[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83067-en":3,"doc-seo-83067-105":30,"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":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83067,1374391975076,"Riley","https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051",8,"Research & Report","Taming Discrete Rough Paths via Strong Lyapunov Functions","Taming discrete rough paths via strong Lyapunov functions studies a tamed numerical scheme for rough differential equations driven by genuinely rough stochastic paths. Using the newly introduced strong Lyapunov function framework, it derives explicit norm estimates for the tamed dynamics that match the continuous system’s behavior. The work proves L1 convergence of the tamed scheme and, under a negative gradient condition, establishes a numerical pullback attractor for the generated random dynamical system with integrability and upper semi-continuity in the step size.","arXiv :2607 .06266v1 [math .NA] 7 Jul 2026  \nTaming discrete rough paths via strong Lyapunov functions  \nLuu Hoang Duca  \nworking paper  \nAbstract  \nBased on the newly introduced concept of strong Lyapunov functions for rough differential equations [2], we study a tamed numerical scheme to approximate the solutions of the continuous system. We derive explicit estimates of solution norms of the tamed system which look similar to those of the continuous system. As a result, we prove the convergence of the tamed scheme in the L 1 sense. For systems with the negative gradient condition, we prove the existence of a numerical pullback attractor for the generated random dynamical system from the tamed numerical scheme which is integrable and upper semi-continuous w.r.t. the scheme step size.  \nKeywords: rough differential equations, tamed numerical scheme, Doss-Sussmann transformation, strong Lyapunov functions, pullback attractors, upper semi-continuity.  \n1 Introduction  \nConsider the rough differential equation  \ndyt = f (yt)dt + g(yt)dxt. (1.1)  \nThe solution of (1.1) is often understood in the sense of either Lyons-Davie [10] or of Friz-Victoir [8], which does not need to specify rough integrals, or in the sense of Gubinelli [9] with rough integrals. The following assumptions are imposed for coefficient functions.  \n(Hf) f : Rd → Rd is locally Lipschitz continuous.  \n(Hg) g is in C3b(Rd , Rd×m ) where we define  \nCg := max n ∥g∥∞ , ∥Dg∥∞ , ∥D2g∥∞ , ∥D3g∥∞o. (1.2)  \n(HX) For a given ν ∈ ( ~~1~~3 , ~~1~~2], Rm ∋ Xt(ω) is a stochastic process with stationary increments, of which almost all realizations x belong to the space Cν(R, Rm ) of ν−H¨older continuous paths, such that x is truly rough and can be lifted into a rough path lift x = (x, X) of a stochastic process (X · (ω), X · , · (ω)) with stationary increments, and the estimate  \n E 􀀐 ∥Xs,t∥p + ∥Xs,t∥q􀀑 ≤ CT,ν|t − s|pν , ∀s, t ∈ [0, T] (1.3)  \na Department of Mathematics, University of Klagenfurt, Austria & Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany & Institute of Mathematics, Vietnam Academy of Science and Technology, Vietnam. [duc.luu@aau.at](duc.luu@aau.at) , [duc.luu@mis.mpg.de](duc.luu@mis.mpg.de) , [lhduc@math.ac.vn](lhduc@math.ac.vn)  \nholds for any [0, T], with pν ≥ 1, q = ~~p~~2 and some constant CT,ν . (Examples of such processes include multi-dimensional fractional Brownian motions) .  \nSystem (1.1) is often solved with the Doss-Sussmann technique [11], namely, under a transformation yt = ϕt,τk (x, zt) where ϕ is the pure rough flow, there is an one-to-one correspondence between a solution yt of (1.1) on a certain interval [τk,τk+1] and a solution zt of the associated ordinary differential equation  \nz˙t = h ϕz (t,τk , x, zt)i−1f (ϕt,τk (x, zt)) = (Id + ψt)f (zt+ ηt), t ∈ [τk,τk+1], zτk = yτk , (1.4) such that we can control η ∈ Rd and ψ ∈ Rd×d so that  \n∥ηt∥, ∥ψt∥ ≤ λ, ∀t ∈ [τk,τk+1] . (1.5)  \nHowever, estimating the solution norms for (1.4) is a challenge which requires a technical condition that is difficult to be applied, namely the drift has to be of linear growth in the perpendicular direction (see e.g. [4]) . Another difficulty is to approximate the solution of (1.1) using the Euler scheme  \nytk+1 = ytk + f(ytk)(tk+1 − tk) + g(ytk)xtk,tk+1 + Dg(ytk)g (ytk)Xtk,tk+1 , k ∈ N. (1.6)  \nIn [4], it is proved that the scheme (1.6) approximates the continuous system (1.1) in the pathwise sense, using the cut-off technique. A drawback of this method is that, while the approximation error is estimated as C (x)|Π|3ν−1, the constant C (x) might not be integrable in general. Later, it is proved in [3] that C (x) ∈ L 1 under additional assumption on global Lipschitz continuity of the drift f.  \nRecently, another condition has been proposed in [2], which introduces the new notion of a strong Lyapunov function V with the existence of a small constant λ0 ∈ (0 , 1) and of constants Cλ0 > 0,δ ∈ R such that  \nsup ⟨∇V(z),(I + ψ)f (z + η)⟩ ≤ C λ0 + δV(z)","cbCaig49vX5c8HDv","https://ap.wps.com/l/cbCaig49vX5c8HDv","pdf",393745,2,1,25,"English","en",105,"# Abstract\n# Introduction\n# Strong Lyapunov functions\n## K spaces","[{\"question\":\"What is the main goal of the paper?\",\"answer\":\"To approximate solutions of a rough differential equation using a tamed numerical scheme and to prove convergence and attractor properties based on strong Lyapunov functions.\"},{\"question\":\"How does the strong Lyapunov function condition help the analysis?\",\"answer\":\"It provides a drift inequality that yields integrable solution-norm estimates, enabling L1 convergence of the tamed scheme and control of dynamics.\"},{\"question\":\"What additional result holds under the negative gradient condition?\",\"answer\":\"The paper proves existence of a numerical pullback attractor for the discrete random dynamical system, together with upper semi-continuity with respect to the step size and noise intensity.\"}]",1784184992,63,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"taming-discrete-rough-paths-via-strong-lyapunov-functions","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/taming-discrete-rough-paths-via-strong-lyapunov-functions/83067/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","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 is the main goal of the paper?","Question",{"text":75,"@type":76},"To approximate solutions of a rough differential equation using a tamed numerical scheme and to prove convergence and attractor properties based on strong Lyapunov functions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the strong Lyapunov function condition help the analysis?",{"text":80,"@type":76},"It provides a drift inequality that yields integrable solution-norm estimates, enabling L1 convergence of the tamed scheme and control of dynamics.",{"name":82,"@type":73,"acceptedAnswer":83},"What additional result holds under the negative gradient condition?",{"text":84,"@type":76},"The paper proves existence of a numerical pullback attractor for the discrete random dynamical system, together with upper semi-continuity with respect to the step size and noise 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