Document Zbl 1498.60288

<!doctype html> <html lang="en"> <head> <meta charset="utf-8"> <title>Document Zbl 1498.60288 - 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="/1498.60288?"> <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:1498.60288" 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/kumar.chaman" title="Author Profile">Kumar, Chaman</a>; <a href="/authors/?q=ai%3Aneelima.c-a%7Cneelima.d" title="Author Profile">Neelima</a>; <a href="/authors/reisinger.christoph" title="Author Profile">Reisinger, Christoph</a>; <a href="/authors/stockinger.wolfgang" title="Author Profile">Stockinger, Wolfgang</a></div> <h2 class="title"> <strong>Well-posedness and tamed schemes for McKean-Vlasov equations with common noise.</strong> <i>(English)</i> <a class="label nowrap" href="/1498.60288">Zbl 1498.60288</a> </h2> <div class="source"> <a href="/serials/1375" title="Journal Profile">Ann. Appl. Probab.</a> <a href="/?q=in%3A489433" title="Articles in this Issue">32, No. 5, 3283-3330 (2022)</a>. </div> <div class="abstract">Summary: In this paper, we first establish well-posedness of McKean-Vlasov stochastic differential equations (McKean-Vlasov SDEs) with common noise, possibly with coefficients of super-linear growth in the state variable. Second, we present stable time-stepping schemes for this class of McKean-Vlasov SDEs. Specifically, we propose an explicit tamed Euler and tamed Milstein scheme for an interacting particle system associated with the McKean-Vlasov equation. We prove stability and strong convergence of order $1/2$ and 1, respectively. To obtain our main results, we employ techniques from calculus on the Wasserstein space. The proof for the strong convergence of the tamed Milstein scheme only requires the coefficients to be once continuously differentiable in the state and measure component. To demonstrate our theoretical findings, we present several numerical examples, including mean-field versions of the stochastic $3/2$ volatility model and the stochastic double well dynamics with multiplicative noise.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7609670">Cited in <strong>10</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><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%3A65C05" title="MSC2020">65C05</a> </td> <td class="space"> Monte Carlo methods </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></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AMcKean-Vlasov+SDEs">McKean-Vlasov SDEs</a>; <a href="/?q=ut%3Aparticle+system">particle system</a>; <a href="/?q=ut%3Asuper-linear+coefficients">super-linear coefficients</a>; <a href="/?q=ut%3Atamed+numerical+schemes">tamed numerical schemes</a>; <a href="/?q=ut%3Acommon+noise">common noise</a>; <a href="/?q=ut%3Amean-field+stochastic+3%2F2+model">mean-field stochastic 3/2 model</a>; <a href="/?q=ut%3Amean-field+stochastic+double+well+dynamics+with+multiplicative+noise">mean-field stochastic double well dynamics with multiplicative noise</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 1498.60288" data-ciurl="/ci/07609670" data-biburl="/bibtex/07609670.bib" data-amsurl="/amsrefs/07609670.bib" data-xmlurl="/xml/07609670.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07609670.pdf" title="Zbl 1498.60288 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/10.1214/21-AAP1760" aria-label="DOI for “Well-posedness and tamed schemes for McKean-Vlasov equations with common noise”" title="10.1214/21-AAP1760">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2006.00463"title="Note: arXiv document may differ from published version">arXiv</a> <a class="btn btn-default btn-xs" type="button" href="https://ora.ox.ac.uk/objects/uuid:a6ea8c7c-8984-4ff8-a840-da35fcaeeb9c" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">ADAMS, D., DOS REIS, G., RAVAILLE, R., SALKELD, W. and TUGAUT, J. (2020). Large deviations and exit-times for reflected McKean-Vlasov equations with self-stabilizing terms and superlinear drifts. Preprint. Available at arXiv:2005.10057.</td> </tr><tr> <td>[2]</td> <td class="space">AMBROSIO, L., GIGLI, N. and SAVARÉ, G. (2008). <span class="zbmathjax-textit">Gradient Flows in Metric Spaces and in the Space of Probability Measures</span>, 2nd ed. <span class="zbmathjax-textit">Lectures in Mathematics ETH Zürich</span>. Birkhäuser, Basel. · <a href="/1145.35001" class="nowrap">Zbl 1145.35001</a></td> </tr><tr> <td>[3]</td> <td class="space">BAO, J. and HUANG, H. (2021). Approximations of McKean-Vlasov SDEs with irregular coefficients. <span class="zbmathjax-textit">J. Theor. Probab.</span> 159. · <a href="/1493.65018" class="nowrap">Zbl 1493.65018</a> · <a href="https://doi.org/10.1007/s10959-021-01082-9" class="nowrap">doi:10.1007/s10959-021-01082-9</a></td> </tr><tr> <td>[4]</td> <td class="space">BAO, J., REISINGER, C., REN, P. and STOCKINGER, W. (2020). Milstein schemes for delay McKean-Vlasov equations and interacting particle systems. Preprint. Available at arXiv:2005.01165.</td> </tr><tr> <td>[5]</td> <td class="space">BAO, J., REISINGER, C., REN, P. and STOCKINGER, W. (2021). First-order convergence of Milstein schemes for McKean-Vlasov equations and interacting particle systems. <span class="zbmathjax-textit">Proc. R. Soc. A</span> 477 Paper No. 20200258, 27 pp.</td> </tr><tr> <td>[6]</td> <td class="space">BAUER, M., MEYER-BRANDIS, T. and PROSKE, F. (2018). Strong solutions of mean-field stochastic differential equations with irregular drift. <span class="zbmathjax-textit">Electron. J. Probab.</span> 23 Paper No. 132, 35 pp. · <a href="/1406.60083" class="nowrap">Zbl 1406.60083</a> · <a href="https://doi.org/10.1214/18-EJP259" class="nowrap">doi:10.1214/18-EJP259</a></td> </tr><tr> <td>[7]</td> <td class="space">BEYN, W.-J., ISAAK, E. and KRUSE, R. (2017). Stochastic C-stability and B-consistency of explicit and implicit Milstein-type schemes. <span class="zbmathjax-textit">J. Sci. Comput.</span> 70 1042-1077. · <a href="/1397.65013" class="nowrap">Zbl 1397.65013</a> · <a href="https://doi.org/10.1007/s10915-016-0290-x" class="nowrap">doi:10.1007/s10915-016-0290-x</a></td> </tr><tr> <td>[8]</td> <td class="space">Bossy, M. and Talay, D. (1997). A stochastic particle method for the McKean-Vlasov and the Burgers equation. <span class="zbmathjax-textit">Math. Comp.</span> 66 157-192. · <a href="/0854.60050" class="nowrap">Zbl 0854.60050</a> · <a href="https://doi.org/10.1090/S0025-5718-97-00776-X" class="nowrap">doi:10.1090/S0025-5718-97-00776-X</a></td> </tr><tr> <td>[9]</td> <td class="space">BUJOK, K., HAMBLY, B. M. and REISINGER, C. (2015). Multilevel simulation of functionals of Bernoulli random variables with application to basket credit derivatives. <span class="zbmathjax-textit">Methodol. Comput. Appl. Probab.</span> 17 579-604. · <a href="/1327.65003" class="nowrap">Zbl 1327.65003</a> · <a href="https://doi.org/10.1007/s11009-013-9380-5" class="nowrap">doi:10.1007/s11009-013-9380-5</a></td> </tr><tr> <td>[10]</td> <td class="space">CARDALIAGUET, P. (2013). Notes on mean-field games, notes from P. L. Lions lectures at Collège de France. Available at https://www.ceremade.dauphine.fr/cardalia/MFG100629.pdf.</td> </tr><tr> <td>[11]</td> <td class="space">CARMONA, R. and DELARUE, F. (2018). <span class="zbmathjax-textit">Probabilistic Theory of Mean Field Games with Applications I</span>: <span class="zbmathjax-textit">Mean Field FBSDEs, Control, and Games</span>. <span class="zbmathjax-textit">Probability Theory and Stochastic Modelling</span> 83. Springer, Cham.</td> </tr><tr> <td>[12]</td> <td class="space">CARMONA, R. and DELARUE, F. (2018). <span class="zbmathjax-textit">Probabilistic Theory of Mean Field Games with Applications II</span>: <span class="zbmathjax-textit">Mean Field Games with Common Noise and Master Equations</span>. <span class="zbmathjax-textit">Probability Theory and Stochastic Modelling</span> 84. Springer, Cham. · <a href="/1422.91015" class="nowrap">Zbl 1422.91015</a></td> </tr><tr> <td>[13]</td> <td class="space">CHEN, X. and DOS REIS, G. (2021). A flexible split-step scheme for MV-SDEs. Preprint. Available at arXiv:2105.09688.</td> </tr><tr> <td>[14]</td> <td class="space">DING, X. and QIAO, H. (2021). Euler-Maruyama approximations for stochastic McKean-Vlasov equations with non-Lipschitz coefficients. <span class="zbmathjax-textit">J. Theoret. Probab.</span> 34 1408-1425. · <a href="/1476.60097" class="nowrap">Zbl 1476.60097</a> · <a href="https://doi.org/10.1007/s10959-020-01041-w" class="nowrap">doi:10.1007/s10959-020-01041-w</a></td> </tr><tr> <td>[15]</td> <td class="space">DOS REIS, G., ENGELHARDT, S. and SMITH, G. (2022). Simulation of McKean-Vlasov SDEs with super-linear growth. <span class="zbmathjax-textit">IMA J. Numer. Anal.</span> 42 874-922. · <a href="/1481.65020" class="nowrap">Zbl 1481.65020</a> · <a href="https://doi.org/10.1093/imanum/draa099" class="nowrap">doi:10.1093/imanum/draa099</a></td> </tr><tr> <td>[16]</td> <td class="space">DOS REIS, G., SALKELD, W. and TUGAUT, J. (2019). Freidlin-Wentzell LDP in path space for McKean-Vlasov equations and the functional iterated logarithm law. <span class="zbmathjax-textit">Ann. Appl. Probab.</span> 29 1487-1540. · <a href="/1466.60056" class="nowrap">Zbl 1466.60056</a> · <a href="https://doi.org/10.1214/18-AAP1416" class="nowrap">doi:10.1214/18-AAP1416</a></td> </tr><tr> <td>[17]</td> <td class="space">Giles, M. B. (2015). Multilevel Monte Carlo methods. <span class="zbmathjax-textit">Acta Numer.</span> 24 259-328. · <a href="/1316.65010" class="nowrap">Zbl 1316.65010</a> · <a href="https://doi.org/10.1017/S096249291500001X" class="nowrap">doi:10.1017/S096249291500001X</a></td> </tr><tr> <td>[18]</td> <td class="space">GOARD, J. and MAZUR, M. (2013). Stochastic volatility models and the pricing of VIX options. <span class="zbmathjax-textit">Math. Finance</span> 23 439-458. · <a href="/1281.91131" class="nowrap">Zbl 1281.91131</a> · <a href="https://doi.org/10.1111/j.1467-9965.2011.00506.x" class="nowrap">doi:10.1111/j.1467-9965.2011.00506.x</a></td> </tr><tr> <td>[19]</td> <td class="space">GYÖNGY, I. and KRYLOV, N. (2003). On the rate of convergence of splitting-up approximations for SPDEs. In <span class="zbmathjax-textit">Stochastic Inequalities and Applications</span>. <span class="zbmathjax-textit">Progress in Probability</span> 56 301-321. Birkhäuser, Basel. · <a href="/1038.60057" class="nowrap">Zbl 1038.60057</a></td> </tr><tr> <td>[20]</td> <td class="space">GYÖNGY, I. and KRYLOV, N. V. (1980/81). On stochastic equations with respect to semimartingales. I. <span class="zbmathjax-textit">Stochastics</span> 4 1-21. · <a href="/0439.60061" class="nowrap">Zbl 0439.60061</a> · <a href="https://doi.org/10.1080/03610918008833154" class="nowrap">doi:10.1080/03610918008833154</a></td> </tr><tr> <td>[21]</td> <td class="space">HAMMERSLEY, W. R. P., ŠIŠKA, D. and SZPRUCH, Ł. (2021). Weak existence and uniqueness for McKean-Vlasov SDEs with common noise. <span class="zbmathjax-textit">Ann. Probab.</span> 49 527-555. · <a href="/1522.60050" class="nowrap">Zbl 1522.60050</a> · <a href="https://doi.org/10.1214/20-aop1454" class="nowrap">doi:10.1214/20-aop1454</a></td> </tr><tr> <td>[22]</td> <td class="space">HAMMERSLEY, W. R. P., ŠIŠKA, D. and SZPRUCH, Ł. (2021). McKean-Vlasov SDEs under measure dependent Lyapunov conditions. <span class="zbmathjax-textit">Ann. Inst. Henri Poincaré Probab. Stat.</span> 57 1032-1057. · <a href="/1489.60099" class="nowrap">Zbl 1489.60099</a> · <a href="https://doi.org/10.1214/20-aihp1106" class="nowrap">doi:10.1214/20-aihp1106</a></td> </tr><tr> <td>[23]</td> <td class="space">HUTZENTHALER, M., JENTZEN, A. and KLOEDEN, P. E. (2012). Strong convergence of an explicit numerical method for SDEs with nonglobally Lipschitz continuous coefficients. <span class="zbmathjax-textit">Ann. Appl. Probab.</span> 22 1611-1641. · <a href="/1256.65003" class="nowrap">Zbl 1256.65003</a> · <a href="https://doi.org/10.1214/11-AAP803" class="nowrap">doi:10.1214/11-AAP803</a></td> </tr><tr> <td>[24]</td> <td class="space">KUMAR, C. and NEELIMA (2021). On explicit Milstein-type scheme for McKean-Vlasov stochastic differential equations with super-linear drift coefficient. <span class="zbmathjax-textit">Electron. J. Probab.</span> 26 Paper No. 111, 32 pp. · <a href="/07424791" class="nowrap">Zbl 07424791</a> · <a href="https://doi.org/10.1214/21-ejp676" class="nowrap">doi:10.1214/21-ejp676</a></td> </tr><tr> <td>[25]</td> <td class="space">KUMAR, C. and SABANIS, S. (2017). On explicit approximations for Lévy driven SDEs with super-linear diffusion coefficients. <span class="zbmathjax-textit">Electron. J. Probab.</span> 22 Paper No. 73, 19 pp. · <a href="/1379.60076" class="nowrap">Zbl 1379.60076</a> · <a href="https://doi.org/10.1214/17-EJP89" class="nowrap">doi:10.1214/17-EJP89</a></td> </tr><tr> <td>[26]</td> <td class="space">KUMAR, C. and SABANIS, S. (2019). On Milstein approximations with varying coefficients: The case of super-linear diffusion coefficients. <span class="zbmathjax-textit">BIT</span> 59 929-968. · <a href="/1505.65011" class="nowrap">Zbl 1505.65011</a> · <a href="https://doi.org/10.1007/s10543-019-00756-5" class="nowrap">doi:10.1007/s10543-019-00756-5</a></td> </tr><tr> <td>[27]</td> <td class="space">LEDGER, S. and SØJMARK, A. (2021). At the mercy of the common noise: Blow-ups in a conditional McKean-Vlasov problem. <span class="zbmathjax-textit">Electron. J. Probab.</span> 26 Paper No. 35, 39 pp. · <a href="/1469.60161" class="nowrap">Zbl 1469.60161</a> · <a href="https://doi.org/10.1214/21-EJP597" class="nowrap">doi:10.1214/21-EJP597</a></td> </tr><tr> <td>[28]</td> <td class="space">LEOBACHER, G., REISINGER, C. and STOCKINGER, W. (2020). Well-posedness and numerical schemes for one-dimensional McKean-Vlasov equations and interacting particle systems with discontinuous drift. Preprint. Available at arXiv:2006.14892.</td> </tr><tr> <td>[29]</td> <td class="space">MAO, X. (1997). <span class="zbmathjax-textit">Stochastic Differential Equations and Their Applications</span>. <span class="zbmathjax-textit">Horwood Publishing Series in Mathematics & Applications</span>. Horwood Publishing Limited, Chichester. · <a href="/0892.60057" class="nowrap">Zbl 0892.60057</a></td> </tr><tr> <td>[30]</td> <td class="space">McKean, H. P. Jr. (1966). A class of Markov processes associated with nonlinear parabolic equations. <span class="zbmathjax-textit">Proc. Natl. Acad. Sci. USA</span> 56 1907-1911. · <a href="/0149.13501" class="nowrap">Zbl 0149.13501</a> · <a href="https://doi.org/10.1073/pnas.56.6.1907" class="nowrap">doi:10.1073/pnas.56.6.1907</a></td> </tr><tr> <td>[31]</td> <td class="space">MEHRI, S., SCHEUTZOW, M., STANNAT, W. and ZANGENEH, B. Z. (2020). Propagation of chaos for stochastic spatially structured neuronal networks with delay driven by jump diffusions. <span class="zbmathjax-textit">Ann. Appl. Probab.</span> 30 175-207. · <a href="/1434.60299" class="nowrap">Zbl 1434.60299</a> · <a href="https://doi.org/10.1214/19-AAP1499" class="nowrap">doi:10.1214/19-AAP1499</a></td> </tr><tr> <td>[32]</td> <td class="space">MISHURA, Y. S. and VERETENNIKOV, A. Y. (2020). Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations. <span class="zbmathjax-textit">Theory Probab. Math. Statist.</span> 103 59-101. · <a href="/1482.60079" class="nowrap">Zbl 1482.60079</a></td> </tr><tr> <td>[33]</td> <td class="space">NEELIMA, BISWAS, S., KUMAR, C., DOS REIS, G. and REISINGER, C. (2020). Well-posedness and tamed Euler schemes for McKean-Vlasov equations driven by Lévy noise. Preprint. Available at arXiv:2010.08585.</td> </tr><tr> <td>[34]</td> <td class="space">PHAM, H. and PHAM, H. (2016). Linear quadratic optimal control of conditional McKean-Vlasov equation with random coefficients and applications. <span class="zbmathjax-textit">Probab. Uncertain. Quant. Risk</span> 1 Paper No. 7, 26 pp. · <a href="/1433.49030" class="nowrap">Zbl 1433.49030</a> · <a href="https://doi.org/10.1186/s41546-016-0008-x" class="nowrap">doi:10.1186/s41546-016-0008-x</a></td> </tr><tr> <td>[35]</td> <td class="space">REISINGER, C. and STOCKINGER, W. (2022). An adaptive Euler-Maruyama scheme for McKean-Vlasov SDEs with super-linear growth and application to the mean-field FitzHugh-Nagumo model. <span class="zbmathjax-textit">J. Comput. Appl. Math.</span> 400 Paper No. 113725, 23 pp. · <a href="/1469.60195" class="nowrap">Zbl 1469.60195</a> · <a href="https://doi.org/10.1016/j.cam.2021.113725" class="nowrap">doi:10.1016/j.cam.2021.113725</a></td> </tr><tr> <td>[36]</td> <td class="space">SABANIS, S. (2013). A note on tamed Euler approximations. <span class="zbmathjax-textit">Electron. Commun. Probab.</span> 18 Paper No. 47, 10 pp. · <a href="/1329.60237" class="nowrap">Zbl 1329.60237</a> · <a href="https://doi.org/10.1214/ECP.v18-2824" class="nowrap">doi:10.1214/ECP.v18-2824</a></td> </tr><tr> <td>[37]</td> <td class="space">SABANIS, S. (2016). Euler approximations with varying coefficients: The case of superlinearly growing diffusion coefficients. <span class="zbmathjax-textit">Ann. Appl. Probab.</span> 26 2083-2105. · <a href="/1352.60101" class="nowrap">Zbl 1352.60101</a> · <a href="https://doi.org/10.1214/15-AAP1140" class="nowrap">doi:10.1214/15-AAP1140</a></td> </tr><tr> <td>[38]</td> <td class="space">ŠIŠKA, D. and SZPRUCH, Ł. (2020). Gradient flows for regularized stochastic control problems. Preprint. Available at arXiv:2006.05956v3.</td> </tr><tr> <td>[39]</td> <td class="space">Sznitman, A.-S. (1991). Topics in propagation of chaos. In <span class="zbmathjax-textit">École D’Été de Probabilités de Saint-Flour XIX—</span>1989. <span class="zbmathjax-textit">Lecture Notes in Math.</span> 1464 165-251. Springer, Berlin. · <a href="/0732.60114" class="nowrap">Zbl 0732.60114</a> · <a href="https://doi.org/10.1007/BFb0085169" class="nowrap">doi:10.1007/BFb0085169</a></td> </tr><tr> <td>[40]</td> <td class="space">ULLNER, E., POLITI, A. and TORCINI, A. (2018). Ubiquity of collective irregular dynamics in balanced networks of spiking neurons. <span class="zbmathjax-textit">Chaos</span> 28 081106, 5 pp. · <a href="https://doi.org/10.1063/1.5049902" class="nowrap">doi:10.1063/1.5049902</a></td> </tr><tr> <td>[41]</td> <td class="space">VILLANI, C. (2011). Optimal transport: Monge meets Riemann and Fourier. <span class="zbmathjax-textit">J. Egyptian Math. Soc.</span> 19 95-96. · <a href="/1246.49041" class="nowrap">Zbl 1246.49041</a> · <a href="https://doi.org/10.1016/j.joems.2011.09.007" class="nowrap">doi:10.1016/j.joems.2011.09.007</a></td> </tr><tr> <td>[42]</td> <td class="space">WANG, F.-Y. (2018). Distribution dependent SDEs for Landau type equations. <span class="zbmathjax-textit">Stochastic Process. Appl.</span> 128 595-621 · <a href="/1380.60077" class="nowrap">Zbl 1380.60077</a> · <a href="https://doi.org/10.1016/j.spa.2017.05.006" class="nowrap">doi:10.1016/j.spa.2017.05.006</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 1498.60288 - zbMATH Open