Document Zbl 08011066

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 08011066 - zbMATH Open</title> <meta name="viewport" content="width=device-width, minimum-scale=0.1, maximum-scale=5.0"> <meta name="robots" content="noarchive, noindex"> <meta name="referrer" content="origin-when-cross-origin"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap/v3.3.7/css/bootstrap-theme.min.css" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.css" rel="stylesheet" media="screen,print"> <link rel="stylesheet" href="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/css/bootstrap-select.min.css"> <link href="/static/css/smoothness/jquery-ui-1.10.1.custom.min.css" rel="stylesheet" media="screen"> <link href="/static/styles.css?v=20241024" rel="stylesheet" media="screen,print"> <link href="https://static.zbmath.org/zbMathJax/v0.1.38/zbmathjax.css" rel="stylesheet" media="screen,print"> <link rel="shortcut icon" href="/static/zbmath.ico"> <script type="application/ld+json"> { "@context": "http://schema.org", "@type": "Organization", "url": "https://zbmath.org/", "logo": "https://zbmath.org/static/zbMATH.png" } </script> </head> <body> <div id="line"></div> <span id="clear" style="cursor: pointer;">×</span> <div id="page"> <div id="head"> <nav id="menu" class="navbar navbar-default"> <div class="container-fluid"> <div class="navbar-header"> <button type="button" class="navbar-toggle collapsed" data-toggle="collapse" data-target="#zbnav" aria-expanded="false"> <span class="sr-only">Toggle navigation</span> <span class="icon-bar"></span> <span class="icon-bar"></span> <span class="icon-bar"></span> </button> <a class="navbar-brand" href="#"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open logo"> </a> </div> <div id="zbnav" class="collapse navbar-collapse"> <ul class="nav navbar-nav pages"> <li class="about"> <a href="/about/">About</a> </li> <li class="frequently-asked-questions"> <a href="/frequently-asked-questions/">FAQ</a> </li> <li class="general-help"> <a href="/general-help/">General Help</a> </li> <li class="reviewer-service"> <a href="https://zbmath.org/reviewer-service/" target="_self" >Reviewer Service</a> </li> <li> <a href="/tools-and-resources/">Tools & Resources</a> </li> <li class="contact"> <a href="/contact/">Contact</a> </li> </ul> <ul class="nav navbar-nav navbar-right prefs"> <li class="preferences dropdown"> <a data-toggle="dropdown" href="#">Preferences <i class="caret"></i></a> <ul class="dropdown-menu preferences"> <li> <form id="preferences" class="navbar-form" method="post" action="/preferences/" onsubmit="return confirm('This website uses cookies for the purposes of storing preference information on your device. Do you agree to this?\n\nPlease refer to our Privacy Policy to learn more about our use of cookies.')" > <input type="hidden" name="path" value="/8011066?"> <span class=""> <label class="title">Search Form</label> <div class="form-group"> <input id="search-multi-line" type="radio" name="search" value="multi-line" checked> <label for="search-multi-line" class="radio">Multi-Line Search (default)</label> </div> <div class="form-group"> <input id="search-one-line" type="radio" name="search" value="one-line"> <label for="search-one-line" class="radio">One-Line Search</label> </div> </span> <span class="count"> <label class="title">Hits per Page</label> <div class="form-group"> <input id="count-10" type="radio" name="count" value="10"> <label for="count-10" class="radio">10</label> </div> <div class="form-group"> <input id="count-20" type="radio" name="count" value="20"> <label for="count-20" class="radio">20</label> </div> <div class="form-group"> <input id="count-50" type="radio" name="count" value="50"> <label for="count-50" class="radio">50</label> </div> <div class="form-group"> <input id="count-100" type="radio" name="count" value="100" checked> <label for="count-100" class="radio">100 (default)</label> </div> <div class="form-group"> <input id="count-200" type="radio" name="count" value="200"> <label for="count-200" class="radio">200</label> </div> </span> <span class="format"> <label class="title">Display Format</label> <div class="form-group"> <input id="format-mathjax" type="radio" name="format" value="mathjax" checked> <label for="format-mathjax" class="radio">MathJax (default)</label> </div> <div class="form-group"> <input id="format-amstex" type="radio" name="format" value="latex"> <label for="format-amstex" class="radio">LaTeX</label> </div> </span> <span class="ranking"> <label class="title">Documents Sorting</label> <div class="form-group"> <input id="documents-ranking-default" type="radio" name="documents_ranking" value="date" checked> <label for="documents-ranking-default" class="radio">Newest first (default)</label> </div> <div class="form-group"> <input id="documents-ranking-references" type="radio" name="documents_ranking" value="references"> <label for="documents-ranking-references" class="radio">Citations</label> </div> <div class="form-group"> <input id="documents-ranking-relevance" type="radio" name="documents_ranking" value="relevance"> <label for="documents-ranking-relevance" class="radio">Relevance</label> </div> </span> <span class="ranking"> <label class="title">Authors Sorting</label> <div class="form-group"> <input id="authors-ranking-default" type="radio" name="authors_ranking" value="alpha" checked> <label for="authors-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="authors-ranking-references" type="radio" name="authors_ranking" value="references"> <label for="authors-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Serials Sorting</label> <div class="form-group"> <input id="serials-ranking-default" type="radio" name="serials_ranking" value="alpha" checked> <label for="serials-ranking-default" class="radio">Alphabetically (default)</label> </div> <div class="form-group"> <input id="serials-ranking-references" type="radio" name="serials_ranking" value="references"> <label for="serials-ranking-references" class="radio">Citations</label> </div> </span> <span class="ranking"> <label class="title">Software Sorting</label> <div class="form-group"> <input id="software-ranking-default" type="radio" name="software_ranking" value="references" checked> <label for="software-ranking-default" class="radio">Citations (default)</label> </div> <div class="form-group"> <input id="software-ranking-alpha" type="radio" name="software_ranking" value="alpha"> <label for="software-ranking-alpha" class="radio">Alphabetically</label> </div> </span> <button type="submit" class="btn btn-default">OK</button> <div class="clearfix"> </form> </li> </ul> </li> </ul> </div> </div> </nav> <div id="tabs"> <h1 class="logo"> <a class="logo" href="/"> <img class="logo" src="/static/zbmath.gif" alt="zbMATH Open — the first resource for mathematics" > </a> </h1> <nav> <ul class="nav nav-tabs"> <li class="tab-documents active"> <a href="/">Documents</a> </li> <li class="tab-authors"> <a href="/authors/">Authors</a> </li> <li class="tab-serials"> <a href="/serials/">Serials</a> </li> <li class="tab-software"> <a href="/software/">Software</a> </li> <li class="tab-classification"> <a href="/classification/">Classification</a> </li> <li class="tab-formulae"> <a href="/formulae/">Formulæ</a> </li> </ul> </nav> <div class="clearfix"></div> </div> <div class="content-fixed"> <div class="content-formular"> <div style="display: none;"> <div class="row ml-0"id="multi-line-new-line" style="display: none;"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-0-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-0-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input"> <span class="multi-line-value" id="multi-line-type-range"><span style="padding-left: 5px;">from</span> <input name="ml-0-v1" class="form-control multi-line-input" type="text" value="" aria-label="value"> until <input name="ml-0-v2" class="form-control multi-line-input" type="text" value="" aria-label="value"></span> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-la" placeholder="use name or ISO code"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-rv" placeholder="enter name or zbMATH reviewer number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-an" placeholder="Zbl, JFM or ERAM number"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-en" placeholder="e.g. DOI, ISBN, arXiv ID"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-sw" placeholder="use * to find all documents using software"> <input name="ml-0-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value" id="multi-line-type-input-br" placeholder="find documents about the life or work of a person"> <span class="multi-line-value" id="multi-line-type-multiselect-db"> <select class="multi-line-selectpicker" data-width="100%" multiple> <option value="zbl">Zbl</option> <option value="arxiv">arXiv</option> <option value="jfm">JFM</option> <option value="eram">ERAM</option> </select> <input type="hidden" class="multi-line-input" name="ml-0-v" value=""> </span> </div> <form name="documents" method="GET" action="/" autocomplete="off"> <div class="documents multi-line" style="display: none;"> <div class="forms"> <ul class="nav forms"> <li class="one-line"> <span tabindex="0" class="glyphicon glyphicon-question-sign" title="One-Line Search allows for free logical combinations of search fields" aria-label="One-Line Search allows for free logical combinations of search fields" data-placement="bottom"></span> <a style="display: inline-block;" href="#">One-Line Search <span class="glyphicon glyphicon-search"></span></a> </li> </ul> </div> <div class="clearfix"></div> <div class="container-fluid"> <input type="hidden" id="multi-line-ml" name="ml" value="3"> <div id="multi-line-row-wrapper"> <div class="row ml-1"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-1-f" aria-label="field"> <option data-type="input" value="any" selected>Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-1-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-1-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-2"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-2-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au" selected>Authors</option> <option data-type="input" value="ti">Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-2-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><select class="form-control multi-line-operators multi-line-selectpicker" name="ml-2-op" aria-label="operator"> <option value="and" selected>AND</option> <option value="andnot">AND NOT</option> <option value="or">OR</option> </select></div> </div> <div class="row ml-3"> <div class="col-xs-12 form-inline multi-line"> <select class="form-control multi-line-field multi-line-selectpicker" name="ml-3-f" aria-label="field"> <option data-type="input" value="any">Anywhere</option> <option data-type="input" value="au">Authors</option> <option data-type="input" value="ti" selected>Title</option> <option data-type="input" value="py">Year</option> <option data-type="range" value="pyr">Year Range</option> <option data-type="input" value="cc">MSC</option> <option data-type="input" value="cc1">MSC Primary</option> <option data-type="input" value="so">Source / Journal</option> <option data-type="input" value="pu">Publisher</option> <option data-type="input" value="la">Language</option> <option data-type="input" value="ab">Summary / Review</option> <option data-type="input" value="rv">Reviewer</option> <option data-type="input" value="an">zbMATH ID</option> <option data-type="input" value="en">External ID</option> <option data-type="input" value="ut">Keywords</option> <option data-type="input" value="sw">Software</option> <option data-type="input" value="br">Biographic Ref</option> <option data-type="input" value="rft">Reference Text</option> <option data-type="multiselect-db" value="db">Database</option> <option data-divider="true"></option> <option data-function="remove-line" data-content='<span class="glyphicon glyphicon-minus" aria-hidden="true"></span> remove line' value="any">remove line</option> </select><input name="ml-3-v" class="form-control multi-line-value multi-line-input" type="text" value="" aria-label="value"><div id="multi-line-plus"> <a href="#"><span class="glyphicon glyphicon-plus" aria-hidden="true"></span> add line</a> </div></div> </div> </div> <div class="row"> <div class="col-xs-12 form-inline"> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-test" style="visibility: hidden; position: fixed;"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label title="search for Articles in Journals"> <input type="checkbox" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label title="search for arXiv Preprints"> <input type="checkbox" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> <div class="form-group field checkboxes-wrapper" id="checkboxes-wrapper-real"> <label>Document Type:</label> <div class="checkboxes"> <div class="slider"> <label for="dt-j" title="search for Articles in Journals"> <input type="checkbox" id="dt-j" name="dt" class="form-control" value="j" checked> <span tabindex="0"><small></small></span> Journal Articles </label> </div> <div class="slider"> <label for="dt-a" title="search for Articles in Conference Proceedings and Collected Volumes"> <input type="checkbox" id="dt-a" name="dt" class="form-control" value="a" checked> <span tabindex="0"><small></small></span> Collection Articles </label> </div> <div class="slider"> <label for="dt-b" title="search for Monographs, Proceedings, Dissertations etc."> <input type="checkbox" id="dt-b" name="dt" class="form-control" value="b" checked> <span tabindex="0"><small></small></span> Books </label> </div> <div class="slider"> <label for="dt-p" title="search for arXiv Preprints"> <input type="checkbox" id="dt-p" name="dt" class="form-control" value="p" checked> <span tabindex="0"><small></small></span> arXiv Preprints </label> </div> </div> </div> </div> </div> <div class="row"> <div class="col-xs-12 buttons"> <a tabindex="0" class="btn btn-default clear-all">Reset all <span class="glyphicon glyphicon-remove"></span></a> <div class="submit"> <button class="btn btn-default search" type="submit">Search <span class="glyphicon glyphicon-search"></span></button> </div> </div> </div> </div> </div> </form> <form class="form-inline" name="documents" method="GET" action="/"> <div class="documents one-line" style="display: block;"> <div class="forms"> <ul class="nav forms"> <li class="multi-line"><a href="#">New Multi-Line Search <span class="glyphicon glyphicon-list"></span></a></li> </ul> </div> <div id="search-row" class="input-group box"> <span> <div id="search-field"> <input class="query form-control" type="text" name="q" value="an:8011066" aria-label="Search for documents" placeholder="Search for documents" autocomplete="off"> </div> <div class="search-buttons input-group-btn"> <div class="btn-group"> <button class="btn btn-default search" type="submit"><span class="virtual">Search</span> <span class="glyphicon glyphicon-search" style="top: 2px;"></span></button> </div> </div> </span> <span> <div class="search-buttons input-group-btn"> <div class="btn-group"> <div class="btn-group fields"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Fields <i class="caret"></i></button> <ul id="fields" class="dropdown-menu pull-right"> <li><a href="#"><span class="token item">any:</span><span> </span>anywhere (default)</a></li> <li><a href="#"><span class="token item">ab:</span><span>  </span>review text</a></li> <li><a href="#"><span class="token item">an:</span><span>  </span>zbmath id</a></li> <li><a href="#"><span class="token item">any:</span><span>  </span>anywhere</a></li> <li><a href="#"><span class="token item">au:</span><span>  </span>contributor name</a></li> <li><a href="#"><span class="token item">br:</span><span>  </span>biographic reference name</a></li> <li><a href="#"><span class="token item">cc:</span><span>  </span>msc title</a></li> <li><a href="#"><span class="token item">dt:</span><span>  </span>document type</a></li> <li><a href="#"><span class="token item">doi:</span><span>  </span>doi</a></li> <li><a href="#"><span class="token item">en:</span><span>  </span>external id</a></li> <li><a href="#"><span class="token item">la:</span><span>  </span>language</a></li> <li><a href="#"><span class="token item">pu:</span><span>  </span>publisher</a></li> <li><a href="#"><span class="token item">py:</span><span>  </span>year</a></li> <li><a href="#"><span class="token item">rv:</span><span>  </span>reviewer name</a></li> <li><a href="#"><span class="token item">so:</span><span>  </span>source</a></li> <li><a href="#"><span class="token item">sw:</span><span>  </span>software name</a></li> <li><a href="#"><span class="token item">ti:</span><span>  </span>title</a></li> <li><a href="#"><span class="token item">ut:</span><span>  </span>keyword</a></li> </ul> </div> <div class="btn-group operators"> <button class="btn btn-default dropdown-toggle" data-toggle="dropdown">Operators <i class="caret"></i></button> <ul id="operators" class="dropdown-menu pull-right"> <li><a href="#"><span class="token">a <span class="item">&</span> b </span><span> </span>logical and (default)</a></li> <li><a href="#"><span class="token">a <span class="item">|</span> b </span><span> </span>logical or</a></li> <li><a href="#"><span class="token"><span class="item">!</span>ab   </span><span> </span>logical not</a></li> <li><a href="#"><span class="token">abc<span class="item">*</span>  </span><span> </span>right wildcard</a></li> <li><a href="#"><span class="token"><span class="item">"</span>ab c<span class="item">"</span></span><span> </span>phrase</a></li> <li><a href="#"><span class="token"><span class="item">(</span>ab c<span class="item">)</span></span><span> </span>parentheses</a></li> </ul> </div> </div> </div> <div class="special"> <ul class="nav help-button"> <li class="dropdown pull-right"> <a href="#">Help <i class="caret"></i></a> </li> </ul> </div> </span> </div> <div class="help"><h2>Examples</h2> <div id="help-terms" role="table"> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Geometry">Geometry</a></span> <span class="search-explanation" role="cell" role="cell">Search for the term <em>Geometry</em> in <strong>any</strong> field. Queries are <strong>case-independent</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Funct%2A">Funct*</a></span> <span class="search-explanation" role="cell"><strong>Wildcard</strong> queries are specified by <strong><u>*</u></strong> (e.g. <em>functions</em>, <em>functorial</em>, etc.). Otherwise the search is <strong>exact</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Topological+group%22">"Topological group"</a></span> <span class="search-explanation" role="cell"><strong>Phrases</strong> (multi-words) should be set in <u>"</u>straight quotation marks<u>"</u>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=au%3A+Bourbaki+%26+ti%3A+Algebra">au: Bourbaki & ti: Algebra</a></span> <span class="search-explanation" role="cell">Search for <strong><u>au</u>thor</strong> and <strong><u>ti</u>tle</strong>. The <strong>and-operator &</strong> is default and can be omitted.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Chebyshev+%7C+Tschebyscheff">Chebyshev | Tschebyscheff</a></span> <span class="search-explanation" role="cell">The <strong>or-operator |</strong> allows to search for <em>Chebyshev</em> or <em>Tschebyscheff</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=Quasi%2A+map%2A+py%3A+1989">Quasi* map* py: 1989</a></span> <span class="search-explanation" role="cell">The resulting documents have <strong><u>p</u>ublication <u>y</u>ear</strong> <em>1989</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=so%3A+Eur%2A+J%2A+Mat%2A+Soc%2A+cc%3A+14">so: Eur* J* Mat* Soc* cc: 14</a></span> <span class="search-explanation" role="cell">Search for publications in a particular <strong><u>so</u>urce</strong> with a <strong>Mathematics Subject <u>C</u>lassification <u>c</u>ode (<u>cc</u>)</strong> in <em>14</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=%22Partial+diff%2A+eq%2A%22+%21+elliptic">"Partial diff* eq*" ! elliptic</a></span> <span class="search-explanation" role="cell">The <strong>not</strong>-operator <strong>!</strong> eliminates all results containing the word <em>elliptic</em>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=dt%3A+b+%26+au%3A+Hilbert">dt: b & au: Hilbert</a></span> <span class="search-explanation" role="cell">The <strong><u>d</u>ocument <u>t</u>ype</strong> is set to books; alternatively: <u>j</u> for <strong>journal articles</strong>, <u>a</u> for <strong>book articles</strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=py%3A+2000-2015+cc%3A+%2894A+%7C+11T%29">py: 2000-2015 cc: (94A | 11T)</a></span> <span class="search-explanation" role="cell">Number <strong>ranges</strong> are accepted. Terms can be grouped within <strong><u>(</u>parentheses<u>)</u></strong>.</span> </div> <div class="help-item" role="row"> <span class="search-example" role="rowheader"><a href="/?q=la%3A+chinese">la: chinese</a></span> <span class="search-explanation" role="cell">Find documents in a given <strong><u>la</u>nguage</strong>. <a href="http://en.wikipedia.org/wiki/ISO_639-1">ISO 639-1</a> language codes can also be used.</span> </div> </div> <div id="help-fields"> <h2>Fields</h2> <table> <tr> <td class="nowrap padding" role="rowheader"><strong>any</strong></td> <td class="padding">anywhere</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>an</strong></td> <td class="padding">internal document identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>au</strong></td> <td class="padding">author, editor</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ai</strong></td> <td class="padding">internal author identifier</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ti</strong></td> <td class="padding">title</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>la</strong></td> <td class="padding">language</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>so</strong></td> <td class="padding">source</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ab</strong></td> <td class="padding">review, abstract</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>py</strong></td> <td class="padding">publication year</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>rv</strong></td> <td class="padding">reviewer</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>cc</strong></td> <td class="padding">MSC code</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>ut</strong></td> <td class="padding">uncontrolled term</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>dt</strong></td> <td class="padding" colspan="4">document type (<strong>j</strong>: journal article; <strong>b</strong>: book; <strong>a</strong>: book article)</td> </tr> </table> </div> <div id="help-operators"> <h2>Operators</h2> <table> <tr> <td class="nowrap padding" role="rowheader">a <strong>&</strong> b</td> <td class="padding">logic and</td> </tr> <tr> <td class="nowrap padding" role="rowheader">a <strong>|</strong> b</td> <td class="padding">logic or</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>!</strong>ab</td> <td class="padding">logic not</td> </tr> <tr> <td class="nowrap padding" role="rowheader">abc<strong>*</strong></td> <td class="padding">right wildcard</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>"</strong>ab c<strong>"</strong></td> <td class="padding">phrase</td> </tr> <tr> <td class="nowrap padding" role="rowheader"><strong>(</strong>ab c<strong>)</strong></td> <td class="padding">parentheses</td> </tr> </table> </div> <p> See also our <a href="/general-help/">General Help</a>. </p></div> </div> </form> <div class="clearfix"></div> </div> <div class="content-shadow"></div> </div> </div> <div id="body"> <div id="main"> <div class="messages"> </div> <div id="documents"> <div class="content-main"> <div class="content-item"><div class="item"> <article> <div class="author"><a href="/authors/chen.xingyuan" title="Author Profile">Chen, Xingyuan</a>; <a href="/authors/dos-reis.goncalo" title="Author Profile">Dos Reis, Gonçalo</a>; <a href="/authors/stockinger.wolfgang" title="Author Profile">Stockinger, Wolfgang</a></div> <h2 class="title"> <strong>Wellposedness, exponential ergodicity and numerical approximation of fully super-linear McKean-Vlasov SDEs and associated particle systems.</strong> <i>(English)</i> <a class="label nowrap" href="/8011066">Zbl 08011066</a> </h2> <div class="source"> <a href="/serials/2165" title="Journal Profile">Electron. J. Probab.</a> <a href="/?q=in%3A523479" title="Articles in this Issue">30, Paper No. 23, 50 p. (2025)</a>. </div> <div class="abstract">Summary: We study a class of McKean-Vlasov Stochastic Differential Equations (MV-SDEs) with drifts and diffusions having super-linear growth in measure and space – the maps have general polynomial form but also satisfy a certain monotonicity condition. The combination of the drift’s super-linear growth in measure (by way of a convolution) and the super-linear growth in space and measure of the diffusion coefficient requires novel technical elements in order to obtain the main results. We establish wellposedness, propagation of chaos (PoC), and under further assumptions on the model parameters, we show an exponential ergodicity property alongside the existence of an invariant distribution. No differentiability or non-degeneracy conditions are required.<br class="zbmathjax-paragraph">Further, we present a particle system based Euler-type split-step scheme (SSM) for the simulation of this type of MV-SDEs. The scheme attains, in stepsize, the strong error rate $1 / 2$ in the non-path-space root-mean-square error metric and we demonstrate the property of mean-square contraction. Our results are illustrated by numerical examples including: estimation of PoC rates across dimensions, preservation of periodic phase-space, and the observation that taming appears to be not a suitable method unless strong dissipativity is present.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A60H10" title="MSC2020">60H10</a> </td> <td class="space"> Stochastic ordinary differential equations (aspects of stochastic analysis) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A60H35" title="MSC2020">60H35</a> </td> <td class="space"> Computational methods for stochastic equations (aspects of stochastic analysis) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65C30" title="MSC2020">65C30</a> </td> <td class="space"> Numerical solutions to stochastic differential and integral equations </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65C35" title="MSC2020">65C35</a> </td> <td class="space"> Stochastic particle methods </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A65C05" title="MSC2020">65C05</a> </td> <td class="space"> Monte Carlo methods </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AMcKean-Vlasov+stochastic+differential+equations">McKean-Vlasov stochastic differential equations</a>; <a href="/?q=ut%3Aergodicity">ergodicity</a>; <a href="/?q=ut%3Awell-posedness">well-posedness</a>; <a href="/?q=ut%3Ainteracting+particle+systems">interacting particle systems</a>; <a href="/?q=ut%3Asplit-step+methods">split-step methods</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 08011066" data-ciurl="/ci/08011066" data-biburl="/bibtex/08011066.bib" data-amsurl="/amsrefs/08011066.bib" data-xmlurl="/xml/08011066.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/08011066.pdf" title="Zbl 08011066 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1214/25-EJP1278" aria-label="DOI for “Wellposedness, exponential ergodicity and numerical approximation of fully super-linear McKean-Vlasov SDEs and associated particle systems”" title="10.1214/25-EJP1278">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2302.05133"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Markus Ableidinger, Evelyn Buckwar, and Harald Hinterleitner, A stochastic version of the Jansen and Rit neural mass model: Analysis and numerics, J. Math. Neurosci. 7 (2017), no. 1, 1-35. MathSciNet: MR3683994 · <a href="/1395.92027" class="nowrap">Zbl 1395.92027</a></td> </tr><tr> <td>[2]</td> <td class="space">Juan A. Acebrón, L. L. Bonilla, Conrad J. Pérez Vicente, Félix Ritort, and Renato Spigler, The Kuramoto model: A simple paradigm for synchronization phenomena, Rev. Mod. Phys. 77 (2005), 137-185.</td> </tr><tr> <td>[3]</td> <td class="space">Daniel Adams, Gonçalo dos Reis, Romain Ravaille, William Salkeld, and Julian Tugaut, Large Deviations and Exit-times for reflected McKean-Vlasov equations with self-stabilising terms and superlinear drifts, Stochastic Processes and their Applications 146 (2022), 264-310. MathSciNet: MR4374937 · <a href="/1484.60027" class="nowrap">Zbl 1484.60027</a></td> </tr><tr> <td>[4]</td> <td class="space">Shin Mi Ahn and Seung-Yeal Ha, Stochastic flocking dynamics of the Cucker-Smale model with multiplicative white noises, J. Math. Phys. 51 (2010), no. 10, 103301, 17. MathSciNet: MR2761313 · <a href="/1314.92019" class="nowrap">Zbl 1314.92019</a></td> </tr><tr> <td>[5]</td> <td class="space">Jianhai Bao, Christoph Reisinger, Panpan Ren, and Wolfgang Stockinger, First-order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems, Proceedings of the Royal Society A 477 (2021), no. 2245, 20200258. MathSciNet: MR4212406</td> </tr><tr> <td>[6]</td> <td class="space">Jianhai Bao, Christoph Reisinger, Panpan Ren, and Wolfgang Stockinger, Milstein schemes and antithetic multilevel Monte Carlo sampling for delay McKean-Vlasov equations and interacting particle systems, IMA J. Numer. Anal. 44 (2024), no. 4, 2437-2479. MathSciNet: MR4780841 · <a href="/1550.65019" class="nowrap">Zbl 1550.65019</a></td> </tr><tr> <td>[7]</td> <td class="space">Denis Belomestny, Vytautė Pilipauskaitė, and Mark Podolskij, Semiparametric estimation of McKean-Vlasov SDEs, Annales de l’Institut Henri Poincare (B) Probabilites et statistiques 59 (2023), no. 1, 79-96. MathSciNet: MR4533721 · <a href="/07657644" class="nowrap">Zbl 07657644</a></td> </tr><tr> <td>[8]</td> <td class="space">Denis Belomestny and John Schoenmakers, Projected particle methods for solving McKean-Vlasov stochastic differential equations, SIAM Journal on Numerical Analysis 56 (2018), no. 6, 3169-3195. MathSciNet: MR3871063 · <a href="/1406.60086" class="nowrap">Zbl 1406.60086</a></td> </tr><tr> <td>[9]</td> <td class="space">S. Benachour, B. Roynette, D. Talay, and P. Vallois, Nonlinear self-stabilizing processes. i. existence, invariant probability, propagation of chaos, Stochastic Processes and their Applications 75(2) (1998), 173-201. MathSciNet: MR1632193 · <a href="/0932.60063" class="nowrap">Zbl 0932.60063</a></td> </tr><tr> <td>[10]</td> <td class="space">Wolf-Jürgen Beyn, Elena Isaak, and Raphael Kruse, Stochastic C-Stability and B-Consistency of Explicit and Implicit Euler-Type Schemes, Journal of Scientific Computing 67 (2015), no. 3, 955-987. MathSciNet: MR3493491 · <a href="/1362.65011" class="nowrap">Zbl 1362.65011</a></td> </tr><tr> <td>[11]</td> <td class="space">Sani Biswas, Chaman Kumar, Gonçalo dos Reis, Christoph Reisinger, et al., Well-posedness and tamed Euler schemes for McKean-.Vlasov equations driven by Lévy noise, arXiv preprint arXiv:2010.08585, 2020.</td> </tr><tr> <td>[12]</td> <td class="space">Sani Biswas, Chaman Kumar, Neelima, Gonçalo dos Reis, and Christoph Reisinger, An explicit Milstein-type scheme for interacting particle systems and McKean-Vlasov SDEs with common noise and non-differentiable drift coefficients, Ann. Appl. Probab. 34 (2024), no. 2, 2326-2363. MathSciNet: MR4728171 · <a href="/1545.65027" class="nowrap">Zbl 1545.65027</a></td> </tr><tr> <td>[13]</td> <td class="space">François Bolley, José A. Cañizo, and José A. Carrillo, Stochastic mean-field limit: non-Lipschitz forces and swarming, Math. Models Methods Appl. Sci. 21 (2011), no. 11, 2179-2210. MathSciNet: MR2860672 · <a href="/1273.82041" class="nowrap">Zbl 1273.82041</a></td> </tr><tr> <td>[14]</td> <td class="space">Giacomo Borghi, Michael Herty, and Lorenzo Pareschi, Constrained consensus-based optimization, SIAM Journal on Optimization 33 (2023), no. 1, 211-236. MathSciNet: MR4538901 · <a href="/1517.35227" class="nowrap">Zbl 1517.35227</a></td> </tr><tr> <td>[15]</td> <td class="space">Mireille Bossy and Denis Talay, A stochastic particle method for the McKean-Vlasov and the Burgers equation, Math. Comp. 66 (1997), no. 217, 157-192. MathSciNet: MR1370849 · <a href="/0854.60050" class="nowrap">Zbl 0854.60050</a></td> </tr><tr> <td>[16]</td> <td class="space">Evelyn Buckwar, Adeline Samson, Massimiliano Tamborrino, and Irene Tubikanec, A splitting method for SDEs with locally Lipschitz drift: Illustration on the FitzHugh-Nagumo model, Applied Numerical Mathematics 179 (2022), 191-220. MathSciNet: MR4422320 · <a href="/1491.65014" class="nowrap">Zbl 1491.65014</a></td> </tr><tr> <td>[17]</td> <td class="space">Evelyn Buckwar, Massimiliano Tamborrino, and Irene Tubikanec, Spectral density-based and measure-preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs, Stat. Comput. 30 (2020), no. 3, 627-648. MathSciNet: MR4065223 · <a href="/1505.62081" class="nowrap">Zbl 1505.62081</a></td> </tr><tr> <td>[18]</td> <td class="space">René Carmona, Fran ̧cois Delarue, and Daniel Lacker, Mean field games with common noise, Ann. Probab. 44 (2016), no. 6, 3740-3803. MathSciNet: MR3572323 · <a href="/1422.91083" class="nowrap">Zbl 1422.91083</a></td> </tr><tr> <td>[19]</td> <td class="space">Rene Carmona and Francois Delarue, Probabilistic theory of mean field games with applications I, 1st ed., Probability Theory and Stochastic Modelling, vol. 84, Springer International Publishing, 2017. MathSciNet: MR3753660</td> </tr><tr> <td>[20]</td> <td class="space">José A Carrillo, Young-Pil Choi, Claudia Totzeck, and Oliver Tse, An analytical framework for consensus-based global optimization method, Mathematical Models and Methods in Applied Sciences 28 (2018), no. 06, 1037-1066. MathSciNet: MR3804923 · <a href="/1397.35311" class="nowrap">Zbl 1397.35311</a></td> </tr><tr> <td>[21]</td> <td class="space">José A Carrillo, Robert J McCann, and Cédric Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Revista Matematica Iberoamericana 19 (2003), no. 3, 971-1018. MathSciNet: MR2053570 · <a href="/1073.35127" class="nowrap">Zbl 1073.35127</a></td> </tr><tr> <td>[22]</td> <td class="space">Paul-Eric Chaudru de Raynal and Noufel Frikha, Well-posedness for some non-linear sdes and related pde on the wasserstein space, Journal de Mathématiques Pures et Appliquées 159 (2022), 1-167. MathSciNet: MR4377993 · <a href="/1494.60063" class="nowrap">Zbl 1494.60063</a></td> </tr><tr> <td>[23]</td> <td class="space">Xingyuan Chen and Gonçalo dos Reis, Euler simulation of interacting particle systems and McKean-Vlasov SDEs with fully super-linear growth drifts in space and interaction, IMA J. Numer. Anal. 44 (2024), no. 2, 751-796. MathSciNet: MR4727110 · <a href="/07935837" class="nowrap">Zbl 07935837</a></td> </tr><tr> <td>[24]</td> <td class="space">Xingyuan Chen, Gonçalo dos Reis, Wolfgang Stockinger, and Zac Wilde, Improved weak convergence for the long time simulation of mean-field langevin equations, arXiv preprint arXiv:2405.01346, 2024.</td> </tr><tr> <td>[25]</td> <td class="space">Xingyuan Chen and Gonçalo dos Reis, A flexible split-step scheme for solving McKean-Vlasov stochastic differential equations, Appl. Math. Comput. 427 (2022), Paper No. 127180. MathSciNet: MR4413221 · <a href="/1510.65011" class="nowrap">Zbl 1510.65011</a></td> </tr><tr> <td>[26]</td> <td class="space">Fabienne Comte and Valentine Genon-Catalot, Nonparametric adaptive estimation for interacting particle systems, Scand. J. Stat. 50 (2023), no. 4, 1716-1755. MathSciNet: MR4677374 · <a href="/07782059" class="nowrap">Zbl 07782059</a></td> </tr><tr> <td>[27]</td> <td class="space">Paul-Eric Chaudru de Raynal, Manh Hong Duong, Pierre Monmarché, Milica Tomašević, and Julian Tugaut, Reducing exit-times of diffusions with repulsive interactions, ESAIM: Probability and Statistics 27 (2023), 723-748. MathSciNet: MR4621420 · <a href="/1520.60045" class="nowrap">Zbl 1520.60045</a></td> </tr><tr> <td>[28]</td> <td class="space">François Delarue, Daniel Lacker, and Kavita Ramanan, From the master equation to mean field game limit theory: a central limit theorem, Electron. J. Probab. 24 (2019), Paper No. 51, 54. MathSciNet: MR3954791 · <a href="/1508.60032" class="nowrap">Zbl 1508.60032</a></td> </tr><tr> <td>[29]</td> <td class="space">François Delarue and Alvin Tse, Uniform in time weak propagation of chaos on the torus, arXiv preprint arXiv:2104.14973, 2021.</td> </tr><tr> <td>[30]</td> <td class="space">Gonçalo dos Reis, Stefan Engelhardt, and Greig Smith, Simulation of McKean-Vlasov SDEs with super-linear growth, IMA J. Numer. Anal. 42 (2022), no. 1, 874-922. MathSciNet: MR4367675 · <a href="/1481.65020" class="nowrap">Zbl 1481.65020</a></td> </tr><tr> <td>[31]</td> <td class="space">Gonçalo dos Reis, William Salkeld, and Julian Tugaut, Freidlin-Wentzell LDP in path space for McKean-Vlasov equations and the functional iterated logarithm law, Ann. Appl. Probab. 29 (2019), no. 3, 1487-1540. MathSciNet: MR3914550 · <a href="/1466.60056" class="nowrap">Zbl 1466.60056</a></td> </tr><tr> <td>[32]</td> <td class="space">Renjun Duan, Massimo Fornasier, and Giuseppe Toscani, A kinetic flocking model with diffusion, Comm. Math. Phys. 300 (2010), no. 1, 95-145. MathSciNet: MR2725184 · <a href="/1213.35395" class="nowrap">Zbl 1213.35395</a></td> </tr><tr> <td>[33]</td> <td class="space">Xavier Erny, Eva Löcherbach, and Dasha Loukianova, Strong error bounds for the convergence to its mean field limit for systems of interacting neurons in a diffusive scaling, Ann. Appl. Probab. 33 (2023), no. 5, 3563-3586. MathSciNet: MR4663491 · <a href="/1540.92041" class="nowrap">Zbl 1540.92041</a></td> </tr><tr> <td>[34]</td> <td class="space">Nicolas Fournier and Arnaud Guillin, On the rate of convergence in Wasserstein distance of the empirical measure, Probability Theory and Related Fields 162 (2015), no. 3, 707-738. MathSciNet: MR3383341 zbMATH: 1325.60042 · <a href="/1325.60042" class="nowrap">Zbl 1325.60042</a></td> </tr><tr> <td>[35]</td> <td class="space">Valentine Genon-Catalot and Catherine Larédo, Parametric inference for small variance and long time horizon Mckean-Vlasov diffusion models, Electronic Journal of Statistics 15 (2021), no. 2, 5811-5854. MathSciNet: MR4355698 · <a href="/1482.60107" class="nowrap">Zbl 1482.60107</a></td> </tr><tr> <td>[36]</td> <td class="space">Valentine Genon-Catalot and Catherine Larédo, Probabilistic properties and parametric inference of small variance nonlinear self-stabilizing stochastic differential equations, Stochastic Process. Appl. 142 (2021), 513-548. MathSciNet: MR4324348 · <a href="/1476.60140" class="nowrap">Zbl 1476.60140</a></td> </tr><tr> <td>[37]</td> <td class="space">Valentine Genon-Catalot and Catherine Larédo, Inference for ergodic McKean-Vlasov stochastic differential equations with polynomial interactions, Ann. Inst. Henri Poincaré Probab. Stat. 60 (2024), no. 4, 2668-2693. MathSciNet: MR4828854 · <a href="/07967659" class="nowrap">Zbl 07967659</a></td> </tr><tr> <td>[38]</td> <td class="space">Arnaud Guillin, Wei Liu, Liming Wu, and Chaoen Zhang, Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems, Ann. Appl. Probab. 32 (2022), no. 3, 1590-1614. MathSciNet: MR4429996 · <a href="/1503.60150" class="nowrap">Zbl 1503.60150</a></td> </tr><tr> <td>[39]</td> <td class="space">Seung-Yeal Ha, Jiin Jeong, Se Eun Noh, Qinghua Xiao, and Xiongtao Zhang, Emergent dynamics of Cucker-Smale flocking particles in a random environment, J. Differential Equations 262 (2017), no. 3, 2554-2591. MathSciNet: MR3582237 · <a href="/1352.74014" class="nowrap">Zbl 1352.74014</a></td> </tr><tr> <td>[40]</td> <td class="space">Samuel Herrmann, Peter Imkeller, and Dierk Peithmann, Large deviations and a Kramers’ type law for self-stabilizing diffusions, Ann. Appl. Probab. 18 (2008), no. 4, 1379-1423. MathSciNet: MR2434175 · <a href="/1149.60020" class="nowrap">Zbl 1149.60020</a></td> </tr><tr> <td>[41]</td> <td class="space">Samuel Herrmann and Julian Tugaut, Self-stabilizing processes: uniqueness problem for stationary measures and convergence rate in the small-noise limit, ESAIM Probab. Stat. 16 (2012), 277-305. MathSciNet: MR2956576 · <a href="/1302.60112" class="nowrap">Zbl 1302.60112</a></td> </tr><tr> <td>[42]</td> <td class="space">Desmond J Higham, Xuerong Mao, and Andrew M Stuart, Strong convergence of Euler-type methods for nonlinear stochastic differential equations, SIAM Journal on Numerical Analysis 40 (2002), no. 3, 1041-1063. MathSciNet: MR1949404 · <a href="/1026.65003" class="nowrap">Zbl 1026.65003</a></td> </tr><tr> <td>[43]</td> <td class="space">Shan-Shan Hu, Long-time behaviour for distribution dependent SDEs with local Lipschitz coefficients, arXiv preprint arXiv:2103.13101, 2021.</td> </tr><tr> <td>[44]</td> <td class="space">Xing Huang, Panpan Ren, and Feng-Yu Wang, Distribution dependent stochastic differential equations, Front. Math. China 16 (2021), no. 2, 257-301. MathSciNet: MR4254653 · <a href="/1475.60033" class="nowrap">Zbl 1475.60033</a></td> </tr><tr> <td>[45]</td> <td class="space">Martin Hutzenthaler and Arnulf Jentzen, Numerical approximations of stochastic differential equations with non-globally Lipschitz continuous coefficients, Mem. Amer. Math. Soc. 236 (2015), no. 1112, 99. MathSciNet: MR3364862 · <a href="/1330.60084" class="nowrap">Zbl 1330.60084</a></td> </tr><tr> <td>[46]</td> <td class="space">Shi Jin, Lei Li, and Jian-Guo Liu, Random batch methods (RBM) for interacting particle systems, Journal of Computational Physics 400 (2020), 108877. MathSciNet: MR4019084 · <a href="/1453.82065" class="nowrap">Zbl 1453.82065</a></td> </tr><tr> <td>[47]</td> <td class="space">Alexander Kalinin, Thilo Meyer-Brandis, and Frank Proske, Stability, uniqueness and existence of solutions to Mckean-Vlasov Sdes in arbitrary moments, arXiv preprint arXiv:2205.02176, 2022. MathSciNet: MR4805033</td> </tr><tr> <td>[48]</td> <td class="space">Chaman Kumar and Neelima, On explicit Milstein-type scheme for McKean-Vlasov stochastic differential equations with super-linear drift coefficient, Electron. J. Probab. 26 (2021), Paper No. 111, 32. MathSciNet: MR4302574 · <a href="/07424791" class="nowrap">Zbl 07424791</a></td> </tr><tr> <td>[49]</td> <td class="space">Chaman Kumar, Neelima, Christoph Reisinger, and Wolfgang Stockinger, Well-posedness and tamed schemes for McKean-Vlasov equations with common noise, Ann. Appl. Probab. 32 (2022), no. 5, 3283-3330. MathSciNet: MR4497846 · <a href="/1498.60288" class="nowrap">Zbl 1498.60288</a></td> </tr><tr> <td>[50]</td> <td class="space">Daniel Lacker, On a strong form of propagation of chaos for McKean-Vlasov equations, Electron. Commun. Probab. 23 (2018), Paper No. 45, 11. MathSciNet: MR3841406 zbMATH: 1396.65013 · <a href="/1396.65013" class="nowrap">Zbl 1396.65013</a></td> </tr><tr> <td>[51]</td> <td class="space">Yun Li, Xuerong Mao, Qingshuo Song, Fuke Wu, and George Yin, Strong convergence of Euler-Maruyama schemes for McKean-Vlasov stochastic differential equations under local Lipschitz conditions of state variables, IMA J. Numer. Anal. 43 (2023), no. 2, 1001-1035. MathSciNet: MR4568438 · <a href="/1527.65005" class="nowrap">Zbl 1527.65005</a></td> </tr><tr> <td>[52]</td> <td class="space">Zhenxin Liu and Jun Ma, Existence, uniqueness and exponential ergodicity under Lyapunov conditions for McKean-Vlasov SDEs with Markovian switching, Journal of Differential Equations 337 (2022), 138-167. MathSciNet: MR4464543 · <a href="/1497.60113" class="nowrap">Zbl 1497.60113</a></td> </tr><tr> <td>[53]</td> <td class="space">Florent Malrieu, Convergence to equilibrium for granular media equations and their Euler schemes, Ann. Appl. Probab. 13 (2003), no. 2, 540-560. MathSciNet: MR1970276 · <a href="/1031.60085" class="nowrap">Zbl 1031.60085</a></td> </tr><tr> <td>[54]</td> <td class="space">Xuerong Mao, Stochastic differential equations and applications, Horwood, 2008. MathSciNet: MR2380366</td> </tr><tr> <td>[55]</td> <td class="space">Sima Mehri, Michael Scheutzow, Wilhelm Stannat, and Bian Z Zangeneh, Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusions, Annals of Applied Probability 30 (2020), no. 1, 175-207. MathSciNet: MR4068309 · <a href="/1434.60299" class="nowrap">Zbl 1434.60299</a></td> </tr><tr> <td>[56]</td> <td class="space">Sylvie Méléard, Asymptotic behaviour of some interacting particle systems; McKean-Vlasov and Boltzmann models, Probabilistic models for nonlinear partial differential equations, Springer, 1996, pp. 42-95. MathSciNet: MR1431299</td> </tr><tr> <td>[57]</td> <td class="space">Mohamed Amine Mezerdi, On the convergence of Carathéodory numerical scheme for McKean-Vlasov equations, Stoch. Anal. Appl. 39 (2021), no. 5, 804-818. MathSciNet: MR4310347 · <a href="/1474.60152" class="nowrap">Zbl 1474.60152</a></td> </tr><tr> <td>[58]</td> <td class="space">Yuliya Mishura and Alexander Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, Theory Probab. Math. Statist. (2020), no. 103, 59-101. MathSciNet: MR4421344 · <a href="/1482.60079" class="nowrap">Zbl 1482.60079</a></td> </tr><tr> <td>[59]</td> <td class="space">Neelima, Sani Biswas, Chaman Kumar, Gonçalo dos Reis, and Christoph Reisinger, Well-posedness and tamed Euler schemes for McKean-Vlasov equations driven by Lévy noise, arXiv preprint arXiv:2010.08585, 2020.</td> </tr><tr> <td>[60]</td> <td class="space">Gonçalo dos Reis and Zac Wilde, Malliavin differentiability of McKean-Vlasov SDEs with locally lipschitz coefficients, arXiv preprint arXiv:2310.13400, 2023.</td> </tr><tr> <td>[61]</td> <td class="space">Christoph Reisinger and Wolfgang Stockinger, An adaptive Euler-Maruyama scheme for McKean-Vlasov SDEs with super-linear growth and application to the mean-field Fitzhugh-Nagumo model, Journal of Computational and Applied Mathematics 400 (2022), 113725. MathSciNet: MR4293705 · <a href="/1469.60195" class="nowrap">Zbl 1469.60195</a></td> </tr><tr> <td>[62]</td> <td class="space">Panpan Ren and Feng-Yu Wang, Exponential convergence in entropy and Wasserstein for McKean-Vlasov SDEs, Nonlinear Analysis 206 (2021), 112259. MathSciNet: MR4206077 · <a href="/1458.60014" class="nowrap">Zbl 1458.60014</a></td> </tr><tr> <td>[63]</td> <td class="space">Michael Röckner and Xicheng Zhang, Well-posedness of distribution dependent SDEs with singular drifts, Bernoulli 27 (2021), no. 2, 1131-1158. MathSciNet: MR4255229 · <a href="/1480.60171" class="nowrap">Zbl 1480.60171</a></td> </tr><tr> <td>[64]</td> <td class="space">Sotirios Sabanis and Ying Zhang, On explicit order 1.5 approximations with varying coefficients: The case of super-linear diffusion coefficients, Journal of Complexity 50 (2019), 84-115. MathSciNet: MR3907365 · <a href="/1403.60048" class="nowrap">Zbl 1403.60048</a></td> </tr><tr> <td>[65]</td> <td class="space">Alain-Sol Sznitman, Topics in propagation of chaos, Ecole d’Eté de Probabilités de Saint-Flour XIX — 1989 (1991), 165-251. MathSciNet: MR1108185 · <a href="/0732.60114" class="nowrap">Zbl 0732.60114</a></td> </tr><tr> <td>[66]</td> <td class="space">Lukasz Szpruch and Alvin Tse, Antithetic multilevel sampling method for nonlinear functionals of measure, The Annals of Applied Probability 31 (2021), no. 3, 1100-1139. MathSciNet: MR4278780 · <a href="/1479.60005" class="nowrap">Zbl 1479.60005</a></td> </tr><tr> <td>[67]</td> <td class="space">Julian Tugaut, Convergence to the equilibria for self-stabilizing processes in double-well landscape, Ann. Probab. 41 (2013), no. 3A, 1427-1460. MathSciNet: MR3098681 · <a href="/1292.60060" class="nowrap">Zbl 1292.60060</a></td> </tr><tr> <td>[68]</td> <td class="space">Julian Tugaut, Self-stabilizing processes in multi-wells landscape in $\mathbb{R}^d -convergence $, Stochastic Processes and Their Applications 123 (2013), no. 5, 1780-1801. MathSciNet: MR3027901 · <a href="/1263.60053" class="nowrap">Zbl 1263.60053</a></td> </tr><tr> <td>[69]</td> <td class="space">Cédric Villani, Optimal transport, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338, Springer-Verlag, Berlin, 2009, Old and new. MathSciNet: MR2459454 · <a href="/1156.53003" class="nowrap">Zbl 1156.53003</a></td> </tr><tr> <td>[70]</td> <td class="space">Feng-Yu Wang, Distribution dependent SDEs for Landau type equations, Stochastic Process. Appl. 128 (2018), no. 2, 595-621. MathSciNet: MR3739509 · <a href="/1380.60077" class="nowrap">Zbl 1380.60077</a></td> </tr><tr> <td>[71]</td> <td class="space">Feng-Yu Wang, Distribution dependent reflecting stochastic differential equations, Sci. China Math. 66 (2023), no. 11, 2411-2456. MathSciNet: MR4658659 · <a href="/1523.60105" class="nowrap">Zbl 1523.60105</a></td> </tr><tr> <td>[72]</td> <td class="space">Feng-Yu Wang, Exponential ergodicity for singular reflecting McKean-Vlasov SDEs, Stochastic Process. Appl. 160 (2023), 265-293. MathSciNet: MR4567526 · <a href="/1511.60091" class="nowrap">Zbl 1511.60091</a></td> </tr><tr> <td>[73]</td> <td class="space">Wei Wei, Ting Gao, Xiaoli Chen, and Jinqiao Duan, An optimal control method to compute the most likely transition path for stochastic dynamical systems with jumps, Chaos 32 (2022), no. 5, Paper No. 051102, 10. MathSciNet: MR4422807 · <a href="/1540.60157" class="nowrap">Zbl 1540.60157</a></td> </tr><tr> <td>[74]</td> <td class="space">Eberhard Zeidler, Nonlinear functional analysis and its applications. II/A, Springer-Verlag, New York, 1990, Linear monotone operators, Translated from the German by the author and Leo F. Boron. MathSciNet: MR1033497 · <a href="/0684.47028" class="nowrap">Zbl 0684.47028</a></td> </tr><tr> <td>[75]</td> <td class="space">Shao-Qin Zhang, Existence and non-uniqueness of stationary distributions for distribution dependent SDEs, Electron. J. Probab. 28 (2023). MathSciNet: MR4613856 · <a href="/1543.60076" class="nowrap">Zbl 1543.60076</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

CINXE.COM

Document Zbl 08011066 - zbMATH Open