[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85397-en":3,"doc-seo-85397-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85397,1099513958607,"Jiven","https://ap-avatar.wpscdn.com/avatar/100002390cf8733938c?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778829742770036399",8,"Research & Report","Geometric Ergodicity and Strong Error Estimates for Tamed Schemes of Super-linear SODEs","Constructs a family of explicit tamed Euler–Maruyama (TEM) schemes for super-linear stochastic ordinary differential equations (SODEs) with multiplicative noise, designed to preserve the same Lyapunov function as the underlying SODE. The resulting numerical Markov processes inherit geometric ergodicity from the continuous dynamics and attain optimal strong convergence orders. Non-degeneracy conditions ensure equivalence of transition probabilities. Numerical experiments validate the theoretical ergodicity and strong error estimates.","arXiv :2411 .06049v2 [math .NA] 13 Jul 2026  \nNoname manuscript No.  \n(will be inserted by the editor)  \nGeometric Ergodicity and Strong Error Estimates for Tamed Schemes of Super-linear SODEs  \nZhihui LIU · Xiaoming WU  \nReceived: date / Accepted: date  \nAbstract We construct a family of explicit tamed Euler–Maruyama (TEM) schemes, which can preserve the same Lyapunov function for super-linear stochastic ordinary differential equations (SODEs) driven by multiplicative noise. These TEM schemes are shown to inherit the geometric ergodicity of the considered SODEs and converge with optimal strong convergence orders. Numerical experiments verify our theoretical results.  \nKeywords super-linear stochastic ordinary differential equation · numerical invariant measure · numerical ergodicity · strong error estimate Mathematics Subject Classification (2020) 65C30 · 60H35 · 60H10  \n1 Introduction  \nThe long-time behavior of the Wiener process-driven SODE  \ndX(t) = b(X(t))dt + σ(X(t))dW(t), t ≥ 0 , (SDE)  \nplays a vital role in many scientific areas. As a significant long-time behavior, the ergodicity characterizes the identity of the temporal average and spatial average for a Markov process or chain generated by Eq. (SDE) and its numerical discretization, respectively, which has a lot of applications in quantum mechanics, fluid dynamics, financial mathematics, and many other fields [4, 6] .  \nZ. LIU  \nDepartment of Mathematics & National Center for Applied Mathematics Shenzhen (NCAMS) & Shenzhen International Center for Mathematics, Southern University of Science and Technology, Shenzhen, 518055, P.R.China  \nE-mail: [liuzh3@sustech.edu.cn](liuzh3@sustech.edu.cn) (Corresponding author)  \nX. WU  \nDepartment of Mathematics, Southern University of Science and Technology, Shenzhen 518055, P.R. China  \nE-mail: [12331004@mail.sustech.edu.cn](12331004@mail.sustech.edu.cn)  \nIt is known that the coefficients of most nonlinear SODEs arising in applications do not satisfy the traditional, but restrictive, Lipschitz continuity assumptions. This paper analyzes integrators for the super-linear Eq. (SDE) whose solution is uniquely ergodic with respect to an equilibrium distribution that  \n1. are ergodic with respect to an invariant measure on infinite time intervals, 2. strongly converge with optimal convergence rate to the solutions of Eq.  \n(SDE) on any finite time intervals, and  \n3. involve negligible computational expense.  \nIt is known, even for the particular Langevin system, that pure sampling methods can accomplish (1), but they typically do not approximate the solution to Eq. (SDE); see, e.g., [1] . Integrators for Eq. (SDE) certainly satisfy (2), but they are often divergent on infinite-time intervals or ergodic concerning a different equilibrium distribution [17, 18] . The backward Euler method and the stochastic theta method (see, e.g., [10, 13 , 14 , 17]) satisfy (1) and (2), but they usually entail high computational cost. We will show that a family of TEM methods constructed in this paper, as explicit schemes, can simultaneously accomplish these three goals. Similar arguments were successfully applied in [15] to construct Galerkin-based fully discrete tamed schemes for superlinear stochastic PDEs.  \nTo construct explicit methods that inherit the unique ergodicity of Eq.(SDE), including the tamed and truncated methods studied, e.g., in [2, 8 , 9 , 19], the numerical Lyapunov structure plays a key role. However, it was shown in [7] that the classical Euler–Maruyama (EM) scheme applied to Eq. (SDE) with super-linear growth coefficients would blow up in p-th moment for all p ≥ 2. In particular, the square function is not an appropriate Lyapunov function of the EM scheme, even though it is a natural Lyapunov function of the considered monotone Eq. (SDE) .  \nOur main aim in this paper is to construct a family of explicit TEM schemes to preserve the same Lyapunov structure of the super-linear Eq. (SDE) .  \nIt is shown that, under certain non-","cbCaidXUroJkCkP5","https://ap.wps.com/l/cbCaidXUroJkCkP5","pdf",478184,1,25,"English","en",105,"# Introduction\n# Geometric Ergodicity of Tamed Euler–Maruyama Schemes\n## Lyapunov structure and probabilistic regularity","[{\"question\":\"What problem does the paper address in numerical discretization of super-linear SODEs?\",\"answer\":\"It addresses that the classical Euler–Maruyama method can blow up in moments for super-linear growth coefficients and may fail to preserve the correct ergodic behavior.\"},{\"question\":\"How do the proposed TEM schemes preserve long-time behavior?\",\"answer\":\"They are constructed to preserve the same Lyapunov function structure as the original SODE, and under non-degeneracy assumptions they inherit geometric ergodicity.\"},{\"question\":\"What strong convergence orders are proved for the TEM schemes?\",\"answer\":\"The paper shows an expected optimal strong order of 1/2 in the multiplicative noise case and order 1 in the additive noise case.\"}]",1784203124,63,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":27},"geometric-ergodicity-and-strong-error-estimates-for-tamed-schemes-of-super-linear-sodes","",{"@graph":35,"@context":85},[36,53,68],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/geometric-ergodicity-and-strong-error-estimates-for-tamed-schemes-of-super-linear-sodes/85397/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What problem does the paper address in numerical discretization of super-linear SODEs?","Question",{"text":75,"@type":76},"It addresses that the classical Euler–Maruyama method can blow up in moments for super-linear growth coefficients and may fail to preserve the correct ergodic behavior.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How do the proposed TEM schemes preserve long-time behavior?",{"text":80,"@type":76},"They are constructed to preserve the same Lyapunov function structure as the original SODE, and under non-degeneracy assumptions they inherit geometric ergodicity.",{"name":82,"@type":73,"acceptedAnswer":83},"What strong convergence orders are proved for the TEM schemes?",{"text":84,"@type":76},"The paper shows an expected optimal strong order of 1/2 in the multiplicative noise case and order 1 in the additive noise case.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":45,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":45,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":45,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":45,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":45,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":45,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]