[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82758-en":3,"doc-seo-82758-105":30,"detail-sidebar-cat-0-en-105":92},{"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},82758,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","An AI-Assisted Solution to the Signed BAR Conjecture","For a multidimensional reflected diffusion, the basic adjoint relationship (BAR) determines stationary behavior, but whether it uniquely identifies the stationary distribution has stayed open for over 35 years. This paper resolves finite-signed uniqueness for stable Harrison–Reiman data under a nonsingular M-matrix reflection matrix, using pathwise differentiability and resolvent directional derivatives to show boundary invariance. It also proves the nonsingular M-matrix condition is structural: in a wider completely-S class, singular proper principal blocks generate nonzero zero-mass signed BAR tuples and an explicit 3D family obstruction.","arXiv :2607 .03639v1 [math .PR] 3 Jul 2026  \nAn AI-Assisted Solution to the Signed BAR Conjecture: Uniqueness in the Harrison–Reiman Class and a Completely-S  \nClass Obstruction  \nYiping Lu 1 and Youheng Zhu 1  \n1 Department of Industrial Engineering and Management Sciences, McCormick School of Engineering, Northwestern University.  \nAbstract. For a multidimensional reflected diffusion, determining whether the associated basic adjoint relationship (BAR) uniquely characterizes the stationary distribution is a basic uniqueness problem in the BAR approach. The problem has remained unresolved for more than 35 years since the introduction of the BAR approach. In this paper, we resolve the finite-signed uniqueness problem for stable Harrison–Reiman data with a nonsingular M-matrix reflection matrix. The proof uses pathwise differentiability of the reflected diffusion implies feasible directional differentiability of the probabilistic resolvent to show that, at boundary points, its one-sided initial-state derivative factors through the tangent projection and vanishes along active reflection directions. An interior one-sided convolution then yields smooth test functions whose oblique derivatives are uniformly bounded and converge pointwise to zero on each closed face. The interior signed measure is consequently invariant for the reflected semigroup. A Jordan-decomposition argument identifies it as a scalar multiple of the unique invariant probability, and an induction over boundary strata, using invertibility of the principal reflection blocks, identifies the boundary measures. The proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors.  \nWe also show that the nonsingular M-matrix assumption is structural. In the larger completely-S class, a nonsingular reflection matrix with a singular proper principal block admits boundary gauges supported on lower-dimensional strata. Under standard exponential ergodicity and a mild one-step regulator bound, these gauges produce nonzero zero-mass signed BAR tuples; indeed the zero-mass interior BAR coordinates contain an infinite-dimensional subspace. A four-parameter three-dimensional family, including an explicit rational example, verifies the obstruction. Thus the finite signed version of the Dai–Dieker question has a positive answer in the Harrison–Reiman M-matrix class and a negative answer in a natural completely-S extension.  \nMSC2020 subject classifications: Primary 60J60; 60J55; secondary 35J25; 46A20; 60K25  \nKeywords: semimartingale reflected Brownian motion; basic adjoint relationship; signed measure; Skorokhod map; pathwise derivative; resolvent; completely S matrix  \n1. Introduction  \nSemimartingale reflected Brownian motions (SRBMs) in the nonnegative orthant are diffusion approximations for stochastic networks in heavy traffic. In the interior of the orthant the process behaves as a Brownian motion with drift and covariance matrix; when it reaches a face, it is pushed back into the state space in an oblique direction prescribed by the corresponding column of a reflection matrix. The Harrison–Reiman construction [23, 24] is the canonical orthant model behind open queueing networks in heavy traffic [21, 22, 25, 32, 35]; it is the main positive setting of this paper.  \nA central analytic object for such reflected diffusions is the basic adjoint relationship (BAR) . It appears in the early stationary analysis and product-form theory for RBM/SRBM [24–26], underlies numerical methods for orthant SRBMs [6, 7], has been used in steady-state heavy-traffic approximation through the BAR approach [3, 4], and is one of the standard weak formulations used to characterize stationary distributions of reflected diffusions [5, 27] . If π is an interior measure and νi is a boundary measure on the face Fi = {xi = 0}, the BAR has the form  \nd  \n(1.1) ZE Lf dπ +Xi=1 ZFi Dif dνi = 0, f ∈ C2b(E),  \nwhere L is the interior diffusion generator and Di is the direct","cbCaiayR8oF1XUiC","https://ap.wps.com/l/cbCaiayR8oF1XUiC","pdf",707311,5,1,32,"English","en",105,"# Abstract\n# 1. Introduction\n## Semimartingale reflected Brownian motion and the Harrison–Reiman model\n## Basic adjoint relationship and the uniqueness conjecture","[{\"question\":\"What uniqueness problem does the paper address for reflected diffusions?\",\"answer\":\"Whether a BAR solution necessarily has interior part equal to the stationary distribution, even when BAR measures are allowed to be finite signed measures.\"},{\"question\":\"What condition is proved sufficient in the Harrison–Reiman class?\",\"answer\":\"Finite-signed uniqueness holds for stable Harrison–Reiman data with a nonsingular M-matrix reflection matrix.\"},{\"question\":\"How does the completely-S extension change the conclusion?\",\"answer\":\"In the larger completely-S class, nonsingular reflection matrices with singular proper principal blocks produce nonzero zero-mass signed BAR tuples, giving a negative answer in that extension.\"}]",1784182747,81,{"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":87,"head_meta":89,"extra_data":91,"updated_unix":28},"an-ai-assisted-solution-to-the-signed-bar-conjecture","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/an-ai-assisted-solution-to-the-signed-bar-conjecture/82758/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What uniqueness problem does the paper address for reflected diffusions?","Question",{"text":76,"@type":77},"Whether a BAR solution necessarily has interior part equal to the stationary distribution, even when BAR measures are allowed to be finite signed measures.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What condition is proved sufficient in the Harrison–Reiman class?",{"text":81,"@type":77},"Finite-signed uniqueness holds for stable Harrison–Reiman data with a nonsingular M-matrix reflection matrix.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the completely-S extension change the conclusion?",{"text":85,"@type":77},"In the larger completely-S class, nonsingular reflection matrices with singular proper principal blocks produce nonzero zero-mass signed BAR tuples, giving a negative answer in that extension.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":20,"slug":138},19,"General","general"]