is, pathwise by using a suitable stochastic area process. Our construction of the area, which generalizes previous works of Lyons–Stoica and then Lejay, is based on Dirichlet forms associated to subellitpic operators. This enables us in particular to discuss large deviations and support descriptions in suitable rough path topologies. As typical rough path corollary, Freidlin–Wentzell theory and the Stroock–Varadhan support theorem remain valid for stochastic differential equations driven by X a. More... »

PAGES

475-523

References to SciGraph publications

  • 2005-12-29. A note on the notion of geometric rough paths in PROBABILITY THEORY AND RELATED FIELDS
  • 2000. Second-Order Subelliptic Operators on Lie Groups II: Real Measurable Principal Coefficients in SEMIGROUPS OF OPERATORS: THEORY AND APPLICATIONS
  • 1998. Large Deviations Techniques and Applications in NONE
  • 1986-12. A new proof of Moser's parabolic harnack inequality using the old ideas of Nash in ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
  • 1994. A simple proof of the support theorem for diffusion processes in SÉMINAIRE DE PROBABILITÉS XXVIII
  • 1995. On the Geometry Defined by Dirichlet Forms in SEMINAR ON STOCHASTIC ANALYSIS, RANDOM FIELDS AND APPLICATIONS
  • 1988. Diffusion semigroups corresponding to uniformly elliptic divergence form operators in SÉMINAIRE DE PROBABILITÉS XXII
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/s00440-007-0113-y

    DOI

    http://dx.doi.org/10.1007/s00440-007-0113-y

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1001934547


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0102", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Applied Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, CB3 0WB, Cambridge, UK", 
              "id": "http://www.grid.ac/institutes/grid.5335.0", 
              "name": [
                "Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, CB3 0WB, Cambridge, UK"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Friz", 
            "givenName": "Peter", 
            "id": "sg:person.010663776615.90", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010663776615.90"
            ], 
            "type": "Person"
          }, 
          {
            "affiliation": {
              "alternateName": "Hong Kong, Hong Kong", 
              "id": "http://www.grid.ac/institutes/None", 
              "name": [
                "Hong Kong, Hong Kong"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Victoir", 
            "givenName": "Nicolas", 
            "id": "sg:person.014642077545.17", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014642077545.17"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/bf00251802", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1021016997", 
              "https://doi.org/10.1007/bf00251802"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-0348-8417-4_10", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1003598472", 
              "https://doi.org/10.1007/978-3-0348-8417-4_10"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bfb0073832", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1011328779", 
              "https://doi.org/10.1007/bfb0073832"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bfb0084145", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1024290631", 
              "https://doi.org/10.1007/bfb0084145"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-3-0348-7026-9_17", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1021472546", 
              "https://doi.org/10.1007/978-3-0348-7026-9_17"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/s00440-005-0487-7", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1006769344", 
              "https://doi.org/10.1007/s00440-005-0487-7"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/978-1-4612-5320-4", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1109705063", 
              "https://doi.org/10.1007/978-1-4612-5320-4"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "2007-11-27", 
        "datePublishedReg": "2007-11-27", 
        "description": "Let X a be a Markov process with generator \u2211i,j\u2202i( aij\u2202j\u00b7 ) where a is a uniformly elliptic symmetric matrix. Thanks to the fundamental works of T. Lyons, stochastic differential equations driven by X a can be solved in the \u201crough path sense\u201d; that is, pathwise by using a suitable stochastic area process. Our construction of the area, which generalizes previous works of Lyons\u2013Stoica and then Lejay, is based on Dirichlet forms associated to subellitpic operators. This enables us in particular to discuss large deviations and support descriptions in suitable rough path topologies. As typical rough path corollary, Freidlin\u2013Wentzell theory and the Stroock\u2013Varadhan support theorem remain valid for stochastic differential equations driven by X a.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/s00440-007-0113-y", 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1053886", 
            "issn": [
              "0178-8051", 
              "1432-2064"
            ], 
            "name": "Probability Theory and Related Fields", 
            "publisher": "Springer Nature", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "3-4", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "142"
          }
        ], 
        "keywords": [
          "stochastic differential equations", 
          "differential equations", 
          "Stroock\u2013Varadhan support theorem", 
          "rough paths sense", 
          "Freidlin-Wentzell theory", 
          "stochastic area", 
          "T. Lyons", 
          "support theorem", 
          "subelliptic operators", 
          "symmetric matrices", 
          "Markov process", 
          "Dirichlet forms", 
          "path topology", 
          "equations", 
          "area process", 
          "large deviations", 
          "support description", 
          "operators", 
          "fundamental work", 
          "theorem", 
          "previous work", 
          "corollary", 
          "topology", 
          "theory", 
          "matrix", 
          "description", 
          "generator", 
          "work", 
          "sense", 
          "deviation", 
          "construction", 
          "thanks", 
          "form", 
          "process", 
          "Lyon", 
          "area"
        ], 
        "name": "On uniformly subelliptic operators and stochastic area", 
        "pagination": "475-523", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1001934547"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/s00440-007-0113-y"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/s00440-007-0113-y", 
          "https://app.dimensions.ai/details/publication/pub.1001934547"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-09-02T15:53", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20220902/entities/gbq_results/article/article_436.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/s00440-007-0113-y"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/s00440-007-0113-y'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/s00440-007-0113-y'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/s00440-007-0113-y'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/s00440-007-0113-y'


     

    This table displays all metadata directly associated to this object as RDF triples.

    131 TRIPLES      21 PREDICATES      67 URIs      52 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/s00440-007-0113-y schema:about anzsrc-for:01
    2 anzsrc-for:0102
    3 schema:author N87980421b2044c4e90230fe0b09073a4
    4 schema:citation sg:pub.10.1007/978-1-4612-5320-4
    5 sg:pub.10.1007/978-3-0348-7026-9_17
    6 sg:pub.10.1007/978-3-0348-8417-4_10
    7 sg:pub.10.1007/bf00251802
    8 sg:pub.10.1007/bfb0073832
    9 sg:pub.10.1007/bfb0084145
    10 sg:pub.10.1007/s00440-005-0487-7
    11 schema:datePublished 2007-11-27
    12 schema:datePublishedReg 2007-11-27
    13 schema:description Let X a be a Markov process with generator ∑i,j∂i( aij∂j· ) where a is a uniformly elliptic symmetric matrix. Thanks to the fundamental works of T. Lyons, stochastic differential equations driven by X a can be solved in the “rough path sense”; that is, pathwise by using a suitable stochastic area process. Our construction of the area, which generalizes previous works of Lyons–Stoica and then Lejay, is based on Dirichlet forms associated to subellitpic operators. This enables us in particular to discuss large deviations and support descriptions in suitable rough path topologies. As typical rough path corollary, Freidlin–Wentzell theory and the Stroock–Varadhan support theorem remain valid for stochastic differential equations driven by X a.
    14 schema:genre article
    15 schema:isAccessibleForFree true
    16 schema:isPartOf N87e4b791fbe54c85af99b265f82b6a54
    17 Neb672c57938048c0b53a6aade7ab8386
    18 sg:journal.1053886
    19 schema:keywords Dirichlet forms
    20 Freidlin-Wentzell theory
    21 Lyon
    22 Markov process
    23 Stroock–Varadhan support theorem
    24 T. Lyons
    25 area
    26 area process
    27 construction
    28 corollary
    29 description
    30 deviation
    31 differential equations
    32 equations
    33 form
    34 fundamental work
    35 generator
    36 large deviations
    37 matrix
    38 operators
    39 path topology
    40 previous work
    41 process
    42 rough paths sense
    43 sense
    44 stochastic area
    45 stochastic differential equations
    46 subelliptic operators
    47 support description
    48 support theorem
    49 symmetric matrices
    50 thanks
    51 theorem
    52 theory
    53 topology
    54 work
    55 schema:name On uniformly subelliptic operators and stochastic area
    56 schema:pagination 475-523
    57 schema:productId N1aceb7abd24f4ec393a5dbf5d9949a2e
    58 Na1b6c1e8c92a4f1aa7868f0687a3448b
    59 schema:sameAs https://app.dimensions.ai/details/publication/pub.1001934547
    60 https://doi.org/10.1007/s00440-007-0113-y
    61 schema:sdDatePublished 2022-09-02T15:53
    62 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    63 schema:sdPublisher N869187e385044d5d855b6abb60331efe
    64 schema:url https://doi.org/10.1007/s00440-007-0113-y
    65 sgo:license sg:explorer/license/
    66 sgo:sdDataset articles
    67 rdf:type schema:ScholarlyArticle
    68 N1aceb7abd24f4ec393a5dbf5d9949a2e schema:name doi
    69 schema:value 10.1007/s00440-007-0113-y
    70 rdf:type schema:PropertyValue
    71 N4ad925ad6e45424b847e6fd05df37365 rdf:first sg:person.014642077545.17
    72 rdf:rest rdf:nil
    73 N869187e385044d5d855b6abb60331efe schema:name Springer Nature - SN SciGraph project
    74 rdf:type schema:Organization
    75 N87980421b2044c4e90230fe0b09073a4 rdf:first sg:person.010663776615.90
    76 rdf:rest N4ad925ad6e45424b847e6fd05df37365
    77 N87e4b791fbe54c85af99b265f82b6a54 schema:issueNumber 3-4
    78 rdf:type schema:PublicationIssue
    79 Na1b6c1e8c92a4f1aa7868f0687a3448b schema:name dimensions_id
    80 schema:value pub.1001934547
    81 rdf:type schema:PropertyValue
    82 Neb672c57938048c0b53a6aade7ab8386 schema:volumeNumber 142
    83 rdf:type schema:PublicationVolume
    84 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    85 schema:name Mathematical Sciences
    86 rdf:type schema:DefinedTerm
    87 anzsrc-for:0102 schema:inDefinedTermSet anzsrc-for:
    88 schema:name Applied Mathematics
    89 rdf:type schema:DefinedTerm
    90 sg:journal.1053886 schema:issn 0178-8051
    91 1432-2064
    92 schema:name Probability Theory and Related Fields
    93 schema:publisher Springer Nature
    94 rdf:type schema:Periodical
    95 sg:person.010663776615.90 schema:affiliation grid-institutes:grid.5335.0
    96 schema:familyName Friz
    97 schema:givenName Peter
    98 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.010663776615.90
    99 rdf:type schema:Person
    100 sg:person.014642077545.17 schema:affiliation grid-institutes:None
    101 schema:familyName Victoir
    102 schema:givenName Nicolas
    103 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.014642077545.17
    104 rdf:type schema:Person
    105 sg:pub.10.1007/978-1-4612-5320-4 schema:sameAs https://app.dimensions.ai/details/publication/pub.1109705063
    106 https://doi.org/10.1007/978-1-4612-5320-4
    107 rdf:type schema:CreativeWork
    108 sg:pub.10.1007/978-3-0348-7026-9_17 schema:sameAs https://app.dimensions.ai/details/publication/pub.1021472546
    109 https://doi.org/10.1007/978-3-0348-7026-9_17
    110 rdf:type schema:CreativeWork
    111 sg:pub.10.1007/978-3-0348-8417-4_10 schema:sameAs https://app.dimensions.ai/details/publication/pub.1003598472
    112 https://doi.org/10.1007/978-3-0348-8417-4_10
    113 rdf:type schema:CreativeWork
    114 sg:pub.10.1007/bf00251802 schema:sameAs https://app.dimensions.ai/details/publication/pub.1021016997
    115 https://doi.org/10.1007/bf00251802
    116 rdf:type schema:CreativeWork
    117 sg:pub.10.1007/bfb0073832 schema:sameAs https://app.dimensions.ai/details/publication/pub.1011328779
    118 https://doi.org/10.1007/bfb0073832
    119 rdf:type schema:CreativeWork
    120 sg:pub.10.1007/bfb0084145 schema:sameAs https://app.dimensions.ai/details/publication/pub.1024290631
    121 https://doi.org/10.1007/bfb0084145
    122 rdf:type schema:CreativeWork
    123 sg:pub.10.1007/s00440-005-0487-7 schema:sameAs https://app.dimensions.ai/details/publication/pub.1006769344
    124 https://doi.org/10.1007/s00440-005-0487-7
    125 rdf:type schema:CreativeWork
    126 grid-institutes:None schema:alternateName Hong Kong, Hong Kong
    127 schema:name Hong Kong, Hong Kong
    128 rdf:type schema:Organization
    129 grid-institutes:grid.5335.0 schema:alternateName Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, CB3 0WB, Cambridge, UK
    130 schema:name Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, CB3 0WB, Cambridge, UK
    131 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...