Nonlocal balance equation: representation of solution and Markov approximation

Nonlocal balance equation: representation of solution and Markov approximation | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Nonlocal balance equation: representation of solution and Markov approximation Yurii Averboukh This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-3953641/v1 This work is licensed under a CC BY 4.0 License Status: Under Review Version 1 posted 7 You are reading this latest preprint version Abstract We study the nonlocal balance equation that describes the evolution of a system consisting of infinitely many identical particles those move along a deterministic dynamics and can also either disappear or give a spring. In this case, the solution of the balance equation is considered in the space of nonnegative measures. We prove the superposition principle for the examined nonlocal balance equation. Furthermore, we interpret the source/sink term as a probability rate of jumps from/to a remote point. Using this idea and replacing the deterministic dynamics of each particle by a nonlinear Markov chain, we approximate the solution of the balance equation is approximated by a solution of a system of ODEs and evaluate the corresponding approximation rate. MSC Classification: 35R06, 70F45, 60J27 balance equation system of particles superposition principle Markov approximation Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Under Review Version 1 posted Editorial decision: Revision requested 05 Apr, 2024 Reviews received at journal 31 Mar, 2024 Reviewers agreed at journal 08 Mar, 2024 Reviewers invited by journal 18 Feb, 2024 Submission checks completed at journal 14 Feb, 2024 Editor assigned by journal 14 Feb, 2024 First submitted to journal 13 Feb, 2024 You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-3953641","acceptedTermsAndConditions":true,"allowDirectSubmit":false,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":273520452,"identity":"1ffc7809-e287-43b7-9586-ec9729433858","order_by":0,"name":"Yurii Averboukh","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA0UlEQVRIiWNgGAWjYJACZiA2MGBvAFEWpGjhOQCiJEjRIpEAYhOhhX9G8rHPBRWHjc0ln1/d8KNAgoG/vTsBrxaJG2nJs2ecOWxmOTun7GYP0GESZ85uwG/NmTPGzLxth20Mbuek3eABajGQyMWvRf7M+c8QLTfPpN38Q4wWg+M9zCAtZgY32I/dJsoWw+NtxswzzqQbG5zJYbstYyDBQ9AvcoeZHzMXVFgbbjh+/NnNN39s5Pjbewl4HwF4DMAkscpBgP0BKapHwSgYBaNgBAEANYFHBjjKnwIAAAAASUVORK5CYII=","orcid":"","institution":"Krasovskii Institute of Mathematics and Mechanics","correspondingAuthor":true,"prefix":"","firstName":"Yurii","middleName":"","lastName":"Averboukh","suffix":""}],"badges":[],"createdAt":"2024-02-13 13:50:10","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-3953641/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-3953641/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":51362613,"identity":"52058984-fb5e-4853-a5c8-461e31616931","added_by":"auto","created_at":"2024-02-20 09:09:44","extension":"pdf","order_by":1,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":450935,"visible":true,"origin":"","legend":"","description":"","filename":"averboukhbalanceab.pdf","url":"https://assets-eu.researchsquare.com/files/rs-3953641/v1_covered_72267b24-0cfa-494e-a07b-62ea11c3a2ed.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Nonlocal balance equation: representation of solution and Markov approximation","fulltext":[],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":false,"highlight":"","institution":"","isAcceptedByJournal":true,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":true,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"journal-of-dynamics-and-differential-equations","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":false,"externalIdentity":"jdde","sideBox":"Learn more about [Journal of Dynamics and Differential Equations](http://link.springer.com/journal/10884)","snPcode":"10884","submissionUrl":"https://submission.nature.com/new-submission/10884/3","title":"Journal of Dynamics and Differential Equations","twitterHandle":"","acdcEnabled":true,"dfaEnabled":true,"editorialSystem":"em","reportingPortfolio":"Springer Hybrid","inReviewEnabled":true,"inReviewRevisionsEnabled":false},"keywords":"balance equation, system of particles, superposition principle, Markov approximation","lastPublishedDoi":"10.21203/rs.3.rs-3953641/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-3953641/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eWe study the nonlocal balance equation that describes the evolution of a system consisting of infinitely many identical particles those move along a deterministic dynamics and can also either disappear or give a spring. 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