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An AI-Assisted Solution to the Signed BAR Conjecture: Uniqueness in the Harrison--Reiman Class and a Completely-$\mathcal{S}$ Class Obstruction
Yiping Lu, Youheng Zhu
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
Last extracted: 7/7/2026, 8:53:32 AM
Summary
This paper resolves the long-standing finite-signed uniqueness problem for the Basic Adjoint Relationship (BAR) associated with Semimartingale Reflected Brownian Motions (SRBMs). The authors prove that the BAR uniquely characterizes the stationary distribution within the stable Harrison-Reiman class (nonsingular M-matrix reflection), while demonstrating a structural obstruction in the broader completely-S class where uniqueness fails. The proof, discovered with AI assistance from ChatGPT 5.5 Pro, utilizes pathwise differentiability, probabilistic resolvents, and boundary strata induction. It answers the Dai-Dieker question positively for the M-matrix class and negatively for the completely-S extension.
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Relation Signals (8)
Yiping Lu → affiliatedwith → Northwestern University
confidence 98% · Yiping Lu Youheng Zhu Department of Industrial Engineering and Management Sciences, McCormick School of Engineering, Northwestern University.
Youheng Zhu → affiliatedwith → Northwestern University
confidence 98% · Yiping Lu Youheng Zhu Department of Industrial Engineering and Management Sciences, McCormick School of Engineering, Northwestern University.
Basic Adjoint Relationship (BAR) → characterizes → Stationary Distribution
confidence 95% · determining whether the associated basic adjoint relationship (BAR) uniquely characterizes the stationary distribution
Harrison-Reiman Class → satisfies → Finite-Signed Uniqueness
confidence 94% · resolve the finite-signed uniqueness problem for stable Harrison–Reiman data with a nonsingular M-matrix reflection matrix
Completely-S Class → exhibits → Signed Uniqueness Obstruction
confidence 93% · In the larger completely-S class... these gauges produce nonzero zero-mass signed BAR tuples
Dai-Dieker Question → has → Positive Answer
confidence 92% · the finite signed version of the Dai–Dieker question has a positive answer in the Harrison–Reiman M-matrix class
M-matrix → isstructuralassumptionfor → Harrison-Reiman Class
confidence 91% · We also show that the nonsingular M-matrix assumption is structural.
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Abstract
Abstract: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. The proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors. We also show that the nonsingular $M$-matrix assumption is structural. In the larger completely-$\mathcal{S}$ 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-$\mathcal{S}$ extension.
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- Source: https://arxiv.org/abs/2607.03639v1
- Canonical: https://arxiv.org/abs/2607.03639v1
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An AI-Assisted Solution to the Signed BAR Conjecture: Uniqueness in the Harrison–Reiman Class and a Completely-S Class Obstruction Yiping Lu Youheng Zhu Department of Industrial Engineering and Management Sciences, McCormick School of Engineering, Northwestern University. Abstract 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. We 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. completely S matrix, keywords: [class=MSC] keywords: 1 Introduction Semimartingale 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 [21, 25] is the canonical orthant model behind open queueing networks in heavy traffic [32, 22, 24, 26, 35]; it is the main positive setting of this paper. A 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 [25, 23, 22], 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 _i is a boundary measure on the face Fi=xi=0F_i=\x_i=0\, the BAR has the form ∫ELfπ+∑i=1d∫FiDifνi=0,f∈Cb2(E), _ELf\,dπ+ _i=1^d _F_iD_if\,d _i=0, f∈ C_b^2(E), (1.1) where L is the interior diffusion generator and DiD_i is the directional derivative in the iith reflection direction. The stationary distribution π0 _0, together with its stationary boundary occupation measures νi0 _i^0, satisfies (1.1). The basic uniqueness question is whether the converse holds: does a BAR solution necessarily have interior part equal to the stationary distribution? The issue has persisted for more than three decades, remaining an open problem since the inception of the BAR approach. The open problem was first stated as a conjecture in [6] for SRBMs in a two dimensional rectangle and in [7] for SRBMs in a d-dimensional orthant. Dai and Dieker [5] describe the fundamental open problem concerning the Basic Adjoint Relationship (BAR) for multidimensional diffusion processes. Specifically, for both Semimartingale Reflecting Brownian Motions (SRBMs) and piecewise Ornstein–Uhlenbeck (OU) processes. Dai and Dieker [5, Proposition 1 and Open Problem 1] formulated the BAR characterization with bounded C2C^2 tests, proved the corresponding characterization in the positive-measure setting, and asked for the signed analogue. The compactly supported C2C^2 formulation leads to the same finite-signed uniqueness problem. The bounded-test identity immediately implies the compactly supported one. Conversely, let f∈Cb2(E)f∈ C_b^2(E) and choose χn∈Cc∞(ℝd) _n∈ C_c^∞(R^d) with 0≤χn≤10≤ _n≤ 1, χn=1 _n=1 on |x|≤n\|x|≤ n\, and ‖∇χn‖∞+‖D2χn‖∞→0\|∇ _n\|_∞+\|D^2 _n\|_∞→ 0. Applying the compactly supported identity to χnf _nf and expanding L(χnf)L( _nf) and Di(χnf)D_i( _nf) gives the bounded-test identity after passage to the limit, because χn→1 _n→ 1 pointwise and all error terms are uniformly bounded by constants times ‖∇χn‖∞+‖D2χn‖∞\|∇ _n\|_∞+\|D^2 _n\|_∞ against finite signed measures. Throughout the paper we therefore use the bounded-test class Cb2(E)C_b^2(E), which is the formulation needed to insert the one-sided smoothings of the probabilistic resolvent without an artificial spatial cutoff. In the signed problem one allows π and the νi _i to be finite signed measures. The question then becomes linear: is every finite signed BAR tuple a scalar multiple of the stationary tuple? This signed formulation is more delicate than the positive one. Positive recurrence identifies invariant probabilities, but the BAR permits cancellation between signed interior and boundary terms. Moreover, the natural functions that identify invariant measures are probabilistic resolvents, which are not classical BAR tests at the corners. Related work BAR characterization of stationary probabilities As shown in the the original BAR calculations for SRBMs [23, 22], positive-measure BAR characterizations identify stationary probabilities, and in many formulations also the associated boundary occupation measures, once the reflected diffusion and its stationary regime are already well posed [6, 7, 27]. These results do not, by themselves, exclude sign-changing finite measures whose interior and boundary terms cancel in the BAR. Our positive theorem addresses exactly that finite-signed nullspace question in the stable Harrison–Reiman nonsingular-M-matrix class, and it identifies the full boundary tuple as well as the interior coordinate. Much of the stationary SRBM literature concerns explicit formulas, transforms, asymptotics, or numerical computation rather than signed uniqueness. Product-form and skew-symmetry results originate with Harrison and Williams [23]; numerical and approximation methods based on the BAR go back at least to Dai and Harrison [6, 7] and continue in the steady-state heavy-traffic BAR approach for queueing networks [3, 4]; two-dimensional and wedge analyses have been developed through sum-of-exponentials, geometric, and boundary-value/functional-equation methods [11, 8, 9, 18, 19]. The present proof uses none of these explicit analytic representations. Its role is instead structural: it proves that, in the stated M-matrix class, the finite signed BAR has no hidden zero-mass directions. Skorokhod-map Differentiability Lipschitz, convex-duality and differentiability properties of oblique reflection maps were developed in deterministic form by Dupuis–Ishii, Dupuis–Ramanan, Mandelbaum–Ramanan, and Lipshutz–Ramanan [13, 14, 15, 31, 28]. We use the reflected-diffusion version of this theory, namely the pathwise differentiability and sensitivity results of Lipshutz and Ramanan [29, 30], only after verifying their assumptions for the normalized Harrison–Reiman data. The negative result is complementary to the existence and stability literature for completely-S data: Taylor–Williams and Dai–Williams give the relevant SRBM existence frameworks [34, 10], while Lyapunov and recurrence criteria for SRBMs are developed for example in [16, 2, 33]. Section 6 shows that existence and recurrence alone do not replace invertibility of every active principal block. Technical Overview Our positive result answers the signed Dai–Dieker problem for stable Harrison–Reiman data with a nonsingular M-matrix reflection matrix. The proof is organized around a resolvent invariant identity. Let Rλh=∫0∞e−λtPthtR_λh= _0^∞e^-λ tP_th\,dt be the probabilistic resolvent of the reflected semigroup. Our core contribution is proving the fact that every finite signed BAR tuple satisfies ∫E(λRλh−h)π¯=0,h∈C0(E),λ>0. _E(λ R_λh-h)\,d π=0, h∈ C_0(E), λ>0. (RI) This identity says exactly that the interior signed measure is invariant under the reflected semigroup. Indeed, using Rλh=∫0∞e−λtPthtR_λh= _0^∞e^-λ tP_th\,dt, (RI) says that the Laplace transform of t↦π¯(Pth)−π¯(h)t π(P_th)- π(h) vanishes for every h∈C0(E)h∈ C_0(E). Strong continuity of the Feller semigroup upgrades this to π¯Pt=π¯ πP_t= π for all t≥0t≥ 0. If π¯=π¯+−π¯− π= π^+- π^- is the Jordan decomposition, positivity of the Markov kernel gives |π¯Pt|≤|π¯|Pt| πP_t|≤| π|P_t; equality of total masses then makes |π¯|| π| invariant, and hence both Jordan components are invariant positive finite measures. After normalization, every nonzero component is an invariant probability, so uniqueness of the invariant probability gives π¯=cπ0 π=c _0. Subtracting c times the stationary BAR leaves a pure boundary identity, and the nonsingular principal reflection blocks identify the boundary measures by an induction over strata. The only nontrivial point in this chain is the derivation of (RI). Formally, if g=Rλhg=R_λh were an admissible Cb2C_b^2 test satisfying Dig=0D_ig=0 on FiF_i, then (RI) would follow by inserting g into the BAR and using (λ−L)g=h(λ-L)g=h. This formal argument is misleading because at corners the resolvent need not be a classical C2C^2 function on the closed orthant; Appendix A gives a stable Harrison–Reiman example where such C2C^2 regularity is impossible. The proof therefore works in the topology actually seen by finite signed measures: uniform convergence of the interior equation and vanishing of the boundary terms after integration against arbitrary finite signed boundary measures. The approximation used in the proof is intentionally simple. We do not insert g=Rλhg=R_λh itself into the BAR. Instead we replace it by the one-sided smoothing gε(x)=∫ρ(w)g(x+εw)w.g_ (x)= ρ(w)g(x+ w)\,dw. The mollifier is supported strictly inside the positive orthant, so the value of gε(x)g_ (x) only uses values of g at interior points x+εwx+ w. This smoothing supplies the required bounded C2C^2 regularity for each fixed ε . The only delicate point is to show that these legitimate Cb2C_b^2 tests have asymptotically zero boundary contribution. The projected boundary derivative of the resolvent gives Digε(x)⟶0,x∈Fi,D_ig_ (x) 0, x∈ F_i, with a uniform bound sufficient for dominated convergence against an arbitrary finite signed boundary measure. Thus the functions gεg_ approximate the resolvent in exactly the topology seen by the BAR: the interior equation converges to (λ−L)Rλh=h(λ-L)R_λh=h, while all boundary terms vanish. The paper also explains why the M-matrix hypothesis is not merely a proof artifact. In the completely-S existence class, a singular proper principal block may cancel all active normal components of a boundary gauge supported on a lower-dimensional stratum. The remaining tangential derivative produces a centered interior source. Under a quantitative recurrence assumption, the zero potential of this source gives a nonzero signed BAR tuple with zero interior mass. Thus signed uniqueness fails in a natural completely-S extension. The Role of AI-assistance The proof given here was not produced by an AI system in a single pass; it is the outcome of an extended, human-directed collaboration (for 3 weeks) in which large language models served as an exploratory and organizational aid, while every mathematical decision and all verification rested with the authors. By shifting the focus from merely verifying the conjecture to characterizing the specific domain where it holds, this study not only reveals the essential divergence between Harrison-Reiman Class and Completely-S Class but also demonstrates the vital role of human-AI collaboration in advancing complex mathematical research. Following the program in Dai and Dieker’s open-problem note [5], we first attacked uniqueness in the completely-S class, where the crux is the low regularity of the solution at the boundary. Over many rounds of interaction the model carried out the boundary-layer expansion and tested whether the boundary contribution is sign-definite and whether it can be absorbed by the interior solution. When this cancellation repeatedly failed for d>3d>3, the authors chose to abandon the direct route and to construct a counterexample in the singular regime; the construction presented here is our own, and it delimits the regime in which signed uniqueness can be expected. We then turned to signed-measure uniqueness in the Harrison–Reiman class. Our first attempt proceeded through a Kato-type inequality, where the obstruction is the boundary term produced by the integration by parts; to organize the inductive cancellation of this term across the boundary strata, we prompted the model to adopt a homological-algebra–style bookkeeping. This yielded a long (roughly 150-page, see https://drive.google.com/file/d/1QEMTMYR9d0l3ToJtdVHEeYT9TF5Cudui/view?usp=sharing) proof outline that passed an initial screening by an ensemble of ten independent model/agent reviewers. Such consensus is not a proof, and we treated it only as a filter: the argument was subsequently checked by the authors, conclusion by conclusion, with each regularity hypothesis verified for mutual consistency. In the course of this verification the model surfaced the pathwise-differentiability results of Lipshutz and Ramanan [28], which considerably simplified the argument and, after further iteration, produced the proof in its present form. The authors have verified every step and are solely responsible for the correctness of the results. Additionally, we attempted to generate a positive proof via one-shot prompting, leveraging the premise that the conjecture holds true within the Harrison-Reiman class. However, both ChatGPT 5.5 Pro-extended and Claude Opus 4.8 max failed this task. The chat logs are available at: https://chatgpt.com/share/6a44a502-d034-83ea-9608-eecb9ecc898d and https://claude.ai/share/25a16238-360a-4649-935f-b23b4ec500f(Attempts https://chatgpt.com/share/6a44b084-91dc-83ea-8fc2-49b06770025d to solve the problem, even when prompted with the literature [28, 29], proved unsuccessful.). Surprisingly, contemporary AI approaches even fail to leverage the specific properties of the Harrison–Reiman class, which are essential for the proof of positivity established via the counterexample in the general Completely-S class presented in this paper. We hypothesize that the AI derived meaningful insights from the first 150 pages version of computations, even though these results were not explicitly incorporated into the final proof. This outcome highlights the potential of AI assistance in tackling open mathematical problems, while simultaneously underscoring the indispensable role of human verification and guidance throughout the process. Organization of the Paper We organize the paper as follows: Section 2 states the SRBM and BAR setting, states the main theorem, and reduces the proof to the resolvent identity (RI). Section 3 establishes the two technical properties of g=Rλhg=R_λh needed later for the approximation: the interior resolvent equation and the projected boundary derivative that will make DigεD_ig_ vanish on FiF_i. Section 4 carries out the one-sided smoothing construction, inserts gε∈Cb2(E)g_ ∈ C_b^2(E) directly into the BAR, and proves (RI). Section 5 proves the implication deferred in Section 2: the identity (RI) implies the signed BAR uniqueness conjecture, thus finishing the proof of the main theorem. Section 6 explains why the nonsingular M-matrix condition is structural by giving the completely-S obstruction and an explicit three-dimensional family. Section 7 repackages the positive and negative arguments through a common BAR homotopy lemma and separates the remaining issue into local boundary algebra. 2 Setting, main theorem, and reduction to the resolvent identity This section fixes the data, states the signed-measure theorem, and isolates the central reduction. The conversion of the present standing assumptions into the hypotheses of the reflected-diffusion results is carried out inline, at the point of use, inside the proof of Theorem 3.3: there each source hypothesis is recalled in the present orthant specialization and verified. 2.1 Notation and standing conventions Let J=1,…,dJ=\1,…,d\, E=ℝ+dE=R_+^d, and E∘=(0,∞)dE =(0,∞)^d. For i∈Ji∈ J write Fi=x∈E:xi=0.F_i=\x∈ E:x_i=0\. For nonempty A⊂JA⊂ J, define the relative boundary stratum SA=x∈E:xi=0(i∈A),xj>0(j∉A).S_A=\x∈ E:x_i=0\ (i∈ A),\ x_j>0\ (j∉ A)\. The sets SAS_A form a disjoint Borel decomposition of ∂E∂ E. For a locally compact space B, C0(B)C_0(B) denotes the continuous real-valued functions vanishing at infinity, and ℳ(B)M(B) denotes the finite signed Radon measures on B. For η∈ℳ(B)η (B), |η||η| is its total variation measure and ‖η‖TV=|η|(B)\|η\|_TV=|η|(B). We write suppηsuppη for the support of a measure and suppfsuppf for the support of a function. The symbol B1_B denotes the indicator of a set B. We use the closed-domain C2C^2 convention. Thus C2(E)C^2(E) consists of functions f:E→ℝf:E such that f∈C2(E∘)f∈ C^2(E ) and all partial derivatives ∂αf∂^αf, |α|≤2|α|≤ 2, extend continuously from E∘E to E. The class Cc2(E)C_c^2(E) consists of the functions in C2(E)C^2(E) with compact support as a subset of E. The class Cb2(E)C_b^2(E) consists of the functions in C2(E)C^2(E) for which f, ∇f∇ f and D2fD^2f are bounded. Since E is the orthant, this closed-domain convention is equivalent to saying that every f∈C2(E)f∈ C^2(E) is the restriction to E of some F∈C2(U)F∈ C^2(U) on an open neighborhood U⊃EU⊃ E. For open subsets of Euclidean space, Cc∞C_c^∞ has its usual meaning. For the reflected semigroup we write Pth(x)=[h(Ztx)]P_th(x)=E[h(Z_t^x)] and Rλh(x)=∫0∞e−λtPth(x)t,λ>0,R_λh(x)= _0^∞e^-λ tP_th(x)\,dt,\;λ>0, whenever the integral is finite. We call the semigroup PtP_t Feller if (Pt)t≥0(P_t)_t≥ 0 satisfies PtC0(E)⊂C0(E)P_tC_0(E)⊂ C_0(E), and is strongly continuous, i.e. ‖Pth−h‖∞→0\|P_th-h\|_∞→ 0 as t↓0t 0 for all h∈C0(E)h∈ C_0(E). 2.2 SRBM, BAR, and finite signed BAR tuples A semimartingale reflected Brownian motion in E is specified by a drift vector μ∈ℝdμ ^d, a symmetric positive definite covariance matrix Σ , and a reflection matrix R=(R1,…,Rd)R=(R_1,…,R_d) whose iith column is the direction of reflection on FiF_i. Put Q=Σ/2Q= /2 and Lf=μ⋅∇f+Q:D2f,Dif=Ri⋅∇f.Lf=μ·∇ f+Q:D^2f, D_if=R_i·∇ f. Throughout the positive part of the paper we work under the following stable nonsingular M-matrix data. The covariance matrix Σ is symmetric positive definite. The reflection matrix R satisfies Rii>0,Rij≤0(i≠j),R−1≥0.R_i>0, R_ij≤ 0\ (i≠ j), R^-1≥ 0. (2.1) The drift satisfies R−1μ<0R^-1μ<0 (2.2) componentwise. The phrase “stable” in this paper means exactly (2.2). The linear-algebra consequences of (2.1) are proved in Section 3.1; the stochastic consequences used later are stated in Theorem 3.3 and justified in its proof, where every source hypothesis is recalled and checked. A finite signed BAR tuple is a tuple (π¯,ν¯1,…,ν¯d)∈ℳ(E)×∏i=1dℳ(Fi)( π, ν_1,…, ν_d) (E)× _i=1^dM(F_i) of finite signed Radon measures satisfying ∫ELfπ¯+∑i=1d∫FiDifν¯i=0,f∈Cb2(E). _ELf\,d π+ _i=1^d _F_iD_if\,d ν_i=0, f∈ C_b^2(E). (2.3) The stationary regulator defines finite boundary occupation measures νi0 _i^0, and the stationary BAR is ∫ELfπ0+∑i=1d∫FiDifνi0=0,f∈Cb2(E). _ELf\,d _0+ _i=1^d _F_iD_if\,d _i^0=0, f∈ C_b^2(E). (2.4) Under (2.1)–(2.2), the normalized reflection matrix is of Harrison–Reiman form, and the associated deterministic Skorokhod problem drains to the origin. Hence [16, Theorem 2.6] and [30, Theorem 3.5] gives provides the existences and the uniqueness of stationary distribution π0 _0 to the SRBM. Obviously, the stationary distribution and the finite stationary boundary measure characterized by the following Section 2.2 together provide a solution to the BAR equation (2.4). Proposition 2.1 (Finite stationary boundary measures and stationary BAR). Start the SRBM with Z0∼π0Z_0 _0 and write it in the original normalization as Zt=Z0+μt+Σ1/2Wt+RYt,Z_t=Z_0+μ t+ ^1/2W_t+RY_t, (2.5) where each YiY_i is continuous, nondecreasing, starts from zero, and increases only on FiF_i. Define, for Borel B⊂FiB⊂ F_i, νi0(B)=π0∫01B(Zs)Yi(s). _i^0(B)=E_ _0 _0^11_B(Z_s)\,dY_i(s). (2.6) Then each νi0 _i^0 is a finite measure supported on FiF_i, and (2.4) holds. Proof. Let a=R−Ta=R^-T1. Since R−1≥0R^-1≥ 0 and no column of the invertible matrix R−1R^-1 is zero, a>0a>0; moreover RTa=R^Ta=1. For α>0α>0, set Φα(x)=−∑k=1dake−αxk. _α(x)=- _k=1^da_ke^-α x_k. The function and its first two derivatives are bounded. If x∈Fix∈ F_i, then, using Rki≤0R_ki≤ 0 for k≠ik≠ i, xi=0x_i=0, and e−αxk≤1e^-α x_k≤ 1, DiΦα(x)=α∑k=1dRkiake−αxk≥α∑k=1dRkiak=α.D_i _α(x)=α _k=1^dR_kia_ke^-α x_k≥α _k=1^dR_kia_k=α. Itô’s formula on [0,1][0,1] gives, pathwise, Φα(Z1)−Φα(Z0)=∫01LΦα(Zs)s+M1+∑i=1d∫01DiΦα(Zs)Yi(s), _α(Z_1)- _α(Z_0)= _0^1L _α(Z_s)\,ds+M_1+ _i=1^d _0^1D_i _α(Z_s)\,dY_i(s), where M is a square-integrable martingale because ∇Φα∇ _α is bounded. The first two terms on the right and the left side are integrable. The boundary sum is nonnegative, so the identity itself shows that it is integrable. Taking expectations and using stationarity therefore yields α∑i=1dπ0Yi(1)≤−π0∫01LΦα(Zs)s≤‖LΦα‖∞.α _i=1^dE_ _0Y_i(1)≤-E_ _0 _0^1L _α(Z_s)\,ds≤\|L _α\|_∞. Thus (2.6) is finite. Its support is contained in FiF_i because YiY_i increases only there. Finally, apply Itô’s formula to f∈Cb2(E)f∈ C_b^2(E). The boundedness of f, ∇f∇ f and D2fD^2f makes the Brownian and drift terms integrable on [0,1][0,1], and the boundary integrals are integrable by the preceding estimate. Stationarity gives 0=∫ELfπ0+∑i=1d∫FiDifνi0.0= _ELf\,d _0+ _i=1^d _F_iD_if\,d _i^0. ∎ Although Section 2.2 provides (π0,ν10,…,νd0)( _0, _1^0,…, _d^0) as a solution to the BAR equation, it remains an open question whether the BAR uniquely characterizes the stationary distribution of the diffusion process. 2.3 Signed BAR uniqueness in the Harrison-Reiman Class In the Harrison-Reiman Class, i.e. under the standing assumptions (2.1)–(2.2), we show that the associated BAR uniquely characterizes the stationary distribution of the diffusion process. Theorem 2.2 (Signed BAR uniqueness). Under the standing assumptions (2.1)–(2.2), let π0 _0 and (νi0)i=1d( _i^0)_i=1^d be the stationary distribution of the process and the corresponding boundary measure constructed in Section 2.2. Every finite signed BAR tuple is a scalar multiple of the stationary BAR tuple. More precisely, if (2.3) holds, then there exists c∈ℝc such that π¯=cπ0,ν¯i=cνi0,i=1,…,d. π=c _0, ν_i=c _i^0, i=1,…,d. (2.7) Consequently the vector space of finite signed BAR tuples is one-dimensional. To prove uniqueness of finite signed BAR tuples, we first show that every BAR tuple satisfies a resolvent identity (RI); we call this identity resolvent insertion. The resolvent insertion identity implies invariance of the interior signed measure under the reflected semigroup, and hence π¯=cπ0 π=c _0. After subtracting the interior stationary BAR, the remaining identity is purely on the boundary, and pure boundary injectivity gives ν¯i=cνi0 ν_i=c _i^0 for all i=1,…,di=1,…,d. Proposition 2.3 (Resolvent identity criterion). Assume that for every finite signed BAR tuple, every h∈C0(E)h∈ C_0(E), and every λ>0λ>0, ∫E(λRλh−h)π¯=0. _E(λ R_λh-h)\,d π=0. (RI) Then the conclusion of Theorem 2.2 holds. The proof of Section 2.3 is given in Section 5. Why the resolvent insertion (RI) should hold The reason for targeting (RI) is transparent from the classical Neumann calculation. Let g=Rλhg=R_λh. If g were an admissible Cb2C_b^2 test and if it satisfied Dig=0D_ig=0 on FiF_i for i=1,…,di=1,…,d, then inserting g into the BAR would give 0=∫ELgπ¯+∑i∫FiDigν¯i=∫ELgπ¯.0= _ELg\,d π+ _i _F_iD_ig\,d ν_i= _ELg\,d π. The resolvent equation (λ−L)g=h(λ-L)g=h would therefore imply ∫E(λRλh−h)π¯=0. _E(λ R_λh-h)\,d π=0. This is only an informal guide. The closed-domain C2C^2 regularity required for this insertion may fail even in the stable Harrison–Reiman class. Appendix A gives an explicit stable nonsingular M-matrix example and a smooth compactly supported h for which Rλh∉C2(E)R_λh∉ C^2(E). The proof below therefore does not try to show that the resolvent belongs to a classical oblique-Neumann core. 2.4 Making the resolvent insertion rigorous The replacement for the formal insertion is a measure-level Neumann approximation. For smooth compactly supported h we construct tests gε∈Cb2(E)g_ ∈ C_b^2(E) such that, as ε↓0 0, gε→Rλh,(λ−L)gε→h,g_ → R_λh, (λ-L)g_ → h, against every finite signed interior measure, while ∫FiDigεν¯i→0,i=1,…,d, _F_iD_ig_ \,d ν_i→ 0, i=1,…,d, for every finite signed boundary measure. This is exactly what is needed to pass to the limit in the BAR. The convergence is not a pointwise assertion that RλhR_λh admits a classical oblique derivative DiRλhD_iR_λh on FiF_i; it is an assertion that the boundary pairings seen by the BAR vanish. Section 4 gives the precise statement, and a density argument then extends (RI) from smooth compactly supported h to all h∈C0(E)h∈ C_0(E). The next two sections supply the projected derivative input and the one-sided smoothing construction. Figure 1 summarizes where this approximation sits in the proof: the analytic work proves the target resolvent identity, while the remaining steps are the soft semigroup and boundary-identification arguments. π¯(h)=λπ¯(Rλh) π(h)=λ π(R_λh) leads to signed BAR uniqueness Proving π¯(h)=λπ¯(Rλh) π(h)=λ π(R_λh): how the missing resolvent boundary regularity is bypassed Target identity: the hard step For every h∈Cc∞(ℝd)h∈ C_c^∞(R^d) and λ>0λ>0, π¯(h)=λπ¯(Rλh). π(h)=λ\, π(R_λh). This is the resolvent form of stationarity. Laplace uniqueness converts resolvents to invariance Define the defect Ah(t):=π¯(Pth)−π¯(h).A_h(t):= π(P_th)- π(h). Using Rλh=∫0∞e−λtPthtR_λh= _0^∞e^-λ tP_th\,dt, Fubini turns the target identity into ∫0∞e−λtAh(t)t=0(λ>0). _0^∞e^-λ tA_h(t)\,dt=0 (λ>0). Semigroup invariance Thus π¯(Pth)=π¯(h) π(P_th)= π(h) for all t≥0t≥ 0. Density of Cc∞|EC_c^∞|_E in C0(E)C_0(E), plus contraction of PtP_t, extends this to all ϕ∈C0(E)φ∈ C_0(E). Hence π¯Pt=π¯ πP_t= π. Interior measure is then forced For a finite signed invariant measure, positivity gives |μPt|≤|μ|Pt|μ P_t|≤|μ|P_t. Total mass equality makes the Jordan parts invariant. Uniqueness of π0 _0 gives π¯=cπ0 π=c _0. Boundary measures then follow Subtract c times the stationary BAR. On each stratum SAS_A, RAA−TR_A^-T prescribes the active oblique jets (Dif)i∈A(D_if)_i∈ A. Induction over |A||A| gives ν¯i=cνi0 ν_i=c _i^0. Signed BAR uniqueness (π¯,ν¯1,…,ν¯d)=c(π0,ν10,…,νd0). ( π, ν_1,…, ν_d)=c( _0, _1^0,…, _d^0). π¯=cπ0 π=c _0 Why direct insertion fails The natural test is g=Rλhg=R_λh, because (λ−L)g=h(λ-L)g=h in the interior. But the BAR accepts bounded C2C^2 tests and boundary terms DifD_if. At corners, g need not have a classical ambient gradient, so Dig=0D_ig=0 is unavailable. Projected derivative replaces a boundary gradient For feasible inward directions, ∂w+g(x)=Λx(xw). _w^+g(x)= _x( L_xw). If x∈Fix∈ F_i, then xRi=0 L_xR_i=0, hence the ambient linear extension satisfies ℓx(Ri)=0 _x(R_i)=0. This is the usable oblique information. One-sided smoothing turns it into BAR tests Smooth only from inside: gε(x)=∫ρ(w)g(x+εw)w.g_ (x)= ρ(w)g(x+ w)\,dw. The support suppρ⋐(0,∞)dsuppρ (0,∞)^d keeps every sampled direction feasible. Integration by parts and domination give Digε→0D_ig_ → 0 on FiF_i. Measure–Neumann approximation Use gε∈Cb2(E)g_ ∈ C_b^2(E) directly. Apply the signed BAR and send ε↓0 0. Boundary integrals vanish for every finite signed ν¯i ν_i; interior terms converge to π¯(h)=λπ¯(Rλh) π(h)=λ π(R_λh). Figure 1: Proof architecture for the uniqueness of the Harrison-Reiman class. The lower half is the analytic insertion mechanism: Section 3 supplies the projected derivative used by the one-sided smoothing, and Section 4 turns the smoothed functions into admissible BAR tests. The upper half is the soft reduction: the resulting resolvent identity gives semigroup invariance, then signed uniqueness of the interior measure and finally the boundary measures. The diagram also shows why the proof first studies the nonsmooth resolvent before carrying out the smoothing. For fixed h and λ, let g=Rλhg=R_λh. The smoothed BAR tests used later are gε(x)=∫ρ(w)g(x+εw)w,suppρ⊂(1,2)d.g_ (x)= ρ(w)g(x+ w)\,dw, ρ⊂(1,2)^d. They must approximate g in the interior equation while also satisfying an asymptotic oblique-Neumann condition on each face: Digε→0D_ig_ → 0 in pairings with arbitrary finite signed measures on FiF_i. This is why Section 3 proves a boundary statement for g itself before any smoothing is introduced. Although g need not be C2C^2 on the closed orthant, its feasible one-sided derivatives exist at boundary points and factor through the active tangent projection; the resulting linear extension ℓx _x satisfies ℓx(Ri)=0 _x(R_i)=0 on active faces, acting as an analog to the classical gradient. The one-sided convolution gεg_ in Section 4 is then precisely designed to inherit this first-order oblique flatness in the weaker, measure-level form needed by the BAR. 3 Resolvent regularity and projected boundary derivatives The goal of this section is to prove Section 3, the input that makes the measure–Neumann approximation in Section 4 possible. Section 4 will construct gε∈Cb2(E)g_ ∈ C_b^2(E) from g=Rλhg=R_λh and will need three properties: gε→g_ → g, (λ−L)gε→h(λ-L)g_ → h, and Digε→0D_ig_ → 0 on FiF_i after integration against arbitrary finite signed boundary measures. The first two properties come from interior smoothing and the interior resolvent equation. The third property comes from the boundary information proved here: at a boundary point, the feasible directional derivative of g factors through the active tangent projection. Combining this factorization with the identity xRi=0 L_xR_i=0 gives the usable oblique information ℓx(Ri)=0 _x(R_i)=0 on FiF_i, which is exactly what later forces Digε→0D_ig_ → 0 in boundary-measure pairings. The proof has two ingredients. The algebraic ingredient is the nonsingularity of every active principal reflection block, which gives the explicit projection x L_x. The stochastic ingredient is external: the Lipshutz–Ramanan initial-condition derivative theorem for the normalized Harrison–Reiman reflected diffusion, together with well posedness, strong-continuity property, and the synchronous Lipschitz estimate. The source-to-assumption conversion is carried out in the proof of Theorem 3.3, where each source hypothesis is recalled in the present orthant specialization and verified with a self-contained argument; no unlisted regularity or boundary conclusion is used. For x∈Ex∈ E, define I(x)=i∈J:xi=0,I(x)=\i∈ J:x_i=0\, and put Gx=w∈ℝd:wi≥0 for i∈I(x),Hx=v∈ℝd:vi=0 for i∈I(x). G_x=\w ^d:w_i≥ 0 for i∈ I(x)\, H_x=\v ^d:v_i=0 for i∈ I(x)\. (3.1) Proposition 3.1 (Resolvent regularity and projected boundary derivatives). Let λ>0λ>0 and let h∈Cc∞(ℝd)h∈ C_c^∞(R^d) be regarded as a function on E. Define g(x)=Rλh(x):=[∫0∞e−λth(Ztx)dt],g(x)=R_λh(x):=E [ _0^∞e^-λ th(Z_t^x)\,dt ], then we have: (i) g is bounded and globally Lipschitz on E. (i) g is a classical solution of the resolvent equation in E∘E ; more precisely, g∈C∞(E∘)g∈ C^∞(E ) and (λ−L)g=h(λ-L)g=h in E∘E . (i) At each x∈Ex∈ E, feasible one-sided directional derivatives ∂w+g(x) _w^+g(x) exist for w∈Gxw∈ G_x. (iv) If A=I(x)A=I(x), then the principal-block projection xv=v−RARAA−1vA L_xv=v-R_AR_A^-1v_A maps ℝdR^d onto HxH_x, and there is a linear functional Λx:Hx→ℝ _x:H_x such that ∂w+g(x)=Λx(xw),w∈Gx. _w^+g(x)= _x( L_xw), w∈ G_x. (v) With ℓx(v)=Λx(xv) _x(v)= _x( L_xv), we have ℓx(Ri)=0 _x(R_i)=0, for i∈I(x).i∈ I(x). 3.1 Matrix normalization and active-set projections Normalize the reflection directions by Δ=diag(R11,…,Rdd),R^=RΔ−1,di=R^i=Ri/Rii. =diag(R_11,…,R_d), R=R ^-1, d_i= R_i=R_i/R_i. Positive rescaling of a reflection direction only rescales its regulator and does not change the reflected path. Lemma 3.2 (Principal block projection). The normalized matrix has the Harrison–Reiman form R^=I−PT,P≥0,ρ(P)<1. R=I-P^T, P≥ 0, ρ(P)<1. (3.2) Every principal submatrix RAAR_A is a nonsingular M-matrix and RAA−1≥0R_A^-1≥ 0. In particular, for every nonempty A⊂JA⊂ J, the active directions di:i∈A\d_i:i∈ A\ are linearly independent. For A⊂JA⊂ J, define LAv=v−RARAA−1vA,L_Av=v-R_AR_A^-1v_A, (3.3) with L∅L_ equal to the identity. If A=I(x)A=I(x), then LA=xL_A= L_x is the (unique) linear map from ℝdR^d to HxH_x such that xv−v∈spanRi:i∈A L_xv-v \R_i:i∈ A\. Moreover, LARi=0,i∈A,L_AR_i=0, i∈ A, (3.4) and C:=maxA⊂J‖I−RARAA−1πA‖<∞,C_ L:= _A⊂ J \|I-R_AR_A^-1 _A \|<∞, (3.5) where πAv=vA _Av=v_A and the expression for A=∅A= is the identity. Proof. The diagonal of R R is one and its off-diagonal entries are nonpositive, so PT:=I−R^P^T:=I- R is nonnegative. Also R^−1=ΔR−1≥0. R^-1= R^-1≥ 0. By Perron–Frobenius, in the standard nonnegative-matrix form summarized for example in [1, Chapter 2], PTP^T has a nonzero vector v≥0v≥ 0 with PTv=ρ(P)vP^Tv=ρ(P)v. If ρ(P)=1ρ(P)=1, then R^v=0 Rv=0, contradicting invertibility. If ρ(P)>1ρ(P)>1, then R^v=(1−ρ(P))v≤0 Rv=(1-ρ(P))v≤ 0; multiplying by R^−1≥0 R^-1≥ 0 gives v≤0v≤ 0, again a contradiction. Hence ρ(P)<1ρ(P)<1. For a principal index set A, the principal block (PT)AA(P^T)_A is nonnegative and ρ((PT)AA)≤ρ(PT)<1ρ((P^T)_A)≤ρ(P^T)<1. One direct verification of the inequality is that ((PT)AA)n((P^T)_A)^n is entrywise bounded by the AA block of (PT)n(P^T)^n, after which Gelfand’s formula applies. Therefore (IA−(PT)AA)−1=∑n=0∞((PT)AA)n≥0.(I_A-(P^T)_A)^-1= _n=0^∞((P^T)_A)^n≥ 0. Since RAA=(IA−(PT)AA)ΔAR_A=(I_A-(P^T)_A) _A, it follows that RAA−1=ΔA−1(IA−(PT)AA)−1≥0.R_A^-1= _A^-1(I_A-(P^T)_A)^-1≥ 0. Linear independence of the active normalized columns follows by restricting a relation ∑i∈Aaidi=0 _i∈ Aa_id_i=0 to rows in A. The maximum in (3.5) is finite because the active-set lattice is finite. For the projection claim, a vector of the form v−RAav-R_Aa belongs to HxH_x exactly when vA−RAAa=0v_A-R_Aa=0. The preceding paragraph gives a=RAA−1vAa=R_A^-1v_A. If v=Riv=R_i with i∈Ai∈ A, then vA=RAAeiv_A=R_Ae_i, which proves (3.4). ∎ 3.2 Regularity of SRBM This subsection proves Theorem 3.3, the stochastic regularity statement used in Section 3. More specifically, for g=Rλhg=R_λh, we will need the reflected semigroup on C0(E)C_0(E), a synchronous Lipschitz estimate for paths driven by the same Brownian motion, and a pathwise derivative with respect to the initial condition. The derivative statement is the key boundary input: at a boundary point, the initial perturbation is projected onto the active tangent space, and the active reflection directions are killed by this projection. This is the stochastic origin of the oblique flatness used in the one-sided smoothing argument. These properties follow from the reflected-diffusion results in [29, 30]. Those results are formulated for simple polyhedral domains with normalized reflection directions. Our SRBM is the constant-coefficient orthant case of that framework, after a harmless normalization of the reflection directions. Set Δ=diag(R11,…,Rdd),R^=RΔ−1,di=R^i=RiRii. =diag(R_11,…,R_d), R=R ^-1, d_i= R_i= R_iR_i. Then ⟨di,ei⟩=1 d_i,e_i =1. Replacing RiR_i by the positive multiple di=Ri/Riid_i=R_i/R_i only rescales the iith regulator coordinate and leaves the reflected path unchanged. Therefore pathwise statements proved for the normalized matrix R R apply to the original BAR normalization R. Theorem 3.3 records the regularity consequences needed for the proof of Section 3. Its proof first places the present SRBM into the notation of [29, 30], then verifies the relevant hypotheses under (2.1)–(2.2), and finally applies the corresponding existence, Lipschitz, and derivative results of [29, 30]. Theorem 3.3 (Regularity of SRBM). Under the standing assumptions (2.1)–(2.2), the following hold. (i) For each x∈Ex∈ E and each prescribed Brownian motion there is a pathwise unique SRBM ZxZ^x, and ZxZ^x is strong Markov. (i) The semigroup (Pt)(P_t) maps C0(E)C_0(E) into itself and is strongly continuous there. (i) There is a constant KΓ<∞K_ <∞, depending only on the normalized reflection data, such that synchronous solutions satisfy, for all x,y∈Ex,y∈ E and t≥0t≥ 0, sup0≤s≤t|Zsx−Zsy|≤KΓ|x−y|almost surely. _0≤ s≤ t Z_s^x-Z_s^y ≤ K_ x-y surely. (3.6) (iv) For every x∈Ex∈ E there is an adapted RCLL derivative process tx∈Lin(Hx,ℝd) J_t^x (H_x,R^d), t≥0t≥ 0. For each fixed w∈Gxw∈ G_x, on an event of probability one the derivative ∂wZtx:=limε↓0Ztx+εw−Ztxε _wZ_t^x:= _ 0 Z_t^x+ w-Z_t^x exists for every t≥0t≥ 0. Moreover, for every fixed t>0t>0, ∂wZtx=tx[xw]almost surely. _wZ_t^x= J_t^x[ L_xw] surely. (3.7) (v) For every x∈Ex∈ E and every fixed u∈Hxu∈ H_x, |tx[u]|≤KΓ|u|for dt⊗ℙ-almost every (t,ω). J_t^x[u] ≤ K_ u dt -almost every (t,ω). To prove Theorem 3.3, we use the following results for reflected diffusions in simple polyhedra [29, 30]. The general framework of [29, 30] is a more flexible version of the same reflected-diffusion equation: it allows a simple polyhedral domain, normalized face directions, and parameter-dependent coefficients. Our orthant SRBM is obtained from that framework by taking constant coefficients and normalized columns di=Ri/Riid_i=R_i/R_i; the only difference from the BAR notation is the harmless positive rescaling of the regulator coordinates. Recall that [29, 30] use the following notation for the general SRBM framework, where we specialized the notation to the spatially homogeneous case. Let parameters α∈Uα∈ U, where the parameter family U is open, and let G=⋂i∈Jx:⟨x,ni⟩≥ci,J=1,…,d,G= _i∈ J\x: x,n_i ≥ c_i\, J=\1,…,d\, be a minimally represented simple polyhedron with unit inward normals nin_i, faces Fi=x∈G:⟨x,ni⟩=ciF_i=\x∈ G: x,n_i =c_i\, and active set IG(x)=i:x∈FiI_G(x)=\i:x∈ F_i\. The normalized reflection directions satisfy ⟨di(α),ni⟩=1 d_i(α),n_i =1. In the spatially-homogeneous-coefficient specialization we care about, the family of reflected diffusions parameterized by α is written as Ztα,x=x+b(α)t+σ(α)Wt+∑i∈Jdi(α)Yiα,x(t),Z_t^α,x=x+b(α)t+σ(α)W_t+ _i∈ Jd_i(α)Y_i^α,x(t), where each Yiα,xY_i^α,x is continuous, nondecreasing, starts from zero, and increases only when Zα,x∈FiZ^α,x∈ F_i. Put a(α)=σ(α)σ(α)Ta(α)=σ(α)σ(α)^T, N=(n1,…,nd)N=(n_1,…,n_d), and (α)=(d1(α),…,d(α))D(α)=(d_1(α),…,d_d(α)). For x∈Gx∈ G, define CG(x)=w:⟨w,ni⟩≥0,i∈IG(x)C_G(x)=\w: w,n_i ≥ 0,\ i∈ I_G(x)\ and HG(x)=v:⟨v,ni⟩=0,i∈IG(x)H_G(x)=\v: v,n_i =0,\ i∈ I_G(x)\. In the orthant specialization, G=EG=E, ni=ein_i=e_i, CG(x)=GxC_G(x)=G_x, and HG(x)=HxH_G(x)=H_x. Then, [29, 30] gives the following proposition. Proposition 3.4 (Reflected diffusions in simple polyhedra). In the setting just described, fix α∈Uα∈ U. Assume the following hypotheses. (A1) G is minimally represented and simple, U is open, and α↦di(α)α d_i(α), b(α)b(α), and σ(α)σ(α) are C1C^1 with bounded first derivatives and local Hölder regularity. (A2) a(α)a(α) is uniformly elliptic: vTa(α)v≥θ|v|2v^Ta(α)v≥θ|v|^2 for some θ>0θ>0 and all v∈ℝdv ^d. (A3) NT(α)N^TD(α) is a nonsingular M-matrix. (A4) The reflection matrix is constant in the parameter, or more generally ∂α(α) _αD(α) is bounded. Then the following conclusions are available under the assumptions indicated. (C1) (Well posedness; uses (A1) and (A3), [30, Theorem 2.8].) For each x∈Gx∈ G and each prescribed Brownian motion, there is a pathwise unique reflected diffusion Zα,xZ^α,x, and it is strong Markov. (C2) (Lipschitz extended Skorokhod map; uses (A1) and (A3), [29, Proposition 2.6].) The extended Skorokhod problem associated with (G,di(α))(G,d_i(α)) is well posed, and its extended Skorokhod map Γ¯α ^α is Lipschitz on compact time intervals: sups≤t|Γ¯α(f)(s)−Γ¯α(g)(s)|≤KΓsups≤t|f(s)−g(s)| _s≤ t ^α(f)(s)- ^α(g)(s) ≤ K_ _s≤ t f(s)-g(s) for some KΓ<∞K_ <∞, all continuous inputs f,gf,g, and all t≥0t≥ 0. (C3) (Boundary jitter; uses (A2) in the above setting, [29, Theorem 3.3].) Uniform ellipticity implies the boundary jitter property required for the pathwise derivative theorem. (C4) (Derivative projection; uses (A1) and (A3), [29, Lemma 3.11].) For each x∈Gx∈ G there is a unique linear projection ℒxα:ℝd→HG(x)L_x^α:R^d→ H_G(x) such that ℒxαv−v∈spandi(α):i∈IG(x)L_x^αv-v \d_i(α):i∈ I_G(x)\ for every v∈ℝdv ^d. (C5) (Pathwise differentiability; uses (A1)–(A4) and (C3)–(C4), [29, Theorem 3.13 and Corollary 3.15].) For this theorem, note that (A3) implies Condition 2.10 of [29]: if A⊂JA⊂ J and ∑i∈Acidi(α)=0 _i∈ Ac_id_i(α)=0, then (NT(α))AAcA=0(N^TD(α))_Ac_A=0, and every principal submatrix of a nonsingular M-matrix is nonsingular, so cA=0c_A=0. Hence [29, Lemma 3.9] gives the exceptional set of this theorem α=∅W^α= . Therefore, for each x∈G\α=Gx∈ G ^α=G there is an adapted RCLL derivative process Jtα,x∈Lin(HG(x),ℝd)J_t^α,x (H_G(x),R^d). For every fixed w∈CG(x)w∈ C_G(x), we have almost surely, ∂wZtα,x:=limε↓0ε−1(Ztα,x+εw−Ztα,x) _wZ_t^α,x:= _ 0 ^-1(Z_t^α,x+ w-Z_t^α,x) exists for every t≥0t≥ 0, is continuous at every t>0t>0 such that Ztα,x∈G∘Z_t^α,x∈ G , and its right-continuous regularization satisfies lims↓t∂wZsα,x=Jtα,x[ℒxαw] _s t _wZ_s^α,x=J_t^α,x[L_x^αw] for all t≥0t≥ 0. (C6) (Fixed-time interior statement; uses (A1)–(A4) and (C3)–(C4), [29, Lemma 4.13].) For every fixed t>0t>0, ℙ(Ztα,x∈G∘)=1P(Z_t^α,x∈ G )=1. Proof of Theorem 3.3. We apply Section 3.2 to the constant parameter family G=E,ni=ei,ci=0,di(α)=di=RiRii,b(α)=μ,σ(α)=Σ1/2,α∈U=(−1,1).G=E, n_i=e_i, c_i=0, d_i(α)=d_i= R_iR_i, b(α)=μ, σ(α)= ^1/2, α∈ U=(-1,1). Thus (α)=R^D(α)= R and N=IN=I. Here constant parameter family means that the domain, reflection directions, drift, and dispersion do not depend on α. Now we verify that the Harrison-Reiman Class satisfies all assumptions (A1)-(A4). Verification of (A1). The orthant is the minimally represented simple cone E=⋂ix:⟨x,ei⟩≥0E= _i\x: x,e_i ≥ 0\. Simplicity follows because the coordinate normals are linearly independent on every active set. Minimality follows because, if the iith half-space is removed, then the point −ei-e_i satisfies all remaining half-space inequalities but does not belong to E. The parameter set U=(−1,1)U=(-1,1) is open. The maps di(α)d_i(α), b(α)b(α), and σ(α)σ(α) are constant, hence C1C^1; all first derivatives are zero, and therefore bounded and locally Hölder. The normalization ⟨di,ei⟩=1 d_i,e_i =1 holds by definition. This verifies (A1). Verification of (A2). Here a(α)=Σa(α)= . Since Σ is symmetric positive definite, vTΣv≥λmin(Σ)|v|2,v∈ℝd.v^T v≥ _ ( )|v|^2, v ^d. Thus (A2) holds with θ=λmin(Σ)>0θ= _ ( )>0. Verification of (A3). Here N=IN=I, (α)=R^=I−PTD(α)= R=I-P^T, and hence NT(α)=R^N^TD(α)= R. By Section 3.1, R R is a nonsingular M-matrix. This verifies (A3). Verification of (A4). The reflection matrix (α)=R^D(α)= R is constant in α, so ∂α(α)=0 _αD(α)=0. This verifies (A4). All assumptions (A1)–(A4) of Section 3.2 have now been verified. Conclusion (C1) gives pathwise existence, uniqueness, and the strong Markov property for the normalized reflected diffusion. Since Ri=RiidiR_i=R_id_i with Rii>0R_i>0, replacing the normalized local time by the correspondingly rescaled regulator leaves the reflected path unchanged. Hence the same pathwise existence, uniqueness, and strong Markov conclusions also hold for the original BAR normalization R. This proves assertion (i). For assertion (i), let the two processes start from x and y and be driven by the same Brownian path. Their free inputs are fx(s)=x+μs+Σ1/2Wsf_x(s)=x+μ s+ ^1/2W_s and fy(s)=y+μs+Σ1/2Wsf_y(s)=y+μ s+ ^1/2W_s, so sups≤t|fx(s)−fy(s)|=|x−y| _s≤ t|f_x(s)-f_y(s)|=|x-y|. Applying conclusion (C2) gives (3.6), proving assertion (i). Assertion (i) follows from (3.6) and Brownian continuity. Put Xt=μt+Σ1/2WtX_t=μ t+ ^1/2W_t. Comparing the input x+Xx+X with the constant input x gives sups≤t|Zsx−x|≤KΓsups≤t|Xs|almost surely. _s≤ t Z_s^x-x ≤ K_ _s≤ t X_s surely. (3.8) If h∈C0(E)h∈ C_0(E), then h is uniformly continuous. Hence (3.6) gives continuity of x↦Pth(x)x P_th(x). If h is supported in the ball of radius r, then |Pth(x)|≤‖h‖∞ℙ(KΓsups≤t|Xs|≥|x|−r),|P_th(x)|≤\|h\|_∞P\! (K_ _s≤ t|X_s|≥|x|-r ), which tends to zero as |x|→∞|x|→∞; approximation by compactly supported functions gives PtC0(E)⊂C0(E)P_tC_0(E)⊂ C_0(E). Finally, if ωh _h is the modulus of continuity of h, then (3.8) gives supx∈E|Pth(x)−h(x)|≤ωh(KΓsups≤t|Xs|)⟶0(t↓0) _x∈ E|P_th(x)-h(x)| \, _h\! (K_ _s≤ t|X_s| ) 0 (t 0) by bounded convergence. Thus (Pt)(P_t) is strongly continuous on C0(E)C_0(E). This proves assertion (i). We now prove assertions (iv)–(v). By conclusion (C4), the derivative projection at x is the unique linear map onto HxH_x whose difference from the identity lies in spandi:i∈I(x)span\d_i:i∈ I(x)\. Since spandi:i∈A=spanRi:i∈Aspan\d_i:i∈ A\=span\R_i:i∈ A\ for every A⊂JA⊂ J, uniqueness and Section 3.1 identify this projection with x L_x. Conclusion (C5), applied to the constant parameter family and with parameter direction equal to zero, gives the derivative process tx J_t^x and the directional derivative ∂wZtx _wZ_t^x for every fixed w∈Gxw∈ G_x. It also gives continuity of s↦∂wZsxs _wZ_s^x at every t>0t>0 such that Ztx∈E∘Z_t^x∈ E , together with the projected right-continuous regularization. Conclusion (C6) gives ℙ(Ztx∈E∘)=1P(Z_t^x∈ E )=1 for every fixed t>0t>0. Therefore, for every fixed w∈Gxw∈ G_x and t>0t>0, on an event of probability one, ∂wZtx=lims↓t∂wZsx=tx[xw]. _wZ_t^x= _s t _wZ_s^x= J_t^x[ L_xw]. This proves assertion (iv). For assertion (v), fix u∈Hxu∈ H_x. Then u∈Gxu∈ G_x and xu=u L_xu=u. For small ε>0 >0, x+εu∈Ex+ u∈ E. Applying (3.6) with y=x+εuy=x+ u, dividing by ε , and letting ε↓0 0 gives |∂uZtx|≤KΓ|u| _uZ_t^x ≤ K_ u on the event where the directional derivative exists for all t≥0t≥ 0. Combining this bound with (3.7) gives |tx[u]|≤KΓ|u| J_t^x[u] ≤ K_ u for every fixed t>0t>0, almost surely. Since x J^x is RCLL, Fubini gives the same bound for dt⊗ℙdt -almost every (t,ω)(t,ω). This proves assertion (v) and completes the proof. ∎ 3.3 The probabilistic resolvent and its boundary directional derivative The purpose of this subsection is to convert the pathwise derivative package into a boundary identity for the probabilistic resolvent. We prove only that the resolvent is a classical solution of the resolvent equation in the interior, then differentiate the time integral in feasible directions. The resulting boundary derivative is an algebraic linear functional; no classical gradient at a corner is assumed. Fix λ>0λ>0 and h∈Cc∞(ℝd)h∈ C_c^∞(R^d). We regard h as a function on E and define g(x)=Rλh(x):=[∫0∞e−λth(Ztx)t].g(x)=R_λh(x):=E [ _0^∞e^-λ th(Z_t^x)\,dt ]. (3.9) Lemma 3.5 (Boundedness and Lipschitz continuity). The function g is bounded and globally Lipschitz on E, with ‖g‖∞≤‖h‖∞λ,Lip(g)≤KΓLip(h)λ. g _∞≤ h _∞λ, (g)≤ K_ Lip(h)λ. (3.10) Proof. The first estimate follows immediately from (3.9). For the second, couple ZxZ^x and ZyZ^y with the same Brownian path. By (3.6), |h(Ztx)−h(Zty)|≤Lip(h)KΓ|x−y|. h(Z_t^x)-h(Z_t^y) (h)K_ x-y . Integrating against e−λtdte^-λ t\,dt proves the claim. ∎ Lemma 3.6 (Interior classical solution of the resolvent equation). The function g is a classical solution of the resolvent equation in E∘E ; more precisely, g∈C∞(E∘)g∈ C^∞(E ) and (λ−L)g=hin E∘.(λ-L)g=h E . (3.11) Proof. Fix concentric balls B′⋐B⋐E∘B B E and let τB _B be the first exit time from B. Before τB _B, the reflected process is the unconstrained diffusion with generator L. The strong Markov property gives, for x∈Bx∈ B, g(x)=x[∫0τBe−λth(Zt)t+e−λτBg(ZτB)].g(x)=E_x [ _0 _Be^-λ th(Z_t)\,dt+e^-λ _Bg(Z_ _B) ]. (3.12) Since g∈C(B¯)g∈ C( B) and ∂B∂ B is compact, Stone–Weierstrass applied to the restrictions of polynomials on ℝdR^d to ∂B∂ B gives polynomials pnp_n with ‖pn−g‖L∞(∂B)→0\|p_n-g\|_L^∞(∂ B)→ 0. Setting φn=pn|B¯ _n=p_n|_ B, we have φn∈C∞(B¯)⊂C2,α(B¯) _n∈ C^∞( B)⊂ C^2,α( B) and φn→g|∂B _n→ g|_∂ B uniformly. To solve the interior Dirichlet problems we use the classical Schauder solvability theorem, which we recall in the form used. Theorem 3.7 ([20, Theorem 6.14]). Let Ω⊂ℝd ^d be a bounded C2,αC^2,α domain and let =aijDij+biDi+cA=a^ijD_ij+b^iD_i+c be strictly elliptic on Ω , that is, aij(y)ξiξj≥θ0|ξ|2a^ij(y) _i _j≥ _0 ξ ^2 for some θ0>0 _0>0 and all y∈Ωy∈ , ξ∈ℝdξ ^d, with coefficients aij,bi,c∈Cα(Ω¯)a^ij,b^i,c∈ C^α( ) and c≤0c≤ 0 on Ω . Then for every f∈Cα(Ω¯)f∈ C^α( ) and every φ∈C2,α(Ω¯) ∈ C^2,α( ) the Dirichlet problem u=fAu=f in Ω , u=φu= on ∂Ω∂ , has a unique solution u∈C2,α(Ω¯)u∈ C^2,α( ). We apply this with Ω=B =B, =L−λA=L-λ, so that in coordinates aij=12Σija^ij= 12 _ij, bi=μib^i= _i, c=−λc=-λ, together with f=−hf=-h and φ=ϕn = _n. The four hypotheses hold in the present setting: (a) B is an open Euclidean ball, hence a C∞C^∞ and a fortiori C2,αC^2,α domain. (b) For all ξ∈ℝdξ ^d, aijξiξj=12ξ⊤Σξ≥12λmin(Σ)|ξ|2a^ij _i _j= 12ξ ξ≥ 12 _ ( ) ξ ^2, and λmin(Σ)>0 _ ( )>0 because Σ is symmetric positive definite; thus A is strictly elliptic with θ0=12λmin(Σ) _0= 12 _ ( ). (c) The coefficients 12Σij 12 _ij, μi _i, −λ-λ are constants, hence lie in Cα(B¯)C^α( B) with vanishing Hölder seminorm; and c=−λ<0≤0c=-λ<0≤ 0 since λ>0λ>0. (d) The source f=−h∈Cc∞(ℝd)⊂Cα(B¯)f=-h∈ C_c^∞(R^d)⊂ C^α( B), and each φ=ϕn∈C∞(B¯)⊂C2,α(B¯) = _n∈ C^∞( B)⊂ C^2,α( B). Therefore [20, Theorem 6.14] yields a unique un∈C2,α(B¯)u_n∈ C^2,α( B) satisfying (λ−L)un=hin B,un=ϕnon ∂B.(λ-L)u_n=h B, u_n= _n ∂ B. Apply Itô’s formula to e−λ(t∧τB)un(Zt∧τB)e^-λ(t _B)u_n(Z_t _B). Since unu_n and its first derivatives are bounded on B¯ B, the stopped stochastic integral has mean zero. Letting t→∞t→∞ is justified by bounded convergence. Indeed, before τB _B the process is x+μt+Σ1/2Wtx+μ t+ ^1/2W_t; a nonzero one-dimensional projection is a Brownian motion with drift and exits the bounded projection of B almost surely, so τB<∞ _B<∞ almost surely. This gives the Feynman–Kac representation un(x)=x[∫0τBe−λth(Zt)t+e−λτBϕn(ZτB)].u_n(x)=E_x [ _0 _Be^-λ th(Z_t)\,dt+e^-λ _B _n(Z_ _B) ]. Comparing it with (3.12) yields ‖un−g‖L∞(B)≤‖ϕn−g‖L∞(∂B)⟶0.\|u_n-g\|_L^∞(B)≤\| _n-g\|_L^∞(∂ B) 0. For n,mn,m, the difference w=un−um∈C2,α(B¯)w=u_n-u_m∈ C^2,α( B) solves the homogeneous equation (λ−L)w=h−h=0(λ-L)w=h-h=0 in B. To pass to the limit we use the interior Schauder estimate, recalled in the form used. Next, we use interior Schauder estiamte to prove unu_n is Cauchy in C2,α(B′)C^2,α(B ): Theorem 3.8 (Interior Schauder estimate [20, Theorem 6.2]). Let =aijDij+biDi+cA=a^ijD_ij+b^iD_i+c be strictly elliptic on a domain Ω⊂ℝd ^d with ellipticity constant θ0>0 _0>0 and coefficients bounded in Cα(Ω)C^α( ) by a constant Θ . If u∈C2,α(Ω)u∈ C^2,α( ) satisfies u=fAu=f with f∈Cα(Ω)f∈ C^α( ), then for every subdomain Ω′⋐Ω , ‖u‖C2,α(Ω′¯)≤C(‖u‖L∞(Ω)+‖f‖Cα(Ω)),C=C(d,α,θ0,Θ,dist(Ω′,∂Ω)). u _C^2,α( )≤ C ( u _L^∞( )+ f _C^α( ) ), C=C (d,α, _0, ,dist( ,∂ ) ). The operator =L−λA=L-λ satisfies these hypotheses with the ellipticity constant θ0=12λmin(Σ) _0= 12 _ ( ) of (b) and the coefficient bound Θ=max12‖Σ‖,|μ|,λ = \ 12 ,\ μ ,\ λ\, both independent of n,mn,m. Applying it to w=un−umw=u_n-u_m, with Ω=B =B, Ω′=B′ =B , and f≡0f≡ 0, gives ‖un−um‖C2,α(B′¯)≤CB′,B‖un−um‖L∞(B),CB′,B=C(d,α,12λmin(Σ),Θ,dist(B′,∂B)), u_n-u_m _C^2,α( B )≤ C_B ,B\, u_n-u_m _L^∞(B), C_B ,B=C (d,α, 12 _ ( ), ,dist(B ,∂ B) ), where the constant CB′,BC_B ,B does not depend on n,mn,m. Because un→gu_n→ g uniformly on B, the right-hand side tends to zero as n,m→∞n,m→∞. Thus (un)(u_n) is Cauchy in C2,α(B′)C^2,α(B ) and converges there to some u∈C2,α(B′)u∈ C^2,α(B ). The same sequence converges uniformly to g on B, so the C2,α(B′)C^2,α(B ) limit must be u=gu=g. Passing to the limit in the equations satisfied by the classical solutions unu_n gives g∈C2,α(B′)g∈ C^2,α(B ) and (3.11) on B′B . Since the coefficients of L are constant and h is smooth, standard interior elliptic regularity, equivalently the usual bootstrapping by interior estimates, gives g∈C∞(B′)g∈ C^∞(B ). Since B′⋐E∘B E was arbitrary, the conclusion follows. ∎ Proposition 3.9 (Directional factorization of the resolvent). Let x∈Ex∈ E. There is a bounded linear functional Λx:Hx→ℝ _x:H_x such that, for every w∈Gxw∈ G_x, the one sided directional derivative exists and satisfies ∂w+g(x):=limε↓0g(x+εw)−g(x)ε=Λx(xw). _w^+g(x):= _ 0 g(x+ w)-g(x) = _x( L_xw). (3.13) The linear functional ℓx(v):=Λx(xv) _x(v):= _x( L_xv) is bounded by |ℓx(v)|≤KΓC‖∇h‖∞λ|v|| _x(v)|≤ K_ C_ L\|∇ h\|_∞λ|v| (3.14) for all v∈ℝdv ^d. If i∈I(x)i∈ I(x), then ℓx(Ri)=0. _x(R_i)=0. Note that no continuity or measurability of the map x↦ℓx _x is asserted. Proof. Fix x∈Ex∈ E. For u∈Hxu∈ H_x, define Λx(u):=∫0∞e−λt∇h(Ztx)⋅tx[u]t. _x(u):=E _0^∞e^-λ t∇ h(Z_t^x)· J_t^x[u]\,dt. We first record the measurability and integrability facts needed to define Λx _x. Since x J^x is adapted and RCLL with values in the finite dimensional space Lin(Hx,ℝd)Lin(H_x,R^d), for every fixed u∈Hxu∈ H_x the process (t,ω)↦tx[u](ω)(t,ω) J_t^x[u](ω) is progressively measurable, hence ℬ([0,∞))⊗ℱB([0,∞)) -measurable. By Theorem 3.3, for each fixed u∈Hxu∈ H_x, we have |tx[u]|≤KΓ|u|| J_t^x[u]|≤ K_ |u| holds for dt⊗ℙdt -almost every (t,ω)(t,ω). Hence ∫0∞e−λt|∇h(Ztx)⋅tx[u]|t≤KΓ‖∇h‖∞λ|u|<∞E _0^∞e^-λ t |∇ h(Z_t^x)· J_t^x[u] |\,dt≤ K_ \|∇ h\|_∞λ|u|<∞, i.e. the integral defining Λx(u) _x(u) is absolutely convergent with respect to e−λtdt⊗ℙe^-λ tdt , and |Λx(u)|≤KΓ‖∇h‖∞λ|u|,u∈Hx.| _x(u)|≤ K_ \|∇ h\|_∞λ|u|, u∈ H_x. Since tx∈Lin(Hx,ℝd) J_t^x (H_x,R^d), linearity of Λx _x follows from linearity of tx J_t^x and the preceding bound. Thus Λx _x is a bounded linear functional on HxH_x. Now fix w∈Gxw∈ G_x. For all sufficiently small ε>0 >0, x+εw∈Ex+ w∈ E. By the definition of g, g(x+εw)−g(x)ε=∫0∞e−λth(Ztx+εw)−h(Ztx)εt. g(x+ w)-g(x) =E _0^∞e^-λ t h(Z_t^x+ w)-h(Z_t^x) \,dt. (3.15) The synchronous Lipschitz estimate gives the deterministic domination |h(Ztx+εw)−h(Ztx)ε|≤KΓ‖∇h‖∞|w|, | h(Z_t^x+ w)-h(Z_t^x) |≤ K_ \|∇ h\|_∞|w|, (3.16) uniformly for all sufficiently small ε>0 >0, all t≥0t≥ 0, and all sample paths. On the probability one event in Theorem 3.3 corresponding to this fixed pair (x,w)(x,w), the pathwise directional derivative ∂wZtx=limε↓0Ztx+εw−Ztxε _wZ_t^x= _ 0 Z_t^x+ w-Z_t^x exists for every t≥0t≥ 0. Since h∈Cc∞(ℝd)h∈ C_c^∞(R^d), the mean value formula gives h(Ztx+εw)−h(Ztx)ε⟶∇h(Ztx)⋅∂wZtx h(Z_t^x+ w)-h(Z_t^x) ∇ h(Z_t^x)· _wZ_t^x for dt⊗ℙdt -almost every (t,ω)(t,ω). Dominated convergence with respect to e−λtdt⊗ℙe^-λ t\,dt therefore yields ∂w+g(x)=∫0∞e−λt∇h(Ztx)⋅∂wZtxdt. _w^+g(x)=E _0^∞e^-λ t∇ h(Z_t^x)· _wZ_t^x\,dt. (3.17) For every fixed t>0t>0, Theorem 3.3 gives ∂wZtx=tx[xw] _wZ_t^x= J_t^x[ L_xw] almost surely. For fixed x and w, the map (t,ω)↦∂wZtx(ω)(t,ω) _wZ_t^x(ω) is ℬ([0,∞))⊗ℱB([0,∞)) -measurable, since it is the pointwise limit of the continuous-in-t difference quotients ε−1(Ztx+εw−Ztx). ^-1 (Z_t^x+ w-Z_t^x ). The process t↦tx[xw]t J_t^x[ L_xw] is measurable because x J^x is RCLL. Hence the fixed-time almost sure identity can be integrated in t, giving the identity dt⊗ℙdt -almost everywhere. By Fubini, this identity holds for dt⊗ℙdt -almost every (t,ω)(t,ω). The value at t=0t=0 is irrelevant for the time integral. Substituting into the preceding display gives ∂w+g(x)=∫0∞e−λt∇h(Ztx)⋅tx[xw]t=Λx(xw), _w^+g(x)=E _0^∞e^-λ t∇ h(Z_t^x)· J_t^x[ L_xw]\,dt= _x( L_xw), (3.18) which proves the factorization. Finally, by Lemma 3.1, we have |xv|≤C|v|,| L_xv|≤ C_ L|v|, holds for all v∈ℝd.v ^d. Therefore |ℓx(v)|=|Λx(xv)|≤KΓ‖∇h‖∞λ|xv|≤KΓC‖∇h‖∞λ|v|.| _x(v)|=| _x( L_xv)|≤ K_ \|∇ h\|_∞λ| L_xv|≤ K_ C_ L\|∇ h\|_∞λ|v|. If i∈I(x)i∈ I(x), then Lemma 3.1 gives xRi=0 L_xR_i=0, and hence ℓx(Ri)=Λx(xRi)=0. _x(R_i)= _x( L_xR_i)=0. This completes the proof. ∎ Remark 3.10 (Regularity at the boundary). At a boundary point x, the proposition gives a bounded linear functional ℓx _x extending the feasible one-sided derivative. Since g is Lipschitz and w↦ℓx(w)w _x(w) is linear, the ray derivatives imply the cone-wise first-order expansion g(x+ϵw)=g(x)+ϵℓx(w)+ow(ϵ)as ϵ→0,x+ϵw∈E.g(x+ε w)=g(x)+ε _x(w)+o_w(ε) ε→ 0,\ x+ε w∈ E. This expansion is local at the fixed boundary point. It does not assert that x↦ℓx _x is continuous towards the boundary, nor that the interior gradient ∇g(y)∇ g(y) has a limit as y→xy→ x from E∘E . The smoothing argument uses only the algebraic value ℓx(Ri)=0 _x(R_i)=0 on active reflection directions, after the feasible-direction limit has been averaged against the one-sided mollifier. Proof of Section 3. Boundedness and Lipschitz continuity are Section 3.3. The statement that g is a classical solution of the resolvent equation in E∘E is Section 3.3. The projection formula and the identity xRi=0 L_xR_i=0 for i∈I(x)i∈ I(x) are Section 3.1. Finally, Section 3.3 gives the feasible one-sided derivatives, the factorization through x L_x, and the linear extension ℓx _x satisfying ℓx(Ri)=0 _x(R_i)=0 on active faces. ∎ 4 One-sided smoothing and the measure–Neumann approximation In this section we prove the resolvent insertion theorem, Theorem 4.1. We first prove the identity for h∈Cc∞(ℝd)h∈ C_c^∞(R^d), regarded as a function on E, by a one-sided smoothing argument. At the end of the proof, a density argument extends the identity to all h∈C0(E)h∈ C_0(E). Thus, until this final density step throughout this whole section, we fix λ>0λ>0, h∈Cc∞(ℝd)h∈ C_c^∞(R^d), and write g=Rλhg=R_λh as in (3.9). Theorem 4.1 (Resolvent insertion theorem). For every finite signed BAR tuple (π¯,ν¯1,…,ν¯d)( π, ν_1,…, ν_d), every h∈C0(E)h∈ C_0(E), and every λ>0λ>0, ∫E(λRλh−h)π¯=0. _E(λ R_λh-h)\,d π=0. (RI) First we prove the insertion for smooth compactly supported h, then extend it to C0(E)C_0(E) by uniform approximation. A convolution supported strictly inside the orthant produces bounded C2C^2 functions on a neighborhood of the closed state space. Integration by parts in the convolution variable proves vanishing of every oblique boundary derivative, with a bound uniform in the smoothing scale. Since the BAR is imposed on Cb2(E)C_b^2(E), no spatial cutoff is needed in the resolvent insertion. Now we one-sided smoothing the function g. We seek a mollifier ρ∈Cc∞((1,2)d)ρ∈ C_c^∞((1,2)^d) that satisfies ρ≥0ρ≥ 0 and ∫ℝdρ(w)w=1. _R^dρ(w)\,dw=1. For ε>0 >0, define gε(x)=∫ℝdρ(w)g(x+εw)w,hε(x)=∫ℝdρ(w)h(x+εw)w,x∈E.g_ (x)= _R^dρ(w)g(x+ w)\,dw, h_ (x)= _R^dρ(w)h(x+ w)\,dw, x∈ E. (4.1) Because the support of ρ lies strictly inside the positive orthant, gεg_ is defined and smooth on an open neighborhood of E. Lemma 4.2 (One-sided smoothing). gε∈Cb2(E)g_ ∈ C_b^2(E) for each fixed ε>0 >0, gεg_ is smooth on an open neighborhood of E, gε→g_ → g uniformly, gεg_ is uniformly bounded and Lipschitz, and (λ−L)gε=hεon E.(λ-L)g_ =h_ E. (4.2) Moreover, ‖gε−g‖∞ g_ -g _∞ ≤CρεLip(g), ≤ C_ρ (g), (4.3) ‖hε−h‖∞ h_ -h _∞ ⟶0, 0, (4.4) ‖gε‖∞ g_ _∞ ≤‖g‖∞, ≤ g _∞, (4.5) ‖∇gε‖∞ ∇ g_ _∞ ≤CρLip(g), ≤ C_ρLip(g), (4.6) ‖D2gε‖∞ D^2g_ _∞ ≤Cρε−1Lip(g). ≤ C_ρ ^-1Lip(g). (4.7) Proof. We first justify the smoothness of gεg_ . Let δρ:=dist(suppρ,∂ℝ+d)>0. _ρ:=dist(suppρ, _+^d)>0. For each fixed ε>0 >0, define Uε:=x∈ℝd:x+εw∈E∘ for every w∈suppρ.U_ :=\x ^d:x+ w∈ E for every w ρ\. Then UεU_ is an open neighborhood of E, because suppρ⋐(0,∞)dsuppρ (0,∞)^d. Thus gεg_ is well defined on UεU_ . Although g is only known to be globally Lipschitz on E, the derivatives of gεg_ may be computed by integration by parts in the convolution variable. For every multiindex α, ∂xαgε(x)=(−1)|α|ε−|α|∫ℝd∂wαρ(w)g(x+εw)dw,x∈Uε. _x^αg_ (x)=(-1)^|α| ^-|α| _R^d _w^αρ(w)\,g(x+ w)\,dw, x∈ U_ . This identity is first obtained in the sense of distributions on UεU_ . Since the right hand side is continuous in x, it is the classical derivative. Iterating the same argument gives derivatives of all orders; hence gε∈C∞(Uε)g_ ∈ C^∞(U_ ). In particular, gε∈C2(E)g_ ∈ C^2(E) in the closed domain sense. For fixed x∈Ex∈ E, the compact set x+εsuppρx+ ρ lies in E∘E . Same as the proof of Lemma 3.3, g is a classical solution of (λ−L)g=h(λ-L)g=h on this compact subset of the interior. Since L has constant coefficients, differentiating under the integral on this interior compact set gives (λ−L)gε(x)=∫ℝdρ(w)(λ−L)g(x+εw)w=∫ℝdρ(w)h(x+εw)w=hε(x),(λ-L)g_ (x)= _R^dρ(w)(λ-L)g(x+ w)\,dw= _R^dρ(w)h(x+ w)\,dw=h_ (x), which proves (4.2). Next, the global Lipschitz continuity of g gives |gε(x)−g(x)|≤εLip(g)∫|w|ρ(w)w, g_ (x)-g(x) ≤ (g) w ρ(w)\,dw, which is (4.3). Since h∈Cc∞(ℝd)h∈ C_c^∞(R^d), it is uniformly continuous, and therefore ‖hε−h‖∞→0\|h_ -h\|_∞→ 0 which proves gives (4.4). The bound (4.5) follows from ρ≥0ρ≥ 0 and ∫ρ=1 ρ=1: |gε(x)|≤∫ℝdρ(w)|g(x+εw)|w≤‖g‖∞.|g_ (x)|≤ _R^dρ(w)|g(x+ w)|\,dw≤\|g\|_∞. For the gradient, integration by parts in w and ∫ℝd∂wjρ(w)dw=0 _R^d _w_jρ(w)\,dw=0 gives ∂xjgε(x)=−1ε∫∂wjρ(w)(g(x+εw)−g(x))dw. _x_jg_ (x)=- 1 _w_jρ(w) (g(x+ w)-g(x) )\,dw. (4.8) Hence |∂xjgε(x)|≤Lip(g)∫ℝd|∂wjρ(w)||w|w.| _x_jg_ (x)| (g) _R^d| _w_jρ(w)|\,|w|\,dw. The Lipschitz bound on the difference proves (4.6). Differentiating once more in the same distributional-convolution formula and subtracting the constant g(x)g(x) gives ∂xjxk2gε(x)=1ε2∫∂wjwk2ρ(w)(g(x+εw)−g(x))dw, _x_jx_k^2g_ (x)= 1 ^2 _w_jw_k^2ρ(w) (g(x+ w)-g(x) )\,dw, and therefore |∂xjxk2gε(x)|≤ε−1Lip(g)∫ℝd|∂wjwk2ρ(w)||w|w.| _x_jx_k^2g_ (x)|≤ ^-1Lip(g) _R^d| _w_jw_k^2ρ(w)|\,|w|\,dw. Taking the maximum over j,kj,k proves (4.7). Thus gεg_ , its first derivatives, and its second derivatives are bounded on E, so gε∈Cb2(E)g_ ∈ C_b^2(E). ∎ The next proposition is where the projected derivative information for g is used at the boundary. It proves two facts on each face FiF_i: first, Digε(x)→0D_ig_ (x)→ 0 for every x∈Fix∈ F_i; second, the uniform bound in (4.9) holds. Together these imply the boundary measure convergence in (4.10) by dominated convergence. The pointwise limit is obtained from Section 3.3: if x∈Fix∈ F_i and w∈suppρ⊂(0,∞)dw ρ⊂(0,∞)^d, then w∈Gxw∈ G_x and (g(x+εw)−g(x))/ε→ℓx(w)(g(x+ w)-g(x))/ → _x(w). After integration by parts in the smoothing variable, the limit of Digε(x)D_ig_ (x) becomes ℓx(Ri) _x(R_i), which is zero because xRi=0 L_xR_i=0 on active faces. Proposition 4.3 (Vanishing oblique derivative after one sided smoothing). For every i∈Ji∈ J and every x∈Fix∈ F_i, Digε(x)⟶0D_ig_ (x) 0 as ε↓0. 0. At the same time, the convergence is pointwise in x. There is a constant Cρ,R<∞C_ρ,R<∞, independent of x and ε , such that |Digε(x)|≤Cρ,RLip(g),x∈Fi,0<ε<1.|D_ig_ (x)|≤ C_ρ,RLip(g), x∈ F_i, 0< <1. (4.9) Consequently, for every finite signed measure ηi _i on FiF_i and every bounded Borel function a:Fi→ℝa:F_i fixed independently of ε , we have ∫Fia(x)Digε(x)ηi(x)⟶0. _F_ia(x)D_ig_ (x)\,d _i(x) 0. (4.10) Proof. Since ρ∈Cc∞((1,2)d)ρ∈ C_c^∞((1,2)^d), integration by parts in the w-variable gives, for x∈Ex∈ E, ∇gε(x)=−1ε∫ℝd∇ρ(w)g(x+εw)w.∇ g_ (x)=- 1 _R^d∇ρ(w)\,g(x+ w)\,dw. Also, ∫ℝdRi⋅∇ρ(w)w=0. _R^dR_i·∇ρ(w)\,dw=0. Therefore, for x∈Fix∈ F_i, Digε(x)=−∫ℝd(Ri⋅∇ρ)(w)g(x+εw)−g(x)εw.D_ig_ (x)=- _R^d(R_i·∇ρ)(w) g(x+ w)-g(x) \,dw. (4.11) Fix x∈Fix∈ F_i. Then i∈I(x)i∈ I(x). Since suppρ⊂(1,2)dsuppρ⊂(1,2)^d, every w∈suppρw ρ belongs to GxG_x. Proposition 3.3 gives, for each such fixed w, g(x+εw)−g(x)ε⟶ℓx(w). g(x+ w)-g(x) _x(w). Moreover, |(Ri⋅∇ρ)(w)g(x+εw)−g(x)ε|≤|(Ri⋅∇ρ)(w)|Lip(g)|w|. |(R_i·∇ρ)(w) g(x+ w)-g(x) |≤|(R_i·∇ρ)(w)|\,Lip(g)|w|. The right hand side is integrable over ℝdR^d, because ρ is smooth and compactly supported. Dominated convergence in the mollifier variable w gives limε↓0Digε(x)=−∫ℝd(Ri⋅∇ρ)(w)ℓx(w)w. _ 0D_ig_ (x)=- _R^d(R_i·∇ρ)(w) _x(w)\,dw. (4.12) Since ℓx _x is linear and ρ has compact support, another integration by parts gives −∫ℝd(Ri⋅∇ρ)(w)ℓx(w)w=∫ℝdρ(w)ℓx(Ri)w=ℓx(Ri).- _R^d(R_i·∇ρ)(w) _x(w)\,dw= _R^dρ(w) _x(R_i)\,dw= _x(R_i). (4.13) There is no boundary term because ρ∈Cc∞((1,2)d)ρ∈ C_c^∞((1,2)^d). Since i∈I(x)i∈ I(x), Proposition 3.3 gives ℓx(Ri)=0 _x(R_i)=0. This proves the pointwise convergence. The same representation and the Lipschitz bound give |Digε(x)|≤Lip(g)∫ℝd|(Ri⋅∇ρ)(w)||w|w.|D_ig_ (x)| (g) _R^d|(R_i·∇ρ)(w)|\,|w|\,dw. Thus the uniform estimate holds with Cρ,R:=max1≤k≤d∫ℝd|(Rk⋅∇ρ)(w)||w|w<∞.C_ρ,R:= _1≤ k≤ d _R^d|(R_k·∇ρ)(w)|\,|w|\,dw<∞. Finally, let ηi∈M(Fi) _i∈ M(F_i), and let a:Fi→ℝa:F_i be bounded Borel and fixed independently of ε . Since DigεD_ig_ is continuous on FiF_i, the product aDigεaD_ig_ is Borel. The pointwise convergence just proved and the bound |a(x)Digε(x)|≤‖a‖∞Cρ,RLip(g)|a(x)D_ig_ (x)|≤\|a\|_∞C_ρ,RLip(g) allow dominated convergence with respect to |ηi|| _i|. Hence ∫Fia(x)Digε(x)ηi(x)⟶0. _F_ia(x)D_ig_ (x)\,d _i(x) 0. This completes the proof. ∎ Proposition 4.4 (Measure–Neumann resolvent approximation). Let m be a finite signed measure on E and let ηi _i be finite signed measures on FiF_i. Then, as ε↓0 0, ∫E(λ−L)gεm _E(λ-L)g_ \,dm ⟶∫Ehm, _Eh\,dm, (4.14) ∫Egεm _Eg_ \,dm ⟶∫Egm, _Eg\,dm, (4.15) ∫FiDigεηi _F_iD_ig_ \,d _i ⟶0,i=1,…,d. 0, i=1,…,d. (4.16) Proof. The first assertion follows from (λ−L)gε=hε(λ-L)g_ =h_ and the uniform convergence hε→h_ → h. The second follows from the uniform convergence gε→g_ → g. Since m is finite signed, uniform convergence is sufficient in both cases. For the boundary terms, take the bounded Borel multiplier a≡1a≡ 1 in (4.10). This gives (4.16) for each finite signed boundary measure ηi _i. ∎ Proof of Theorem 4.1. First assume h∈Cc∞(ℝd)h∈ C_c^∞(R^d), regarded as a function on E, and let g=Rλhg=R_λh. By Section 4, gε∈Cb2(E)g_ ∈ C_b^2(E), so it is an admissible BAR test. Applying the BAR to gεg_ and rearranging gives ∫E(λ−L)gεπ¯=λ∫Egεπ¯+∑i=1d∫FiDigεν¯i. _E(λ-L)g_ \,d π=λ _Eg_ \,d π+ _i=1^d _F_iD_ig_ \,d ν_i. (4.17) By Section 4, letting ε↓0 0 yields ∫Ehπ¯=λ∫ERλhπ¯. _Eh\,d π=λ _ER_λh\,d π. Equivalently, ∫E(λRλh−h)π¯=0 _E(λ R_λh-h)\,d π=0 holds for every smooth compactly supported h. Now let h∈C0(E)h∈ C_0(E). The restrictions to E of functions in Cc∞(ℝd)C_c^∞(R^d) are uniformly dense in C0(E)C_0(E): extend a function from the closed set E to C0(ℝd)C_0(R^d), cut it off, and mollify on ℝdR^d. Choose hn∈Cc∞(ℝd)h_n∈ C_c^∞(R^d) with ‖hn−h‖∞→0\|h_n-h\|_∞→ 0 on E. Since (Pt)(P_t) is a contraction on bounded functions, ‖Rλ(hn−h)‖∞≤λ−1‖hn−h‖∞.\|R_λ(h_n-h)\|_∞≤λ^-1\|h_n-h\|_∞. The finiteness of π¯ π therefore permits passage to the limit in the smooth identity, proving (RI) for h∈C0(E)h∈ C_0(E). ∎ 5 Proof of Section 2.3: From the resolvent identity to signed BAR uniqueness We now complete the proof of resolvent identity criterion (Section 2.3) and then complete the proof of the main Theorem (Theorem 2.2). We first use the uniqueness of Laplace transforms to show that the signed measure that satisfies (RI) is invariant for the semigroup (Pt)(P_t). Lemma 5.1 (Uniqueness of Laplace transforms[12, Chapter I]). Let a:[0,∞)→ℝa:[0,∞) be locally integrable and of at most exponential growth. Suppose that its Laplace transform a^(λ)=∫0∞e−λta(t)t a(λ)= _0^∞e^-λ ta(t)\,dt vanishes for every λ in some interval (λ∗,∞)( _ ,∞). Then a(t)=0 for Lebesgue almost every t≥0.a(t)=0 for Lebesgue almost every t≥ 0. Proposition 5.2 (Resolvent identity implies semigroup invariance). Assume that a finite signed measure π¯ π satisfies (RI) for every h∈C0(E)h∈ C_0(E) and every λ>0λ>0. Then π¯ π is invariant for (Pt)(P_t): ∫EPtφπ¯=∫Eφπ¯,t≥0,φ∈C0(E). _EP_t \,d π= _E \,d π, t≥ 0, ∈ C_0(E). (5.1) Proof. From (RI), for every h∈C0(E)h∈ C_0(E) and every λ>0λ>0, ∫ERλhπ¯=λ−1∫Ehπ¯. _ER_λh\,d π=λ^-1 _Eh\,d π. (5.2) Fix h∈C0(E)h∈ C_0(E) and define Fh(t)=∫EPthπ¯ (t>0).F_h(t)= _EP_th\,d π (t>0). We first record the elementary regularity of FhF_h. By the Feller statement (i) in Theorem 3.3, PtC0(E)⊂C0(E)P_tC_0(E)⊂ C_0(E), and ‖Pth−h‖∞→t↓00.\|P_th-h\|_∞ t 00. The semigroup property and the contraction property imply norm continuity of t↦Ptht P_th on all of [0,∞)[0,∞). Indeed, ‖Psh−Pth‖∞=‖Pmins,t(P|s−t|h−h)‖∞≤‖P|s−t|h−h‖∞⟶0\|P_sh-P_th\|_∞=\|P_ \s,t\(P_|s-t|h-h)\|_∞≤\|P_|s-t|h-h\|_∞ 0 as s→ts→ t. Since π¯ π is finite signed, it follows that FhF_h is continuous: |Fh(s)−Fh(t)|≤|π¯|(E)‖Psh−Pth‖∞.|F_h(s)-F_h(t)|≤| π|(E)\,\|P_sh-P_th\|_∞. Moreover, we have |Fh(t)|≤‖h‖∞|π¯|(E),|F_h(t)|≤\|h\|_∞| π|(E), for |Pth|≤‖h‖∞|P_th|≤\|h\|_∞. We now pass from the resolvent identity to a Laplace transform identity. Since |e−λtPth(x)|≤e−λt‖h‖∞|e^-λ tP_th(x)|≤ e^-λ t\|h\|_∞ and ∫E∫0∞e−λt‖h‖∞td|π¯|(x)=λ−1‖h‖∞|π¯|(E)<∞, _E _0^∞e^-λ t\|h\|_∞\,dt\,d| π|(x)=λ^-1\|h\|_∞| π|(E)<∞, Fubini’s theorem gives ∫ERλhπ¯ _ER_λh\,d π =∫E∫0∞e−λtPth(x)tπ¯(x) = _E _0^∞e^-λ tP_th(x)\,dt\,d π(x) =∫0∞e−λt(∫EPthπ¯)t=∫0∞e−λtFh(t)t. = _0^∞e^-λ t ( _EP_th\,d π )dt= _0^∞e^-λ tF_h(t)\,dt. At the same time, since λ−1∫Ehπ¯=∫0∞e−λtFh(0)t,λ^-1 _Eh\,d π= _0^∞e^-λ tF_h(0)\,dt, therefore (5.2) is equivalent to ∫0∞e−λt(Fh(t)−Fh(0))t=0,λ>0. _0^∞e^-λ t (F_h(t)-F_h(0) )\,dt=0, λ>0. Put ah(t)=Fh(t)−Fh(0).a_h(t)=F_h(t)-F_h(0). Then aha_h is continuous and bounded. In particular, aha_h is locally integrable and of at most exponential growth. Indeed, |ah(t)|≤|Fh(t)|+|Fh(0)|≤2‖h‖∞|π¯|(E),t≥0.|a_h(t)|≤|F_h(t)|+|F_h(0)|≤ 2\|h\|_∞| π|(E), t≥ 0. Equation (5.3)(5.3) says precisely that the Laplace transform of aha_h vanishes for every λ>0λ>0. By Section 5, ah(t)=0 for Lebesgue almost every t≥0.a_h(t)=0 for Lebesgue almost every t≥ 0. Since aha_h is continuous, this almost everywhere equality upgrades to equality for every t≥0t≥ 0. Hence Fh(t)=Fh(0)F_h(t)=F_h(0) for t≥0t≥ 0, i.e. ∫EPthπ¯=∫Ehπ¯, _EP_th\,d π= _Eh\,d π, holds for all h∈C0(E)h∈ C_0(E). For a finite signed measure α, we write (αPt)(B)=∫EPt(x,B)α(dx)(α P_t)(B)= _EP_t(x,B)\,α(dx) for B∈ℬ(E).B (E). Then π¯Pt πP_t is a finite signed Radon measure, and for every φ∈C0(E) ∈ C_0(E), ∫Eφd(π¯Pt)=∫EPtφπ¯=∫Eφπ¯. _E \,d( πP_t)= _EP_t \,d π= _E \,d π. Since C0(E)C_0(E) separates finite Radon measures on the locally compact space E, this implies π¯Pt=π¯. πP_t= π. ∎ Section 5 indicates that the signed-BAR interior solution π¯ π is invariant as a signed measure for the SRBM semigroup, i.e. π¯Pt=π¯ πP_t= π. Then we follows the Dai-Dieker [5], using Jordan decomposition to show that π¯ π uniquely characterizes the stationary probability distribution π0 _0 of the diffusion process in the sense that π¯=π¯(E)π0 π= π(E) _0. Proposition 5.3 (Identification of the interior measure). If π¯ π is a finite signed invariant measure for the SRBM semigroup, then π¯=cπ0 π=c _0 where c=π¯(E)c= π(E). Proof. For a Markov kernel P and a finite signed measure π¯ π, positivity gives the measure inequality |π¯P|≤|π¯|P.| πP|≤| π|P. (5.2) Indeed, for every Borel set B, |∫P(x,B)π¯(x)|≤∫P(x,B)d|π¯|(x)| P(x,B)\,d π(x)|≤ P(x,B)\,d| π|(x), and the same domination holds for finite measurable partitions, hence for total variation. If π¯Pt=π¯ πP_t= π, then (5.2) gives |π¯|≤|π¯|Pt| π|≤| π|P_t. Both positive measures have total mass |π¯|(E)| π|(E), because Pt(x,E)=1P_t(x,E)=1. Thus the domination is actually equality: if finite positive measures α≤βα≤β have α(E)=β(E)α(E)=β(E), then β−αβ-α is a positive measure with total mass zero, hence vanishes. Applying this with α=|π¯|α=| π| and β=|π¯|Ptβ=| π|P_t gives |π¯|Pt=|π¯|.| π|P_t=| π|. Consequently the Jordan components π¯+=12(|π¯|+π¯) π^+= 12(| π|+ π) and π¯−=12(|π¯|−π¯) π^-= 12(| π|- π) are invariant positive finite measures. Each nonzero component, after normalization by its total mass, is an invariant probability and therefore equals π0 _0, which is unique. Thus π¯=(π¯+(E)−π¯−(E))π0=π¯(E)π0. π=( π^+(E)- π^-(E)) _0= π(E) _0. ∎ The preceding Sections 5 and 5 identifies the uniqueness of the interior measure, but an exact same assessment for boundary measure is not yet established. Therefore, by subtracting the appropriate scalar multiple of the stationary BAR vector, the remaining signed tuple has zero interior measure. Thus the only possible obstruction to the signed uniqueness of BAR solution is a purely boundary one: a collection of finite signed measures (ηi)i=1d( _i)_i=1^d, with ηi _i supported on FiF_i, whose boundary pairing vanishes, i.e. ∑i=1d∫FiDifηi=0 _i=1^d _F_iD_if\,d _i=0 against every test function f∈Cb2(E)f∈ C_b^2(E). If such boundary measures exist, adding them to an existing solution doesn’t change the validity of the solution. The next proposition shows that no such nontrivial boundary annihilator exists, ruling out any other signed-BAR solutions. The proof is local on the boundary stratification and uses the nonsingular M-matrix assumption only through the invertibility of the principal reflection blocks RAAR_A. On a stratum SAS_A, the active oblique derivatives are determined by the active normal jet: (Dif|SA)i∈A=RAATa(D_if|_S_A)_i∈ A=R_A^Ta, where a=(∂xjf|SA)j∈Aa=( _x_jf|_S_A)_j∈ A. Since RAAR_A is invertible, we can prescribe these oblique derivatives independently. In particular, choosing a=RAA−Tekψa=R_A^-Te_k\,ψ gives Dif|SA=δikψD_if|_S_A= _ikψ for i∈Ai∈ A, and for any test function ψ∈Cc∞(SA)ψ∈ C_c^∞(S_A). This isolates the k-th boundary measure on SAS_A. An induction over the codimension |A||A| removes all lower-stratum contributions and forces ηk|SA=0 _k|_S_A=0. Since both A and k∈Ak∈ A are arbitrary, all boundary measures vanish. Proposition 5.4 (Pure boundary injectivity). Let ηi∈ℳ(Fi) _i (F_i) be finite signed measures. If ∑i=1d∫FiDifηi=0,f∈Cb2(E), _i=1^d _F_iD_if\,d _i=0, f∈ C_b^2(E), (5.3) then ηi=0, _i=0, for all i=1…,d.i=1…,d. Proof. For A⊂JA⊂ J and i∈Ai∈ A, let ηiA=ηi|SA. _i^A= _i|_S_A. Then ηi=∑A∋iηiA _i= _A i _i^A as a finite sum of mutually singular signed measures. We prove by induction on n=|A|n=|A| that ηiA=0for every A⊂J with |A|=n and every i∈A. _i^A=0 every A⊂ J with |A|=n and every i∈ A. (5.4) Assume the claim has been proved for all strata of cardinality less than n, and fix A with |A|=n|A|=n. Let k∈Ak∈ A and let ψ∈Cc∞(SA)ψ∈ C_c^∞(S_A). Write points as x=(xA,y)x=(x_A,y), where y=xAc∈(0,∞)Acy=x_A^c∈(0,∞)^A^c. We identify ψ with its extension by zero to ℝAcR^A^c; this extension is smooth because suppψsuppψ is compact in the open orthant. Choose a tangential cutoff ϑ∈Cc∞((0,∞)Ac) ∈ C_c^∞((0,∞)^A^c) that equals one on a neighborhood of suppψsuppψ. When A=JA=J, interpret the tangential space as a point and put ϑ=1 =1. Choose a normal cutoff ζ∈Cc∞(ℝA)ζ∈ C_c^∞(R^A) that equals one near the origin. Define the A-vector a(y)=RAA−Tekψ(y)a(y)=R_A^-Te_k\,ψ(y) (5.5) and the test function f(xA,y)=ζ(xA)ϑ(y)∑j∈Axjaj(y).f(x_A,y)=ζ(x_A) (y) _j∈ Ax_ja_j(y). (5.6) After shrinking the support of ϑ if necessary, f is supported away from every face FjF_j with j∉Aj∉ A. At a point of SAS_A, the tangential derivatives of f vanish and ∂xjf(0,y)=aj(y),j∈A. _x_jf(0,y)=a_j(y), j∈ A. (5.7) Therefore, for i∈Ai∈ A, Dif|SA=∑j∈ARjiaj=(RAATa)i=δikψ.D_if|_S_A= _j∈ AR_jia_j=(R_A^Ta)_i= _ikψ. (5.8) The support condition implies that the only boundary strata meeting suppfsuppf are SCS_C with ∅≠C⊂A ≠ C⊂ A. The contributions from |C|<n|C|<n vanish by the induction hypothesis. Hence (5.3) and (5.8) give 0=∑i∈A∫SADifηiA=∫SAψηkA.0= _i∈ A _S_AD_if\,d _i^A= _S_Aψ\,d _k^A. Since ψ is arbitrary, ηkA=0 _k^A=0. Since k∈Ak∈ A was arbitrary, the induction step is complete. The base case n=1n=1 is the same argument with no lower strata. Thus all ηiA _i^A vanish and hence all ηi _i vanish. ∎ Now, the uniqueness of Signed BAR solution in the Harrison-Reiman Class is the natural conclusion of Section 5-Section 5. Proof of Section 2.3. Assume (RI) for the finite signed BAR tuple (π¯,ν¯1,…,ν¯d)( π, ν_1,…, ν_d). By Section 5, we know the signed measure π¯ π is invariant under the semigroup, i.e. π¯Pt=π¯ πP_t= π holds for t≥0t≥ 0. Then by Section 5, we have the uniquness of the interior measure π¯=cπ0 π=c _0 for c=π¯(E)c= π(E). Define ηi=ν¯i−cνi0. _i= ν_i-c _i^0. Subtract c times the stationary BAR (2.4) from the signed BAR (2.3). The interior measures cancel, and we obtain ∑i=1d∫FiDifηi=0, _i=1^d _F_iD_if\,d _i=0, for all test function f∈Cb2(E).f∈ C_b^2(E). Section 5 gives ηi=0 _i=0 for every i, hence ν¯i=cνi0 ν_i=c _i^0. This proves Section 2.3. ∎ Finally, we finish the proof of the signed-BAR problem. Proof of Theorem 2.2. Theorem 4.1 proves (RI) for every finite signed BAR tuple. Section 2.3 converts (RI) into full signed BAR uniqueness. Therefore Theorem 2.2 follows. ∎ 6 Failure in the completely S class In this section, we demonstrate a family of counterexamples in the completely S class due to the singular proper active block of the relection matrix. The negative construction is summarized in Fig. 2. It has two parts: a boundary gauge identity on the singular stratum, followed by a zero-potential correction that extends the resulting centered source into the interior and adds the matching boundary occupation potential. Recall that a square matrix A is an S matrix if there exists a vector u>0u>0 such that Au>0Au>0, and is completely-S if every principal submatrix is an S matrix. completely-S reflection matrices are the natural existence class for orthant SRBMs, but they need not have invertible principal blocks. Boundary gauge on a singular stratum Extension into the interior by a zero-potential Singular active block Choose ∅≠A⊊J ≠ A J and 0≠v∈kerRAA.0≠ v∈ R_A. With T=J∖AT=J A, put w:=RTAv≠0.w:=R_TAv≠ 0. The nonzero vector w is tangent to the stratum SAS_A. Boundary gauge on SAS_A For y∈(0,∞)Ty∈(0,∞)^T and ιA(y)∈SA _A(y)∈ S_A, set dζi(ιA(y))=viφ(y)dy,i∈A,d _i( _A(y))=v_i (y)\,dy, i∈ A, and set ζi=0 _i=0 for i∉Ai∉ A. Normal components cancel Combining active faces gives ∑i∈AviRi=(RAAv,RTAv)=(0,w). _i∈ Av_iR_i=(R_Av,R_TAv)=(0,w). Thus the gauge sees only the tangential derivative w⋅∇Tfw· _Tf on SAS_A. Tangential integration by parts Because φ∈Cc∞((0,∞)T) ∈ C_c^∞((0,∞)^T), ∑i∫FiDifζi _i _F_iD_if\,d _i =∫Efχ, = _Ef\,dχ, dχ(ιA(y)) dχ( _A(y)) =−(w⋅∇Tφ(y))dy. =-(w· _T (y))\,dy. The source is supported on SAS_A, nonzero, and centered: χ(E)=0χ(E)=0. Interior zero-potential Spread the centered source by the reflected semigroup: π¯=∫0∞χPtt. π= _0^∞χ P_t\,dt. Exponential ergodicity and χ(E)=0χ(E)=0 make this finite and give π¯(E)=0 π(E)=0. Boundary occupation correction Use the one-unit regulator kernels KiK_i and set θi=∑n≥0(χPn)Ki. _i= _n≥ 0(χ P_n)K_i. The one-step regulator bound makes each θi _i finite on FiF_i. Poisson identity cancels the source Itô’s formula over integer intervals gives ∫ELfπ¯+∑i∫FiDifθi=−∫Efχ. _ELf\,d π+ _i _F_iD_if\,d _i=- _Ef\,dχ. Adding the boundary-gauge identity from the upper half cancels ∫fχ f\,dχ. Signed BAR tuple with zero interior mass ν¯i:=θi+ζi,∫ELfπ¯+∑i∫FiDifν¯i=0,π¯(E)=0. ν_i:= _i+ _i, _ELf\,d π+ _i _F_iD_if\,d ν_i=0, π(E)=0. Since π¯≠0 π≠ 0 but π¯(E)=0 π(E)=0 whereas π0(E)=1 _0(E)=1, the interior coordinate cannot be a scalar multiple of the stationary probability. Varying φ on disjoint supports gives the infinite-dimensional failure. +∫fχ+ f\,dχ Figure 2: Counterexample construction in the completely-S class. The upper group is the boundary algebra of Section 6.1: a gauge supported on a lower-dimensional stratum has its active normal components killed by RAAv=0R_Av=0, leaving a tangential derivative and hence a centered source χ. The lower group is the zero-potential correction of Section 6.2: the semigroup potential π¯=∫0∞χPtt π= _0^∞χ P_t\,dt and the boundary occupation potentials θi _i cancel the source and produce a nonzero zero-mass signed BAR tuple. 6.1 A singular-block boundary gauge In this section, we utilize the singular proper active block of the reflection matrix to construct a nonzero null direction on that block and place a signed boundary gauge on the corresponding lower-dimensional boundary stratum. The singularity makes the active normal reflection components cancel, so the gauge leaves only a tangential derivative along the stratum. After integration by parts, this tangential derivative becomes a finite nonzero centered signed source supported on the same stratum. This source will be cancelled by the zero-potential correction in the next subsection. Let J=1,…,dJ=\1,…,d\. For a nonempty proper set A⊊JA J, put T=J∖AT=J A and write RBCR_BC for the submatrix with rows in B and columns in C. Define the open stratum SA=x∈E:xi=0 for i∈A,xj>0 for j∈T.S_A=\x∈ E:x_i=0 for i∈ A,\ x_j>0 for j∈ T\. We identify SAS_A with (0,∞)T(0,∞)^T through the embedding ιA:(0,∞)T→E _A:(0,∞)^T→ E that inserts zeros in the coordinates indexed by A. Proposition 6.1 (Boundary gauge from a singular principal block). Assume that R is nonsingular and that RAAR_A is singular for some nonempty proper set A⊊JA J. Choose 0≠v∈kerRAA,w=RTAv∈ℝT.0≠ v∈ R_A, w=R_TAv ^T. Then w≠0w≠ 0. Let ϕ∈Cc∞((0,∞)T)φ∈ C_c^∞((0,∞)^T) satisfy w⋅∇Tϕ≢0w· _Tφ ≡ 0. For i∈Ai∈ A, define a finite signed measure ζi _i on FiF_i by ∫Fig(x)ζi(x)=vi∫(0,∞)Tg(ιA(y))ϕ(y)y, _F_ig(x)\,d _i(x)=v_i _(0,∞)^Tg( _A(y))φ(y)\,dy, (6.1) and set ζi=0 _i=0 for i∉Ai∉ A. Define χ∈ℳ(E)χ (E) by ∫Eg(x)χ(x)=−∫(0,∞)Tg(ιA(y))(w⋅∇Tϕ(y))y. _Eg(x)\,dχ(x)=- _(0,∞)^Tg( _A(y)) (w· _Tφ(y) )\,dy. (6.2) Then χ is finite, nonzero, supported on SAS_A, and satisfies χ(E)=0χ(E)=0. Moreover, ∑i=1d∫FiDifζi=∫Efχ,f∈Cb2(E). _i=1^d _F_iD_if\,d _i= _Ef\,dχ, f∈ C_b^2(E). (6.3) Proof. If w=0w=0, extend v to v~∈ℝd v ^d by setting its coordinates in T equal to zero. Then Rv~=(RAAvRTAv)=0,R v= pmatrixR_Av\\ R_TAv pmatrix=0, contradicting the nonsingularity of R. Hence w≠0w≠ 0, and a compactly supported smooth ϕφ with nonzero directional derivative along w exists. For f∈Cb2(E)f∈ C_b^2(E), combine the face labels before integrating: ∑i=1d∫FiDifζi _i=1^d _F_iD_if\,d _i =∫(0,∞)Tϕ(y)(∑i∈AviRi)⋅∇f(ιA(y))y = _(0,∞)^Tφ(y) ( _i∈ Av_iR_i )·∇ f( _A(y))\,dy =∫(0,∞)Tϕ(y)[(RAAv)⋅∇Af(ιA(y))+(RTAv)⋅∇Tf(ιA(y))]y = _(0,∞)^Tφ(y) [(R_Av)· _Af( _A(y))+(R_TAv)· _Tf( _A(y)) ]dy =∫(0,∞)Tϕ(y)w⋅∇Tf(ιA(y))y = _(0,∞)^Tφ(y)w· _Tf( _A(y))\,dy =−∫(0,∞)T(w⋅∇Tϕ(y))f(ιA(y))y. =- _(0,∞)^T (w· _Tφ(y) )f( _A(y))\,dy. There is no boundary term because ϕφ is compactly supported in the open stratum. This proves (6.3). Taking a test function equal to one on a neighborhood of suppϕsuppφ gives χ(E)=0χ(E)=0, and the choice of ϕφ gives χ≠0χ≠ 0. ∎ The singular block cancels the components normal to the active faces. The remaining vector w is tangent to SAS_A, and tangential integration by parts turns the boundary gauge into the centered source χ. 6.2 Interior correction by a zero-potential In this section, we take that centered source and spread it through the reflected Brownian semigroup by a zero potential, while also adding the matching boundary occupation potentials generated by the regulator. Under exponential ergodicity and a one-step regulator bound, these potentials are finite. Ito’s formula then shows that the zero potential contributes exactly the negative of the source created in the previous section. Adding the original boundary gauge cancels the defect and produces a genuine finite signed BAR tuple. Its interior part has total mass zero but is not the zero measure, so it cannot be a scalar multiple of the stationary distribution, whose mass is one. Let (Pt)t≥0(P_t)_t≥ 0 be the transition semigroup of the SRBM, and let Y=(Y1,…,Yd)Y=(Y_1,…,Y_d) be its regulator. For a finite signed measure α, we define the semigroup (αPt)(B)=∫EPt(x,B)α(dx).(α P_t)(B)= _EP_t(x,B)\,α(dx). For each face define the boundary occupation kernel Ki(x,B)=x∫01B(Z(s))Yi(s),K_i(x,B)=E_x _0^11_B(Z(s))\,dY_i(s), (6.4) which is supported on FiF_i. Proposition 6.2 (Interior correction by a zero-potential). Assume that the SRBM is a strong Markov process whose transition semigroup is Feller on C0(E)C_0(E), and that it has stationary probability π0 _0. Suppose that there are a locally bounded function V:E→[1,∞)V:E→[1,∞) and constants M,κ>0M,κ>0 such that ‖Pt(x,⋅)−π0‖TV≤MV(x)e−κt,x∈E,t≥0.\|P_t(x,·)- _0\|_TV≤ MV(x)e^-κ t, x∈ E,\ t≥ 0. (6.5) Suppose also that ci:=supx∈ExYi(1)<∞,i=1,…,d.c_i:= _x∈ EE_xY_i(1)<∞, i=1,…,d. (6.6) Let χ∈ℳ(E)χ (E) satisfy χ(E)=0χ(E)=0 and ∫EVd|χ|<∞, _EV\,d|χ|<∞, and let ζi∈ℳ(Fi) _i (F_i) satisfy ∑i=1d∫FiDifζi=∫Efχ,f∈Cb2(E). _i=1^d _F_iD_if\,d _i= _Ef\,dχ, f∈ C_b^2(E). (6.7) Define π¯ π =∫0∞χPtt, = _0^∞χ P_t\,dt, (6.8) θi _i =∑n=0∞(χPn)Ki, = _n=0^∞(χ P_n)K_i, (6.9) ν¯i ν_i =θi+ζi. = _i+ _i. (6.10) Then all measures in (6.8)–(6.10) are well defined in total variation and finite. They form a signed BAR tuple for all f∈Cb2(E)f∈ C_b^2(E), hence also for all f∈Cc2(E)f∈ C_c^2(E), and π¯(E)=0. π(E)=0. If χ≠0χ≠ 0, then π¯≠0 π≠ 0. Proof. Because χ(E)=0χ(E)=0, χPt=∫E(Pt(x,⋅)−π0)χ(dx).χ P_t= _E (P_t(x,·)- _0 )\,χ(dx). Therefore ‖χPt‖TV≤Me−κt∫EVd|χ|.\|χ P_t\|_TV≤ Me^-κ t _EV\,d|χ|. (6.11) The integral in (6.8) converges in total variation, and π¯(E)=0 π(E)=0 because χPt(E)=χ(E)=0χ P_t(E)=χ(E)=0. For a finite signed measure α and the positive kernel KiK_i, ‖αKi‖TV≤∫EKi(x,E)|α|(dx)≤ci‖α‖TV.\|α K_i\|_TV≤ _EK_i(x,E)\,|α|(dx)≤ c_i\|α\|_TV. (6.12) Combining (6.11) at integer times with (6.12) proves absolute convergence of (6.9). Each θi _i is supported on FiF_i. Fix f∈Cb2(E)f∈ C_b^2(E). Itô’s formula up to an integer time N, followed by integration against the Jordan decomposition of χ, gives χPN(f)−χ(f)=∫0N(χPt)(Lf)t+∑i=1d∫Ex∫0NDif(Z(t))Yi(t)χ(dx).χ P_N(f)-χ(f)= _0^N(χ P_t)(Lf)\,dt+ _i=1^d _EE_x _0^ND_if(Z(t))\,dY_i(t)\,χ(dx). (6.13) Splitting the boundary integral into unit intervals and using the strong Markov property yields ∫Ex∫0NDif(Z(t))Yi(t)χ(dx)=∑n=0N−1((χPn)Ki)(Dif). _EE_x _0^ND_if(Z(t))\,dY_i(t)\,χ(dx)= _n=0^N-1 ((χ P_n)K_i )(D_if). (6.14) The estimates above justify passage to the limit. Since χPN(f)→0χ P_N(f)→ 0, equations (6.13) and (6.14) give ∫ELfπ¯+∑i=1d∫FiDifθi=−∫Efχ. _ELf\,d π+ _i=1^d _F_iD_if\,d _i=- _Ef\,dχ. Adding (6.7) proves the BAR for (π¯,ν¯1,…,ν¯d)( π, ν_1,…, ν_d). It remains to prove that the zero potential is injective on this centered class. Let A be the generator of the Feller semigroup on C0(E)C_0(E). For h∈D()h∈ D(A), semigroup differentiation gives π¯(h)=limT→∞∫0TχPt(h)t=limT→∞χ(PTh−h)=−χ(h), π(Ah)= _T→∞ _0^Tχ P_t(Ah)\,dt= _T→∞χ(P_Th-h)=-χ(h), where the last limit follows from (6.11). If π¯=0 π=0, then χ(h)=0χ(h)=0 for every h∈D()h∈ D(A). To conclude that χ=0χ=0 we use the following density theorem. Theorem 6.3 ([17, Chapter 1, Section 2]). The infinitesimal generator A of a strongly continuous contraction semigroup on a Banach space has domain D()D(A) dense in that space. In particular, for a Feller semigroup on C0(E)C_0(E), the domain D()D(A) is dense in C0(E)C_0(E). Its hypothesis is exactly the standing assumption of the present proposition: (Pt)(P_t) is Feller on C0(E)C_0(E). Hence D()D(A) is dense in C0(E)C_0(E). Since ℳ(E)=C0(E)∗M(E)=C_0(E)^*, the identity χ(h)=0χ(h)=0 on D()D(A) implies χ=0χ=0. Therefore χ≠0χ≠ 0 implies π¯≠0 π≠ 0. ∎ Theorem 6.4 (Failure in the completely-S class). Consider an SRBM in E=ℝ+dE=R_+^d with positive definite covariance matrix, nonsingular completely-S reflection matrix R, and stationary probability π0 _0. Assume that the process is strong Markov, is Feller on C0(E)C_0(E), and satisfies the quantitative recurrence conditions (6.5) and (6.6). If RAAR_A is singular for some nonempty proper set A⊊JA J, then signed BAR uniqueness fails. More precisely, there exists a finite signed BAR tuple (π¯,ν¯1,…,ν¯d)( π, ν_1,…, ν_d) such that π¯(E)=0,π¯≠0. π(E)=0, π≠ 0. (6.15) Consequently π¯ π is not a scalar multiple of π0 _0. Proof. Apply Section 6.1. Its source χ is compactly supported, and the local boundedness of V gives ∫EVd|χ|<∞ _EV\,d|χ|<∞. Section 6.2 then produces the required BAR tuple. If π¯=cπ0 π=c _0, total masses give c=0c=0, contradicting π¯≠0 π≠ 0. ∎ Corollary 6.5 (Infinite-dimensional failure). Under the assumptions of Theorem 6.4, the set of interior BAR coordinates of total mass zero contains an infinite-dimensional linear subspace. In particular, the full vector space of finite signed BAR tuples is infinite-dimensional. Proof. Choose functions ϕm∈Cc∞((0,∞)T) _m∈ C_c^∞((0,∞)^T) with pairwise disjoint supports and w⋅∇Tϕm≢0w· _T _m ≡ 0. Let χm _m and π¯m π_m be the corresponding sources and zero potentials. If ∑m=1Namπ¯m=0 _m=1^Na_m π_m=0, linearity and the injectivity identity in the proof of Section 6.2 imply ∑m=1Namχm=0 _m=1^Na_m _m=0. The sources are nonzero and have pairwise disjoint supports, so every ama_m is zero. ∎ The theorem concerns a singular proper principal block. Singularity of an arbitrary rectangular or nonprincipal submatrix does not yield the cancellation RAAv=0R_Av=0 needed in Section 6.1. Conversely, every principal block of a nonsingular M-matrix is nonsingular by Section 3.1, so this obstruction is absent from the class covered by Theorem 2.2. 6.3 A checkable three-dimensional family The general obstruction is useful only if the recurrence assumptions can be verified without solving the stationary distribution. The next criterion is a direct way to do this for a broad positive-reflection subclass. A Z matrix means a matrix with nonpositive off-diagonal entries. Corollary 6.6 (A checkable completely-S subclass). Assume in addition that Rii=1R_i=1 and Rij≥0R_ij≥ 0 for all i,ji,j. Suppose there is a symmetric positive definite matrix H such that HRHR is a Z matrix and Hμ<0Hμ<0 componentwise. If R is nonsingular, completely S, and has a singular proper principal block, then signed BAR uniqueness fails and the conclusion of Section 6.2 holds. Proof. We verify the standing hypotheses of Theorem 6.4: existence, the strong Markov property, the C0C_0-Feller property, the recurrence certificate (6.5), and the regulator bound (6.6). First, for the existence, the strong Markov property, the C0C_0-Feller property, as well as (6.6), we have Proposition 6.7 ([34]). For a symmetric positive definite covariance Σ , a drift μ, and a reflection matrix R with unit diagonal, the orthant SRBM with data (Σ,μ,R)( ,μ,R) exists and is unique in law if and only if R is completely-S; when it exists it is a Feller continuous strong Markov process, and x↦Pth(x)x P_th(x) is continuous for every h∈Cb(E)h∈ C_b(E). Since Σ is positive definite, R has unit diagonal, and R is completely-S, the orthant SRBM exists, is unique in law, is strong Markov, and x↦Pth(x)x P_th(x) is continuous for every h∈Cb(E)h∈ C_b(E) (Feller continuity). Now we check the C0C_0 Feller property, i.e. PtC0(E)⊂C0(E)P_tC_0(E)⊂ C_0(E) and that ‖Pth−h‖∞→0\|P_th-h\|_∞→ 0 for h∈C0(E)h∈ C_0(E). Write Zx(t)=x+μt+B(t)+RYx(t),Z^x(t)=x+μ t+B(t)+RY^x(t), where B(t)=Σ1/2W(t)B(t)= ^1/2W(t) and each YixY_i^x is nondecreasing. Since Rij≥0R_ij≥ 0 and Yjx≥0Y_j^x≥ 0, the one-dimensional Skorokhod formula gives, for 0≤s≤t0≤ s≤ t, we have Yix(s)≤|μi|t+sup0≤u≤t|Bi(u)|.Y_i^x(s)≤| _i|t+ _0≤ u≤ t|B_i(u)|. (This also verifies (6.6) since the right side has finite expectation independent of the initial state x.) Hence, with Mt:=|μ|∞t+sup0≤u≤t|B(u)|∞ and CR:=1+maxi∑jRij,M_t:=|μ|_∞t+ _0≤ u≤ t|B(u)|_∞ and C_R:=1+ _i _jR_ij, we have the uniform displacement bound sup0≤s≤t|Zx(s)−x|∞≤CRMt. _0≤ s≤ t|Z^x(s)-x|_∞≤ C_RM_t. Let h∈Cc(E)h∈ C_c(E) and suppose supph⊂|y|∞≤asupph⊂\|y|_∞≤ a\. Then |Pth(x)|≤‖h‖∞ℙCRMt≥|x|∞−a⟶0as |x|∞→∞.|P_th(x)|≤\|h\|_∞P\C_RM_t≥|x|_∞-a\ 0 |x|_∞→∞. Thus Pth∈C0(E)P_th∈ C_0(E) for h∈Cc(E)h∈ C_c(E). By contraction and approximation of C0(E)C_0(E) by compactly supported continuous functions, the same holds for every h∈C0(E)h∈ C_0(E). Finally, every h∈C0(E)h∈ C_0(E) is uniformly continuous. If ωh _h is its modulus of continuity, then supx∈E|Pth(x)−h(x)|≤ωh(CRMt). _x∈ E|P_th(x)-h(x)| \, _h(C_RM_t). Since Mt→0M_t→ 0 almost surely as t↓0t 0 and 0≤ωh≤2‖h‖∞0≤ _h≤ 2\|h\|_∞, dominated convergence gives ‖Pth−h‖∞→0.\|P_th-h\|_∞→ 0. Therefore (Pt)(P_t) is a strongly continuous positive contraction semigroup on C0(E)C_0(E). Then we verify (6.5). Proposition 6.8 ([33, Corollary 3.2]). Let (Σ,μ,R)( ,μ,R) be orthant SRBM data with Σ symmetric positive definite and R completely-S. If there is a symmetric positive definite matrix H with HRHR a Z matrix and Hμ<0Hμ<0 componentwise, then the SRBM is positive recurrent with a unique stationary probability π0 _0 and is V-uniformly exponentially ergodic: there exist a locally bounded V:E→[1,∞)V E→[1,∞) and constants M,κ>0M,κ>0 such that ‖Pt(x,⋅)−π0‖TV≤MV(x)e−κt,x∈E,t≥0. P_t(x,·)- _0 _TV≤ MV(x)e^-κ t, x∈ E,\ t≥ 0. The matrix H in the statement of the corollary is exactly such a certificate: it is symmetric positive definite, HRHR is a Z matrix, and Hμ<0Hμ<0 componentwise, while Σ is positive definite and R is completely-S. Hence Sarantsev’s criterion supplies the stationary probability π0 _0 and the recurrence certificate (6.5). Thus Theorem 6.4 applies. ∎ Set R(a,b,c,d)=(11a11bcd1),Σ=I3,μ=(−11−7−11),R(a,b,c,d)= pmatrix1&1&a\\ 1&1&b\\ c&d&1 pmatrix, =I_3, μ= pmatrix-11\\ -7\\ -11 pmatrix, (6.16) and let P be the parameter region 0<b≤1120,56b+2560≤a≤3b+35,0<c≤425,23≤d≤3546. gathered0<b≤ 1120, 56b+2560≤ a≤ 3b+35, 0<c≤ 425, 23≤ d≤ 3546. gathered (6.17) The subset obtained by making all inequalities strict is nonempty, so P contains a genuine four-dimensional region. Theorem 6.9 (Four-parameter family). For every (a,b,c,d)∈(a,b,c,d) , the SRBM data in (6.16) define a nonsingular completely S, exponentially ergodic SRBM with a unique stationary probability. Its reflection matrix has the singular proper principal block R1,2,1,2=(1111).R_\1,2\,\1,2\= pmatrix1&1\\ 1&1 pmatrix. For every such parameter choice, signed BAR interior uniqueness fails, and the space of zero-mass interior BAR coordinates is infinite-dimensional. Proof. Every entry of R(a,b,c,d)R(a,b,c,d) is positive. Hence every principal submatrix is an S matrix, with the all-ones vector as a witness, and R is completely S. A direct calculation gives detR(a,b,c,d)=(b−a)(c−d). R(a,b,c,d)=(b-a)(c-d). The parameter bounds imply a>ba>b and d>cd>c, so the determinant is positive. The block indexed by A=1,2A=\1,2\ is singular, with v=(1−1)∈kerRAA,w=R3,Av=c−d≠0.v= pmatrix1\\ -1 pmatrix∈ R_A, w=R_\3\,Av=c-d≠ 0. Consider the symmetric matrix H=(12−310−310−31072518−3101823100).H= pmatrix 12&- 310&- 310\\ - 310& 725& 18\\ - 310& 18& 23100 pmatrix. (6.18) Its leading principal minors are 12,120,7980000, 12, 120, 7980000, so H is positive definite. Multiplication gives HR=(15−3c1015−3d10a2−3b10−310c8−150d8−150−3a10+7b25+1823c100−74023d100−740−3a10+b8+23100).HR= pmatrix 15- 3c10& 15- 3d10& a2- 3b10- 310\\[5.69054pt] c8- 150& d8- 150&- 3a10+ 7b25+ 18\\[5.69054pt] 23c100- 740& 23d100- 740&- 3a10+ b8+ 23100 pmatrix. (6.19) The six off-diagonal entries are nonpositive by (6.17). Also Hμ=(−110−7200−21200)<0.Hμ= pmatrix- 110\\[2.84526pt] - 7200\\[2.84526pt] - 21200 pmatrix<0. (6.20) Section 6.3 completes the proof. ∎ For this entire family the algebraic source is explicit. Let S=(0,0,y):y>0S=\(0,0,y):y>0\ and choose a nonconstant ϕ∈Cc∞((0,∞))φ∈ C_c^∞((0,∞)). Define ∫F1gζ1 _F_1g\,d _1 =∫0∞g(0,0,y)ϕ(y)y, = _0^∞g(0,0,y)φ(y)\,dy, (6.21) ∫F2gζ2 _F_2g\,d _2 =−∫0∞g(0,0,y)ϕ(y)y,ζ3=0. =- _0^∞g(0,0,y)φ(y)\,dy, _3=0. (6.22) Since R1−R2=(0,0,c−d)TR_1-R_2=(0,0,c-d)^T, ∑i=13∫FiDifζi=(d−c)∫0∞ϕ′(y)f(0,0,y)y. _i=1^3 _F_iD_if\,d _i=(d-c) _0^∞φ (y)f(0,0,y)\,dy. (6.23) Thus χ(dx)=(d−c)ϕ′(x3)dx3δ0(dx1)δ0(dx2)χ(dx)=(d-c)φ (x_3)\,dx_3\, _0(dx_1) _0(dx_2) (6.24) is nonzero and has total mass zero. Its zero potential and the corresponding boundary occupation potentials give the signed BAR counterexample. Corollary 6.10 (Concrete rational counterexample). For R=(11351116215231),Σ=I3,μ=(−11−7−11),R= pmatrix1&1& 35\\ 1&1& 16\\ 215& 23&1 pmatrix, =I_3, μ= pmatrix-11\\ -7\\ -11 pmatrix, (6.25) there is a finite signed BAR tuple (π¯,ν¯1,ν¯2,ν¯3)( π, ν_1, ν_2, ν_3) such that π¯(E)=0 π(E)=0 and π¯≠0 π≠ 0. Proof. The parameter choice belongs to P. Exact arithmetic gives detR=52225>0,R−1μ=(−365104−203104−12013)<0. R= 52225>0, R^-1μ= pmatrix- 365104\\[2.84526pt] - 203104\\[2.84526pt] - 12013 pmatrix<0. (6.26) The matrix H in (6.18) satisfies HR=(4250−120−130019300−1120−4333000−1360017240),HR= pmatrix 425&0&- 120\\[2.84526pt] - 1300& 19300&- 1120\\[2.84526pt] - 4333000&- 13600& 17240 pmatrix, (6.27) and (6.20) holds. Thus all assumptions in Section 6.3 are verified. For complete explicitness, take the standard bump ϕ(y)=exp(−1(y−1)(2−y)),1<y<2,0,otherwise.φ(y)= cases \! (- 1(y-1)(2-y) ),&1<y<2,\\[5.69054pt] 0,&otherwise. cases (6.28) Define ζ1,ζ2,ζ3 _1, _2, _3 by (6.21)–(6.22). Here R1−R2=(00−815),R_1-R_2= pmatrix0\\ 0\\ - 815 pmatrix, so χ(dx)=815ϕ′(x3)dx3δ0(dx1)δ0(dx2),χ(E)=0,χ≠0.χ(dx)= 815φ (x_3)\,dx_3\, _0(dx_1) _0(dx_2), χ(E)=0, χ≠ 0. (6.29) Let PtP_t be the reflected semigroup and let KiK_i be the kernels in (6.4). Set π¯=∫0∞χPtt,ν¯i=ζi+∑n=0∞(χPn)Ki. π= _0^∞χ P_t\,dt, ν_i= _i+ _n=0^∞(χ P_n)K_i. (6.30) By Section 6.2, all measures in (6.30) are finite and satisfy ∫ELfπ¯+∑i=13∫FiDifν¯i=0,f∈Cb2(E). _ELf\,d π+ _i=1^3 _F_iD_if\,d ν_i=0, f∈ C_b^2(E). Moreover π¯(E)=0 π(E)=0 and π¯≠0 π≠ 0. Therefore π¯ π cannot be a scalar multiple of the stationary probability π0 _0. ∎ 7 Unified understanding: RI defects and boundary algebra This section isolates resolvent insertion as the common mechanism behind both the signed BAR uniqueness theorem and the completely-S obstruction. We view a signed BAR tuple through its interior coordinate and measure the failure of this coordinate to satisfy resolvent insertion by its RI defect. After quotienting out the stationary BAR direction and the pure boundary kernel, the remaining part of the BAR kernel is exactly the RI defect quotient. In the Harrison–Reiman nonsingular M-matrix class this quotient vanishes, while in the completely-S singular-block regime the boundary source and its zero-potential lift produce a nonzero class in this quotient. We denote E=ℳ(E),∂=∏i=1dℳ(Fi),=Cb2(E). M_E=M(E), M_∂= _i=1^dM(F_i), =C_b^2(E). For (π,ν)∈E×∂(π,ν)∈ M_E× M_∂, define the BAR functional (π,ν)(f)=∫ELfπ+∑i=1d∫FiDifνi,A(π,ν)(f)= _ELf\,dπ+ _i=1^d _F_iD_if\,d _i, for f∈f . Thus finite signed BAR tuples are precisely ker . Define the pure boundary operator ∂Rζ(f)=∑i=1d∫FiDifζi _Rζ(f)= _i=1^d _F_iD_if\,d _i. Since one can identify a finite signed measure χ∈ℳ(E)χ (E) with the functional f↦∫Efχf _Ef\,dχ on T, we define the relation ∂Rζ=χ _Rζ=χ by the condition that ∑i=1d∫FiDifζi=∫Efχ _i=1^d _F_iD_if\,d _i= _Ef\,dχ holds for all f∈f . 7.1 The RI-defect quotient Let ZBAR:=kerZ_ BAR:= and ΠE(π,ν)=π _E(π,ν)=π, and define the space of interior coordinates of signed BAR tuples by ℐBAR:=ΠEZBAR=π∈ℳ(E): there exists ν∈∂ with (π,ν)∈ker.I_ BAR:= _EZ_ BAR=\π (E): there exists ν∈ M_∂ with (π,ν)∈ \. (7.1) For a finite signed measure m∈ℳ(E)m (E), define its resolvent-insertion defect by (m)(λ,h):=∫E(λRλh−h)m,λ>0,h∈C0(E). r(m)(λ,h):= _E(λ R_λh-h)\,dm, λ>0, h∈ C_0(E). (7.2) Thus m satisfies the resolvent identity precisely when (m)=0 r(m)=0. Set ℐRI:=ℐBAR∩ker,I_ RI:=I_ BAR∩ r, and define the RI-defect quotient RI:=ℐBAR/ℐRI.Q_ RI:=I_ BAR/I_ RI. Equivalently, RI≃(ℐBAR).Q_ RI r(I_ BAR). This quotient records exactly the part of the signed BAR kernel not killed by resolvent insertion. Lemma 7.1 (BAR quotient by RI defects). Assume that the reflected semigroup is C0C_0-Feller and strongly continuous, that it has a unique invariant probability π0 _0, and that s0=(π0,ν0)s_0=( _0,ν^0) is the stationary BAR tuple. Then there is a canonical exact sequence 0⟶ker∂R⟶ker/ℝs0⟶RI⟶0.0 _R /Rs_0 _ RI 0. (7.3) Equivalently, kerℝs0⊕(0×ker∂R)≃RI. Rs_0 (\0\× _R) _ RI. (7.4) Proof. Let HBAR:=ker/ℝs0.H_ BAR:= /Rs_0. Define Γ:HBAR→RI, :H_ BAR _ RI, and Γ([(π,ν)])=[π]. ([(π,ν)])=[π]. Replacing (π,ν)(π,ν) by (π,ν)+c(π0,ν0)(π,ν)+c( _0,ν^0) changes the interior coordinate by cπ0c _0. Since π0 _0 is invariant, ∫ERλhπ0=∫0∞e−λt∫EPthπ0t=λ−1∫Ehπ0, _ER_λh\,d _0= _0^∞e^-λ t _EP_th\,d _0\,dt=λ^-1 _Eh\,d _0, so (π0)=0 r( _0)=0 and π0∈ℐRI _0 _ RI. Hence [π][π] is unchanged in RIQ_ RI. The map Γ is surjective by the definition of RIQ_ RI. We compute its kernel. Suppose Γ([(π,ν)])=0 ([(π,ν)])=0. Then π∈ℐRIπ _ RI, so ∫E(λRλh−h)π=0,h∈C0(E),λ>0. _E(λ R_λh-h)\,dπ=0, h∈ C_0(E), λ>0. By Section 5, πPt=π P_t=π for all t≥0t≥ 0. By Section 5, we have π=cπ0,π=c _0, where c=π(E).c=π(E). Since both (π,ν)(π,ν) and c(π0,ν0)c( _0,ν^0) are BAR tuples, (0,ν−cν0)=(π,ν)−c(π0,ν0)∈ker.(0,ν-cν^0)=(π,ν)-c( _0,ν^0)∈ . Equivalently, ∂R(ν−cν0)=0. _R(ν-cν^0)=0. Thus [(π,ν)]=[(0,η)][(π,ν)]=[(0,η)] for some η∈ker∂Rη∈ _R. Conversely, if η∈ker∂Rη∈ _R, then (0,η)∈ker(0,η)∈ and its interior coordinate has zero RI defect. Hence kerΓ=[(0,η)]:η∈ker∂R. =\[(0,η)]:η∈ _R\. The map η↦[(0,η)]η [(0,η)] is injective because if [(0,η)]=0[(0,η)]=0 in ker/ℝs0 /Rs_0, then (0,η)=c(π0,ν0)(0,η)=c( _0,ν^0) for some c∈ℝc ; the interior coordinate gives cπ0=0c _0=0, hence c=0c=0 and η=0η=0. This proves the exact sequence (7.3). The quotient isomorphism (7.4) is the corresponding first-isomorphism statement. The sum in the denominator is direct by the same interior-coordinate argument. ∎ Remark 7.2 (Strength of the C0C_0 Feller Assumption). The only assumption we make in this abstraction is the C0C_0 Feller property used in the RI defect quotient which is only a soft semigroup input instead of a boundary regularity assumption. It means that PtC0(E)⊂C0(E),‖Pth−h‖∞→0as t↓0,h∈C0(E).P_tC_0(E)⊂ C_0(E), \|P_th-h\|_∞→ 0 t 0,\ h∈ C_0(E). It is much weaker than strong Feller smoothing, the existence of transition densities, differentiability of PthP_th, or closed domain C2C^2 regularity of the probabilistic resolvent. In particular, it does not assert that RλhR_λh has a classical oblique Neumann trace on the boundary. In the present argument this input is used only through standard semigroup consequences. In Section 5, it makes t⟼∫EPthπ¯t _EP_th\,d π bounded and continuous for h∈C0(E)h∈ C_0(E). The Laplace transform identity then upgrades an almost everywhere conclusion to equality for every t≥0t≥ 0, using uniqueness of the Laplace transform [12, Chapter I]. In Section 6.2, strong continuity is used through the density of the generator domain in C0(E)C_0(E), a standard fact for strongly continuous Feller semigroups [17, Chapter 1, Section 2]. Thus the exact sequence in the unified section could be formulated with these consequences directly: RI null interior coordinates must be signed invariant measures, and the zero potential must be injective on the centered source class. For the positive Harrison–Reiman nonsingular M matrix class, the C0C_0 Feller property is not an extra boundary smoothness input. It is proved in Theorem 3.3. The proof uses the synchronous Lipschitz estimate for the Skorokhod map, imported from [29, Proposition 2.6], together with Brownian path continuity. The same mechanism applies more generally to any orthant SRBM for which one has a pathwise unique continuous construction and a finite horizon Lipschitz estimate of the form sup0≤s≤T|Zsx−Zsy|≤CT|x−y| _0≤ s≤ T|Z_s^x-Z_s^y|≤ C_T|x-y| under a synchronous coupling. Then x↦Pth(x)x P_th(x) is continuous, PthP_th vanishes at infinity, and ‖Pth−h‖∞→0\|P_th-h\|_∞→ 0 follow by the same argument as in the proof of Theorem 3.3. For the completely S counterexample regime, the matrix condition R completely S should not be read as a substitute for the analytic inputs in the RI reduction. It is an existence geometry condition. In the checkable subclass of Section 6.3, Taylor–Williams provide the orthant SRBM existence and Feller framework for completely S reflection data [34], while Sarantsev’s Lyapunov criterion supplies the stationary probability and the V uniform total variation exponential ergodicity used in (6.5) [33, Corollary 3.2]. The one step regulator bound is then checked directly in Section 6.3. These assumptions are sufficient for the zero potential construction, but they do not imply the global RI reduction for all signed BAR tuples. Indeed, in the singular block case the boundary gauge and zero potential construction produce a nonzero RI defect class. Thus the completely S hypothesis alone is not a uniqueness regularity assumption. The minimal zero potential input is also source specific. For a given centered source χ, it is enough to have finite signed measures UχUχ and Θiχ _iχ satisfying (Uχ,Θχ)=−χ,A(Uχ, χ)=-χ, and (Uχ)(E)=0(Uχ)(E)=0 together with injectivity Uχ=0⇒χ=0Uχ=0 χ=0 on the source class under consideration. The quantitative recurrence and regulator assumptions in Section 6.2 are a convenient sufficient package: they imply total variation convergence of ∫0∞χPtt and ∑n≥0(χPn)Ki, _0^∞χ P_t\,dt and _n≥ 0(χ P_n)K_i, justify the Poisson identity, and give the injectivity needed to show that a nonzero centered source yields a nonzero zero mass interior BAR coordinate. The positive theorem in the Harrison–Reiman Class In the Harrison–Reiman nonsingular M-matrix setting, Theorem 4.1 says that every signed BAR tuple satisfies the resolvent identity. Equivalently, (ℐBAR)=0 r(I_ BAR)=0 and RI=0.Q_ RI=0. Then Section 7.1 reduces the quotient ker/ℝs0 /Rs_0 to the pure boundary kernel. The latter is killed by Section 5, i.e. ker∂R=0. _R=\0\. Consequently ker=ℝs0, =Rs_0, which is the signed uniqueness conclusion of Theorem 2.2. This formulation separates the two uses of active-block invertibility. First, RAA−1R_A^-1 defines the active projection Av=v−RARAA−1vA,ARi=0,i∈A. L_Av=v-R_AR_A^-1v_A, L_AR_i=0, i∈ A. This is the algebraic input behind the measure–Neumann resolvent insertion. Second, RAA−TR_A^-T prescribes active oblique jets on SAS_A. If the active normal gradient is a, then (Dif|SA)i∈A=RAATa,(D_if|_S_A)_i∈ A=R_A^Ta, so invertibility of RAAR_A permits the choice a=RAA−Tψa=R_A^-Tψ. The induction over strata in Section 5 uses exactly this prescription. The completely-S obstruction. In this section, we advance our understanding of the zero potential construction as a device for cancelling the source term in the BAR but as a way of placing explicit nonzero elements into RIQ_ RI. Starting from the singular boundary gauge, one has a boundary source identity ∂Rζ=χ. _Rζ=χ. Thus the boundary gauge alone has BAR defect +χ+χ. The zero potential Qχ=(Uχ,Θχ), where Uχ=∫0∞χPtt,Q_χ=(U_χ, _χ), where U_χ= _0^∞χ P_t\,dt, is constructed so that its BAR contribution is exactly the opposite defect: A(Qχ)=−χ.A(Q_χ)=-χ. Therefore we have Qχ+(0,ζ)=(Uχ,Θχ+ζ)∈kerA,Q_χ+(0,ζ)=(U_χ, _χ+ζ)∈ A, so the zero potential turns the boundary gauge into a genuine signed BAR tuple. However, this cancellation does not make the source χ disappear. Instead, we shows that χ reappears as the resolvent insertion defect of the interior measure UχU_χ: r(Uχ)(λ,h)=−∫ERλhχ.r(U_χ)(λ,h)=- _ER_λh\,dχ. Hence, if χ≠0χ≠ 0, then UχU_χ cannot satisfy the resolvent insertion identity; otherwise the strong continuity of the semigroup would imply that χ vanishes on all functions in C0(E)C_0(E), forcing χ=0χ=0. Consequently, the zero potential is the mechanism that converts the singular boundary source into a concrete nonzero class. 0≠[Uχ]∈QRI.0≠[U_χ]∈ Q_RI. This is why the completely S counterexample is best understood as an RI defect: the boundary algebra creates the centered source χ, and the zero potential lifts that source into a genuine BAR tuple whose interior coordinate carries a nonzero resolvent insertion defect. Proposition 7.3 (Boundary sources give RI-defect classes). Assume the semigroup is strongly continuous on C0(E)C_0(E) and that the zero-potential construction of Section 6.2 is available for a centered source χ. Write Qχ=(Uχ,Θχ),Uχ=∫0∞χPtt,Θiχ=∑n=0∞(χPn)Ki.Qχ=(Uχ, χ), Uχ= _0^∞χ P_t\,dt, _iχ= _n=0^∞(χ P_n)K_i. If ζ∈∂ζ∈ M_∂ satisfies ∂Rζ=χ _Rζ=χ, then Qχ+(0,ζ)=(Uχ,Θχ+ζ)∈ker.Qχ+(0,ζ)=(Uχ, χ+ζ)∈ . (7.5) Its image under the map in Section 7.1 is the class [Uχ]∈RI[Uχ] _ RI, and (Uχ)(λ,h)=−∫ERλhχ,λ>0,h∈C0(E). r(Uχ)(λ,h)=- _ER_λh\,dχ, λ>0, h∈ C_0(E). (7.6) In particular, if χ≠0χ≠ 0, then [Uχ]≠0[Uχ]≠ 0 in RIQ_ RI. Proof. By Section 6.2 and fact that ∂Rζ=χ _Rζ=χ, thus (Qχ)(f)=−∫EfχA(Qχ)(f)=- _Ef\,dχ and (0,ζ)(f)=∫EfχA(0,ζ)(f)= _Ef\,dχ holds for all f∈f . Adding the two identities gives (7.5). Hence Uχ∈ℐBARUχ _ BAR and its class in the RI-defect quotient is [Uχ][Uχ]. It remains to identify its defect. For h∈C0(E)h∈ C_0(E) put F(t)=∫EPthχ.F(t)= _EP_th\,dχ. The total-variation convergence in Section 6.2 justifies the following Fubini calculation: (Uχ)(λ,h) r(Uχ)(λ,h) =∫0∞∫EPt(λRλh−h)χt=∫0∞(λ∫0∞e−λsF(t+s)s−F(t))t = _0^∞ _EP_t(λ R_λh-h)\,dχ\,dt= _0^∞ (λ _0^∞e^-λ sF(t+s)\,ds-F(t) )dt =∫0∞F(u)(1−e−λu)u−∫0∞F(u)u=−∫0∞e−λuF(u)u=−∫ERλhχ. = _0^∞F(u)(1-e^-λ u)\,du- _0^∞F(u)\,du=- _0^∞e^-λ uF(u)\,du=- _ER_λh\,dχ. If [Uχ]=0[Uχ]=0 in RIQ_ RI, then (Uχ)=0 r(Uχ)=0, hence χ(Rλh)=0χ(R_λh)=0 for all λ>0λ>0 and h∈C0(E)h∈ C_0(E). Since the semigroup is strongly continuous on C0(E)C_0(E), λRλh=∫0∞e−sPs/λhs⟶hin C0(E)λ R_λh= _0^∞e^-sP_s/λh\,ds h C_0(E) as λ→∞λ→∞. Therefore χ(h)=0χ(h)=0 for every h∈C0(E)h∈ C_0(E), and the finite Radon measure χ is zero. Thus χ≠0χ≠ 0 implies [Uχ]≠0[Uχ]≠ 0. ∎ Suppose that R is nonsingular and that RAAR_A is singular for some nonempty proper subset A⊊JA J. Put T=J∖AT=J A and choose 0≠v∈kerRAA,0≠ v∈ R_A, where w=RTAv.w=R_TAv. Since R is nonsingular, w≠0w≠ 0. On SAS_A, the local boundary symbol is ∑i∈AviDif=(RAAv)⋅∇Af+(RTAv)⋅∇Tf. _i∈ Av_iD_if=(R_Av)· _Af+(R_TAv)· _Tf. (7.7) Because RAAv=0R_Av=0, only the tangential derivative w⋅∇Tfw· _Tf remains. Section 6.1 turns this symbol calculation into a boundary source: for a compactly supported smooth density ϕφ on the open stratum with w⋅∇Tϕ≢0w· _Tφ ≡ 0, it constructs boundary measures ζi _i and a finite nonzero centered measure χ supported on SAS_A such that ∂Rζ=χ. _Rζ=χ. Under the recurrence and regulator hypotheses of Section 6.2, Section 7.1 gives (Uχ,Θχ+ζ)∈ker(Uχ, χ+ζ)∈ and places its interior coordinate into the RI-defect quotient as 0≠[Uχ]∈RI.0≠[Uχ] _ RI. Thus signed BAR uniqueness fails. In fact, the singular-block construction does more than produce a BAR tuple with zero total interior mass: it produces a concrete nonzero resolvent-insertion defect, (Uχ)(λ,h)=−χ(Rλh). r(Uχ)(λ,h)=-χ(R_λh). Therefore a global resolvent insertion theorem cannot hold in this singular-block regime once the zero-potential lift is available. Acknowledgment The authors would like to thank Jose Blanchet for bringing this open problem to our attention and encouraging us to pursue an AI-based solution. We are also grateful to Jose Blanchet, Yufan Chen and Wenhao Yang for their valuable feedback on this manuscript. The authors are also grateful to Bin Dong, Xiao Ma, Jiajin Li and Jianfeng Lu for their insightful discussions regarding the application of AI in mathematical proving and the formulation of our AI usage disclosure. Appendix A A stable example with g=Rλh∉C2(E)g=R_λh∉ C^2(E) This appendix gives a concrete example in which the probabilistic resolvent g=Rλhg=R_λh is not in C2(E)C^2(E). The point is that interior smoothness and one-sided oblique flatness on open faces do not guarantee closed-domain C2C^2 regularity at a corner. We choose a smooth nonnegative source h∈Cc∞(E∘)h∈ C_c^∞(E ) such that h≢0h ≡ 0 but h(0)=0h(0)=0. We first justify that g(0)>0g(0)>0. Since h≥0h≥ 0, h≢0h ≡ 0, and supph⊂E∘supph⊂ E , choose z∗∈E∘z_ ∈ E , r>0r>0, and ch>0c_h>0 such that B(z∗,r)¯⊂E∘ B(z_ ,r)⊂ E and h≥chh≥ c_h on B(z∗,r)¯ B(z_ ,r). Let Γ be the Harrison–Reiman Skorokhod map. On [0,2][0,2], Γ is Lipschitz with constant KΓK_ . Define γ(t)=tz∗,0≤t≤1,γ(t)=z∗,1≤t≤2.γ(t)=tz_ , 0≤ t≤ 1, γ(t)=z_ , 1≤ t≤ 2. Since γ stays in E, Γ(γ)=γ (γ)=γ. For the SRBM started from zero, the free input is X(t)=μt+W(t)X(t)=μ t+W(t). Put b(t)=γ(t)−μtb(t)=γ(t)-μ t. Since Brownian motion has full support in C0([0,2],ℝ3)C_0([0,2],R^3), the event A=sup0≤t≤2|W(t)−b(t)|<r/KΓA= \ _0≤ t≤ 2|W(t)-b(t)|<r/K_ \ has positive probability. On A, the free input X is within r/KΓr/K_ of γ, and therefore the reflected path Z0=Γ(X)Z^0= (X) is within r of γ. Since γ(t)=z∗γ(t)=z_ for 1≤t≤21≤ t≤ 2, we have Z0(t)∈B(z∗,r)Z^0(t)∈ B(z_ ,r) throughout [1,2][1,2] on A. Hence g(0)≥ℙ(A)ch∫12e−λtt>0.g(0) (A)c_h _1^2e^-λ t\,dt>0. On the other hand, if g were C2C^2 up to the corner and satisfied the exact face conditions, those conditions would force ∇g(0)=0∇ g(0)=0 and D2g(0)=0D^2g(0)=0. The resolvent equation at the corner would then give 0=h(0)=(λ−L)g(0)=λg(0)0=h(0)=(λ-L)g(0)=λ g(0), contradicting g(0)>0g(0)>0. Consider d=3d=3, Σ=I3 =I_3, and R=(10−12−12100−121),R−1=17(824482248)≥0.R= pmatrix1&0&- 12\\ - 12&1&0\\ 0&- 12&1 pmatrix, R^-1= 17 pmatrix8&2&4\\ 4&8&2\\ 2&4&8 pmatrix≥ 0. Thus R is a nonsingular M-matrix. Let μ=−Rμ=-R1; then R−1μ=−<0R^-1μ=-1<0, so the data are stable in the sense of (2.2). Choose h∈Cc∞(E∘)h∈ C_c^∞(E ) with h≥0h≥ 0 and h≢0h ≡ 0, and set g=Rλhg=R_λh for some λ>0λ>0. Since the Brownian input has full support on compact time intervals and the Harrison–Reiman Skorokhod map is continuous, the SRBM started from the origin has positive probability of entering a ball on which h>0h>0 and then remaining there for a nonzero time interval. Hence g(0)=0∫0∞e−λth(Zt)t>0g(0)=E_0 _0^∞e^-λ th(Z_t)\,dt>0. Also h(0)=0h(0)=0, because h is supported in the interior. Assume, for contradiction, that g∈C2(E)g∈ C^2(E) in the closed-domain sense. On each open face Fi∘F_i , the one-sided derivative identity of Section 3.3 applies in the feasible direction RiR_i. Since xRi=0 L_xR_i=0 for x∈Fi∘x∈ F_i , it gives Dig=0D_ig=0 on Fi∘F_i . By continuity of the first derivatives, these identities extend to the origin as RT∇g(0)=0R^T∇ g(0)=0. Since R is invertible, ∇g(0)=0∇ g(0)=0. Let H=D2g(0)H=D^2g(0). For j≠ij≠ i, differentiating Dig=Ri⋅∇g=0D_ig=R_i·∇ g=0 in the tangential direction eje_j along Fi∘F_i and then letting the tangential point tend to the origin gives RiTHej=0R_i^THe_j=0. Equivalently, offdiag(RTH)=0offdiag(R^TH)=0. Thus RTH=diag(d1,d2,d3)R^TH=diag(d_1,d_2,d_3) for some real numbers d1,d2,d3d_1,d_2,d_3, and therefore H=R−Tdiag(d1,d2,d3)=17(8d14d22d32d18d24d34d12d28d3).H=R^-Tdiag(d_1,d_2,d_3)= 17 pmatrix8d_1&4d_2&2d_3\\ 2d_1&8d_2&4d_3\\ 4d_1&2d_2&8d_3 pmatrix. Since H is symmetric, comparison of the (1,2)(1,2), (2,3)(2,3), and (1,3)(1,3) entries gives d1=2d2d_1=2d_2, d2=2d3d_2=2d_3, and d3=2d1d_3=2d_1. Hence d1=d2=d3=0d_1=d_2=d_3=0, so H=0H=0. The interior resolvent equation gives (λ−L)g=h(λ-L)g=h on E∘E . 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