🚨 We just used AI to solve a 35-year-old open math problem! In queuing theory, BAR is the “master equation” for finding networks reach a equilibrium. However, a core uniqueness conjecture about BAR remain open Here is how human + AI cracked the case👇1/7 https://t.co/YPf3hghyT7 https://t.co/ZGNcdVuSvG

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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

Source: https://arxiv.org/html/2607.03639v1 Youheng ZhuDepartment 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 nonsingularMM-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 nonsingularMM-matrix assumption is structural. In the larger completely-𝒮\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–ReimanMM-matrix class and a negative answer in a natural completely-𝒮\mathcal{S}extension.

completely S matrix,

keywords:

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1Introduction

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]. Ifπ\piis an interior measure andνi\nu_{i}is a boundary measure on the faceFi={xi=0}F_{i}=\{x_{i}=0\}, the BAR has the form

∫EL​f​𝑑π+∑i=1d∫FiDi​f​𝑑νi=0,f∈Cb2​(E),\int_{E}Lf\,d\pi+\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\nu_{i}=0,\qquad f\in C_{b}^{2}(E),(1.1)whereLLis the interior diffusion generator andDiD_{i}is the directional derivative in theiith reflection direction. The stationary distributionπ0\pi_{0}, together with its stationary boundary occupation measuresνi0\nu_{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 add-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 boundedC2C^{2}tests, proved the corresponding characterization in the positive-measure setting, and asked for the signed analogue. The compactly supportedC2C^{2}formulation leads to the same finite-signed uniqueness problem. The bounded-test identity immediately implies the compactly supported one. Conversely, letf∈Cb2​(E)f\in C_{b}^{2}(E)and chooseχn∈Cc∞​(ℝd)\chi_{n}\in C_{c}^{\infty}(\mathbb{R}^{d})with0≤χn≤10\leq\chi_{n}\leq 1,χn=1\chi_{n}=1on{|x|≤n}\{|x|\leq n\}, and‖∇χn‖∞+‖D2​χn‖∞→0\|\nabla\chi_{n}\|_{\infty}+\|D^{2}\chi_{n}\|_{\infty}\to 0. Applying the compactly supported identity toχn​f\chi_{n}fand expandingL​(χn​f)L(\chi_{n}f)andDi​(χn​f)D_{i}(\chi_{n}f)gives the bounded-test identity after passage to the limit, becauseχn→1\chi_{n}\to 1pointwise and all error terms are uniformly bounded by constants times‖∇χn‖∞+‖D2​χn‖∞\|\nabla\chi_{n}\|_{\infty}+\|D^{2}\chi_{n}\|_{\infty}against finite signed measures. Throughout the paper we therefore use the bounded-test classCb2​(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 allowsπ\piand theνi\nu_{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-MM-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 statedMM-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\mathcal{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]. Section6shows 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 nonsingularMM-matrix reflection matrix. The proof is organized around a resolvent invariant identity. LetRλ​h=∫0∞e−λ​t​Pt​h​𝑑tR_{\lambda}h=\int_{0}^{\infty}e^{-\lambda t}P_{t}h\,dtbe the probabilistic resolvent of the reflected semigroup. Our core contribution is proving the fact that every finite signed BAR tuple satisfies

∫E(λ​Rλ​h−h)​𝑑π¯=0,h∈C0​(E),λ>0.\int_{E}(\lambda R_{\lambda}h-h)\,d\bar{\pi}=0,\qquad h\in C_{0}(E),\quad\lambda>0.(RI)This identity says exactly that the interior signed measure is invariant under the reflected semigroup. Indeed, usingRλ​h=∫0∞e−λ​t​Pt​h​𝑑tR_{\lambda}h=\int_{0}^{\infty}e^{-\lambda t}P_{t}h\,dt, (RI) says that the Laplace transform oft↦π¯​(Pt​h)−π¯​(h)t\mapsto\bar{\pi}(P_{t}h)-\bar{\pi}(h)vanishes for everyh∈C0​(E)h\in C_{0}(E). Strong continuity of the Feller semigroup upgrades this toπ¯​Pt=π¯\bar{\pi}P_{t}=\bar{\pi}for allt≥0t\geq 0. Ifπ¯=π¯+−π¯−\bar{\pi}=\bar{\pi}^{+}-\bar{\pi}^{-}is the Jordan decomposition, positivity of the Markov kernel gives|π¯​Pt|≤|π¯|​Pt|\bar{\pi}P_{t}|\leq|\bar{\pi}|P_{t}; equality of total masses then makes|π¯||\bar{\pi}|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 givesπ¯=c​π0\bar{\pi}=c\pi_{0}. Subtractingcctimes 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, ifg=Rλ​hg=R_{\lambda}hwere an admissibleCb2C_{b}^{2}test satisfyingDi​g=0D_{i}g=0onFiF_{i}, then (RI) would follow by insertinggginto the BAR and using(λ−L)​g=h(\lambda-L)g=h. This formal argument is misleading because at corners the resolvent need not be a classicalC2C^{2}function on the closed orthant;AppendixAgives a stable Harrison–Reiman example where suchC2C^{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 insertg=Rλ​hg=R_{\lambda}hitself into the BAR. Instead we replace it by the one-sided smoothing

gε​(x)=∫ρ​(w)​g​(x+ε​w)​𝑑w.g_{\varepsilon}(x)=\int\rho(w)g(x+\varepsilon w)\,dw.The mollifier is supported strictly inside the positive orthant, so the value ofgε​(x)g_{\varepsilon}(x)only uses values ofggat interior pointsx+ε​wx+\varepsilon w. This smoothing supplies the required boundedC2C^{2}regularity for each fixedε\varepsilon. The only delicate point is to show that these legitimateCb2C_{b}^{2}tests have asymptotically zero boundary contribution. The projected boundary derivative of the resolvent gives

Di​gε​(x)⟶0,x∈Fi,D_{i}g_{\varepsilon}(x)\longrightarrow 0,\qquad x\in F_{i},with a uniform bound sufficient for dominated convergence against an arbitrary finite signed boundary measure. Thus the functionsgεg_{\varepsilon}approximate the resolvent in exactly the topology seen by the BAR: the interior equation converges to(λ−L)​Rλ​h=h(\lambda-L)R_{\lambda}h=h, while all boundary terms vanish.

The paper also explains why theMM-matrix hypothesis is not merely a proof artifact. In the completely-𝒮\mathcal{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\mathcal{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\mathcal{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\mathcal{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 ford>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, seehttps://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-eecb9ecc898dandhttps://claude.ai/share/25a16238-360a-4649-935f-b23b4ec500ff(Attemptshttps://chatgpt.com/share/6a44b084-91dc-83ea-8fc2-49b06770025dto 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\mathcal{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:Section2states the SRBM and BAR setting, states the main theorem, and reduces the proof to the resolvent identity (RI).Section3establishes the two technical properties ofg=Rλ​hg=R_{\lambda}hneeded later for the approximation: the interior resolvent equation and the projected boundary derivative that will makeDi​gεD_{i}g_{\varepsilon}vanish onFiF_{i}.Section4carries out the one-sided smoothing construction, insertsgε∈Cb2​(E)g_{\varepsilon}\in C_{b}^{2}(E)directly into the BAR, and proves (RI).Section5proves the implication deferred inSection2: the identity (RI) implies the signed BAR uniqueness conjecture, thus finishing the proof of the main theorem.Section6explains why the nonsingularMM-matrix condition is structural by giving the completely-𝒮\mathcal{S}obstruction and an explicit three-dimensional family.Section7repackages the positive and negative arguments through a common BAR homotopy lemma and separates the remaining issue into local boundary algebra.

2Setting, 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 ofTheorem3.3: there each source hypothesis is recalled in the present orthant specialization and verified.

2.1Notation and standing conventions

LetJ={1,…,d}J=\{1,\ldots,d\},E=ℝ+dE=\mathbb{R}_{+}^{d}, andE∘=(0,∞)dE^{\circ}=(0,\infty)^{d}. Fori∈Ji\in JwriteFi={x∈E:xi=0}.F_{i}=\{x\in E:x_{i}=0\}.For nonemptyA⊂JA\subset J, define the relative boundary stratumSA={x∈E:xi=0​(i∈A),xj>0​(j∉A)}.S_{A}=\{x\in E:x_{i}=0\ (i\in A),\ x_{j}>0\ (j\notin A)\}.The setsSAS_{A}form a disjoint Borel decomposition of∂E\partial E.

For a locally compact spaceBB,C0​(B)C_{0}(B)denotes the continuous real-valued functions vanishing at infinity, andℳ​(B)\mathcal{M}(B)denotes the finite signed Radon measures onBB. Forη∈ℳ​(B)\eta\in\mathcal{M}(B),|η||\eta|is its total variation measure and‖η‖TV=|η|​(B)\|\eta\|_{\mathrm{TV}}=|\eta|(B). We writesupp⁡η\operatorname{supp}\etafor the support of a measure andsupp⁡f\operatorname{supp}ffor the support of a function. The symbol𝟏B\mathbf{1}_{B}denotes the indicator of a setBB.

We use the closed-domainC2C^{2}convention. ThusC2​(E)C^{2}(E)consists of functionsf:E→ℝf:E\to\mathbb{R}such thatf∈C2​(E∘)f\in C^{2}(E^{\circ})and all partial derivatives∂αf\partial^{\alpha}f,|α|≤2|\alpha|\leq 2, extend continuously fromE∘E^{\circ}toEE. The classCc2​(E)C_{c}^{2}(E)consists of the functions inC2​(E)C^{2}(E)with compact support as a subset ofEE. The classCb2​(E)C_{b}^{2}(E)consists of the functions inC2​(E)C^{2}(E)for whichff,∇f\nabla fandD2​fD^{2}fare bounded. SinceEEis the orthant, this closed-domain convention is equivalent to saying that everyf∈C2​(E)f\in C^{2}(E)is the restriction toEEof someF∈C2​(U)F\in C^{2}(U)on an open neighborhoodU⊃EU\supset E. For open subsets of Euclidean space,Cc∞C_{c}^{\infty}has its usual meaning. For the reflected semigroup we writePt​h​(x)=𝔼​[h​(Ztx)]P_{t}h(x)=\mathbb{E}[h(Z_{t}^{x})]andRλ​h​(x)=∫0∞e−λ​t​Pt​h​(x)​𝑑t,λ>0,R_{\lambda}h(x)=\int_{0}^{\infty}e^{-\lambda t}P_{t}h(x)\,dt,\;\lambda>0,whenever the integral is finite. We call the semigroupPtP_{t}Feller if(Pt)t≥0(P_{t})_{t\geq 0}satisfiesPt​C0​(E)⊂C0​(E)P_{t}C_{0}(E)\subset C_{0}(E), and is strongly continuous, i.e.‖Pt​h−h‖∞→0\|P_{t}h-h\|_{\infty}\to 0ast↓0t\downarrow 0for allh∈C0​(E)h\in C_{0}(E).

2.2SRBM, BAR, and finite signed BAR tuples

A semimartingale reflected Brownian motion inEEis specified by a drift vectorμ∈ℝd\mu\in\mathbb{R}^{d}, a symmetric positive definite covariance matrixΣ\Sigma, and a reflection matrixR=(R1,…,Rd)R=(R_{1},\ldots,R_{d})whoseiith column is the direction of reflection onFiF_{i}. PutQ=Σ/2Q=\Sigma/2and

L​f=μ⋅∇f+Q:D2​f,Di​f=Ri⋅∇f.Lf=\mu\cdot\nabla f+Q:D^{2}f,\qquad D_{i}f=R_{i}\cdot\nabla f. Throughout the positive part of the paper we work under the following stable nonsingularMM-matrix data. The covariance matrixΣ\Sigmais symmetric positive definite. The reflection matrixRRsatisfies

Ri​i>0,Ri​j≤0​(i≠j),R−1≥0.R_{ii}>0,\qquad R_{ij}\leq 0\ (i\neq j),\qquad R^{-1}\geq 0.(2.1)The drift satisfies

componentwise. The phrase “stable” in this paper means exactly (2.2). The linear-algebra consequences of (2.1) are proved inSection3.1; the stochastic consequences used later are stated inTheorem3.3and 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)(\bar{\pi},\bar{\nu}_{1},\ldots,\bar{\nu}_{d})\in\mathcal{M}(E)\times\prod_{i=1}^{d}\mathcal{M}(F_{i})of finite signed Radon measures satisfying

∫EL​f​𝑑π¯+∑i=1d∫FiDi​f​𝑑ν¯i=0,f∈Cb2​(E).\int_{E}Lf\,d\bar{\pi}+\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\bar{\nu}_{i}=0,\qquad f\in C_{b}^{2}(E).(2.3)The stationary regulator defines finite boundary occupation measuresνi0\nu_{i}^{0}, and the stationary BAR is

∫EL​f​𝑑π0+∑i=1d∫FiDi​f​𝑑νi0=0,f∈Cb2​(E).\int_{E}Lf\,d\pi_{0}+\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\nu_{i}^{0}=0,\qquad f\in 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 distributionπ0\pi_{0}to the SRBM. Obviously, the stationary distribution and the finite stationary boundary measure characterized by the followingSection2.2together provide a solution to the BAR equation (2.4).

Proposition 2.1(Finite stationary boundary measures and stationary BAR).

Start the SRBM withZ0∼π0Z_{0}\sim\pi_{0}and write it in the original normalization as

Zt=Z0+μ​t+Σ1/2​Wt+R​Yt,Z_{t}=Z_{0}+\mu t+\Sigma^{1/2}W_{t}+RY_{t},(2.5)where eachYiY_{i}is continuous, nondecreasing, starts from zero, and increases only onFiF_{i}. Define, for BorelB⊂FiB\subset F_{i},

νi0​(B)=𝔼π0​∫01𝟏B​(Zs)​𝑑Yi​(s).\nu_{i}^{0}(B)=\mathbb{E}_{\pi_{0}}\int_{0}^{1}\mathbf{1}_{B}(Z_{s})\,dY_{i}(s).(2.6)Then eachνi0\nu_{i}^{0}is a finite measure supported onFiF_{i}, and (2.4) holds.

Proof.

Leta=R−T​𝟏a=R^{-T}\mathbf{1}. SinceR−1≥0R^{-1}\geq 0and no column of the invertible matrixR−1R^{-1}is zero,a>0a>0; moreoverRT​a=𝟏R^{T}a=\mathbf{1}. Forα>0\alpha>0, set

Φα​(x)=−∑k=1dak​e−α​xk.\Phi_{\alpha}(x)=-\sum_{k=1}^{d}a_{k}e^{-\alpha x_{k}}.The function and its first two derivatives are bounded. Ifx∈Fix\in F_{i}, then, usingRk​i≤0R_{ki}\leq 0fork≠ik\neq i,xi=0x_{i}=0, ande−α​xk≤1e^{-\alpha x_{k}}\leq 1,

Di​Φα​(x)=α​∑k=1dRk​i​ak​e−α​xk≥α​∑k=1dRk​i​ak=α.D_{i}\Phi_{\alpha}(x)=\alpha\sum_{k=1}^{d}R_{ki}a_{k}e^{-\alpha x_{k}}\geq\alpha\sum_{k=1}^{d}R_{ki}a_{k}=\alpha.Itô’s formula on[0,1][0,1]gives, pathwise,

Φα​(Z1)−Φα​(Z0)=∫01L​Φα​(Zs)​𝑑s+M1+∑i=1d∫01Di​Φα​(Zs)​𝑑Yi​(s),\Phi_{\alpha}(Z_{1})-\Phi_{\alpha}(Z_{0})=\int_{0}^{1}L\Phi_{\alpha}(Z_{s})\,ds+M_{1}+\sum_{i=1}^{d}\int_{0}^{1}D_{i}\Phi_{\alpha}(Z_{s})\,dY_{i}(s),whereMMis a square-integrable martingale because∇Φα\nabla\Phi_{\alpha}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=1d𝔼π0​Yi​(1)≤−𝔼π0​∫01L​Φα​(Zs)​𝑑s≤‖L​Φα‖∞.\alpha\sum_{i=1}^{d}\mathbb{E}_{\pi_{0}}Y_{i}(1)\leq-\mathbb{E}_{\pi_{0}}\int_{0}^{1}L\Phi_{\alpha}(Z_{s})\,ds\leq\|L\Phi_{\alpha}\|_{\infty}.Thus (2.6) is finite. Its support is contained inFiF_{i}becauseYiY_{i}increases only there. Finally, apply Itô’s formula tof∈Cb2​(E)f\in C_{b}^{2}(E). The boundedness offf,∇f\nabla fandD2​fD^{2}fmakes the Brownian and drift terms integrable on[0,1][0,1], and the boundary integrals are integrable by the preceding estimate. Stationarity gives

0=∫EL​f​𝑑π0+∑i=1d∫FiDi​f​𝑑νi0.0=\int_{E}Lf\,d\pi_{0}+\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\nu_{i}^{0}.∎

AlthoughSection2.2provides(π0,ν10,…,νd0)(\pi_{0},\nu_{1}^{0},\dots,\nu_{d}^{0})as a solution to the BAR equation, it remains an open questionwhether the BAR uniquely characterizes the stationary distribution of the diffusion process.

2.3Signed 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), letπ0\pi_{0}and(νi0)i=1d(\nu_{i}^{0})_{i=1}^{d}be the stationary distribution of the process and the corresponding boundary measure constructed inSection2.2. Every finite signed BAR tuple is a scalar multiple of the stationary BAR tuple. More precisely, if (2.3) holds, then there existsc∈ℝc\in\mathbb{R}such that

π¯=c​π0,ν¯i=c​νi0,i=1,…,d.\bar{\pi}=c\pi_{0},\qquad\bar{\nu}_{i}=c\nu_{i}^{0},\quad i=1,\ldots,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 identityresolvent insertion. The resolvent insertion identity implies invariance of the interior signed measure under the reflected semigroup, and henceπ¯=c​π0\bar{\pi}=c\pi_{0}. After subtracting the interior stationary BAR, the remaining identity is purely on the boundary, and pure boundary injectivity givesν¯i=c​νi0\bar{\nu}_{i}=c\nu_{i}^{0}for alli=1,…,di=1,\ldots,d.

Proposition 2.3(Resolvent identity criterion).

Assume that for every finite signed BAR tuple, everyh∈C0​(E)h\in C_{0}(E), and everyλ>0\lambda>0,

∫E(λ​Rλ​h−h)​𝑑π¯=0.\int_{E}(\lambda R_{\lambda}h-h)\,d\bar{\pi}=0.(RI)Then the conclusion ofTheorem2.2holds.

The proof ofSection2.3is given inSection5.

Why the resolvent insertion (RI) should hold

The reason for targeting (RI) is transparent from the classical Neumann calculation. Letg=Rλ​hg=R_{\lambda}h. Ifggwere an admissibleCb2C_{b}^{2}test and if it satisfiedDi​g=0D_{i}g=0onFiF_{i}fori=1,…,di=1,\ldots,d, then insertinggginto the BAR would give

0=∫EL​g​𝑑π¯+∑i∫FiDi​g​𝑑ν¯i=∫EL​g​𝑑π¯.0=\int_{E}Lg\,d\bar{\pi}+\sum_{i}\int_{F_{i}}D_{i}g\,d\bar{\nu}_{i}=\int_{E}Lg\,d\bar{\pi}.The resolvent equation(λ−L)​g=h(\lambda-L)g=hwould therefore imply∫E(λ​Rλ​h−h)​𝑑π¯=0.\int_{E}(\lambda R_{\lambda}h-h)\,d\bar{\pi}=0.This is only an informal guide. The closed-domainC2C^{2}regularity required for this insertion may fail even in the stable Harrison–Reiman class.AppendixAgives an explicit stable nonsingularMM-matrix example and a smooth compactly supportedhhfor whichRλ​h∉C2​(E)R_{\lambda}h\notin C^{2}(E). The proof below therefore does not try to show that the resolvent belongs to a classical oblique-Neumann core.

2.4Making the resolvent insertion rigorous

The replacement for the formal insertion is a measure-level Neumann approximation. For smooth compactly supportedhhwe construct testsgε∈Cb2​(E)g_{\varepsilon}\in C_{b}^{2}(E)such that, asε↓0\varepsilon\downarrow 0,

gε→Rλ​h,(λ−L)​gε→h,g_{\varepsilon}\to R_{\lambda}h,\qquad(\lambda-L)g_{\varepsilon}\to h,against every finite signed interior measure, while

∫FiDi​gε​𝑑ν¯i→0,i=1,…,d,\int_{F_{i}}D_{i}g_{\varepsilon}\,d\bar{\nu}_{i}\to 0,\qquad i=1,\ldots,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 thatRλ​hR_{\lambda}hadmits a classical oblique derivativeDi​Rλ​hD_{i}R_{\lambda}honFiF_{i}; it is an assertion that the boundary pairings seen by the BAR vanish.Section4gives the precise statement, and a density argument then extends (RI) from smooth compactly supportedhhto allh∈C0​(E)h\in C_{0}(E). The next two sections supply the projected derivative input and the one-sided smoothing construction.

Figure1summarizes 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)=λ​π¯​(Rλ​h)\bar{\pi}(h)=\lambda\bar{\pi}(R_{\lambda}h)leads to signed BAR uniquenessProvingπ¯​(h)=λ​π¯​(Rλ​h)\bar{\pi}(h)=\lambda\bar{\pi}(R_{\lambda}h): how the missing resolvent boundary regularity is bypassedTarget identity: the hard stepFor everyh∈Cc∞​(ℝd)h\in C_{c}^{\infty}(\mathbb{R}^{d})andλ>0\lambda>0,π¯​(h)=λ​π¯​(Rλ​h).\bar{\pi}(h)=\lambda\,\bar{\pi}(R_{\lambda}h).This is the resolvent form of stationarity.Laplace uniqueness converts resolvents to invarianceDefine the defectAh​(t):=π¯​(Pt​h)−π¯​(h).A_{h}(t):=\bar{\pi}(P_{t}h)-\bar{\pi}(h).UsingRλ​h=∫0∞e−λ​t​Pt​h​𝑑tR_{\lambda}h=\int_{0}^{\infty}e^{-\lambda t}P_{t}h\,dt, Fubini turns the target identity into∫0∞e−λ​t​Ah​(t)​𝑑t=0(λ>0).\int_{0}^{\infty}e^{-\lambda t}A_{h}(t)\,dt=0\qquad(\lambda>0).Semigroup invarianceThusπ¯​(Pt​h)=π¯​(h)\bar{\pi}(P_{t}h)=\bar{\pi}(h)for allt≥0t\geq 0.Density ofCc∞|EC_{c}^{\infty}|_{E}inC0​(E)C_{0}(E), plus contraction ofPtP_{t}, extends this to allϕ∈C0​(E)\phi\in C_{0}(E). Henceπ¯​Pt=π¯\bar{\pi}P_{t}=\bar{\pi}.Interior measure is then forcedFor a finite signed invariant measure, positivity gives|μ​Pt|≤|μ|​Pt|\mu P_{t}|\leq|\mu|P_{t}. Total mass equality makes the Jordan parts invariant.Uniqueness ofπ0\pi_{0}givesπ¯=c​π0\bar{\pi}=c\pi_{0}.Boundary measures then followSubtractcctimes the stationary BAR. On each stratumSAS_{A},RA​A−TR_{AA}^{-T}prescribes the active oblique jets(Di​f)i∈A(D_{i}f)_{i\in A}.Induction over|A||A|givesν¯i=c​νi0\bar{\nu}_{i}=c\nu_{i}^{0}.Signed BAR uniqueness(π¯,ν¯1,…,ν¯d)=c​(π0,ν10,…,νd0).\displaystyle(\bar{\pi},\bar{\nu}_{1},\ldots,\bar{\nu}_{d})=c(\pi_{0},\nu_{1}^{0},\ldots,\nu_{d}^{0}).π¯=c​π0\bar{\pi}=c\pi_{0}Why direct insertion failsThe natural test isg=Rλ​hg=R_{\lambda}h, because(λ−L)​g=h(\lambda-L)g=hin the interior.But the BAR accepts boundedC2C^{2}tests and boundary termsDi​fD_{i}f. At corners,ggneed not have a classical ambient gradient, soDi​g=0D_{i}g=0is unavailable.Projected derivative replaces a boundary gradientFor feasible inward directions,∂w+g​(x)=Λx​(𝖫x​w).\partial_{w}^{+}g(x)=\Lambda_{x}(\mathsf{L}_{x}w).Ifx∈Fix\in F_{i}, then𝖫x​Ri=0\mathsf{L}_{x}R_{i}=0, hence the ambient linear extension satisfiesℓx​(Ri)=0\ell_{x}(R_{i})=0. This is the usable oblique information.One-sided smoothing turns it into BAR testsSmooth only from inside:gε​(x)=∫ρ​(w)​g​(x+ε​w)​𝑑w.g_{\varepsilon}(x)=\int\rho(w)g(x+\varepsilon w)\,dw.The supportsupp⁡ρ⋐(0,∞)d\operatorname{supp}\rho\Subset(0,\infty)^{d}keeps every sampled direction feasible. Integration by parts and domination giveDi​gε→0D_{i}g_{\varepsilon}\to 0onFiF_{i}.Measure–Neumann approximationUsegε∈Cb2​(E)g_{\varepsilon}\in C_{b}^{2}(E)directly. Apply the signed BAR and sendε↓0\varepsilon\downarrow 0. Boundary integrals vanish for every finite signedν¯i\bar{\nu}_{i}; interior terms converge toπ¯​(h)=λ​π¯​(Rλ​h)\bar{\pi}(h)=\lambda\bar{\pi}(R_{\lambda}h).

Figure 1:Proof architecture for the uniqueness of the Harrison-Reiman class.The lower half is the analytic insertion mechanism:Section3supplies the projected derivative used by the one-sided smoothing, andSection4turns 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 fixedhhandλ\lambda, letg=Rλ​hg=R_{\lambda}h. The smoothed BAR tests used later are

gε​(x)=∫ρ​(w)​g​(x+ε​w)​𝑑w,supp⁡ρ⊂(1,2)d.g_{\varepsilon}(x)=\int\rho(w)g(x+\varepsilon w)\,dw,\qquad\operatorname{supp}\rho\subset(1,2)^{d}.They must approximateggin the interior equation while also satisfying an asymptotic oblique-Neumann condition on each face:Di​gε→0D_{i}g_{\varepsilon}\to 0in pairings with arbitrary finite signed measures onFiF_{i}. This is whySection3proves a boundary statement forggitself before any smoothing is introduced. Althoughggneed not beC2C^{2}on the closed orthant, its feasible one-sided derivatives exist at boundary points and factor through the active tangent projection; the resulting linear extensionℓx\ell_{x}satisfiesℓx​(Ri)=0\ell_{x}(R_{i})=0on active faces, acting as an analog to the classical gradient. The one-sided convolutiongεg_{\varepsilon}inSection4is then precisely designed to inherit this first-order oblique flatness in the weaker, measure-level form needed by the BAR.

3Resolvent regularity and projected boundary derivatives

The goal of this section is to proveSection3, the input that makes the measure–Neumann approximation inSection4possible. Section4will constructgε∈Cb2​(E)g_{\varepsilon}\in C_{b}^{2}(E)fromg=Rλ​hg=R_{\lambda}hand will need three properties:gε→gg_{\varepsilon}\to g,(λ−L)​gε→h(\lambda-L)g_{\varepsilon}\to h, andDi​gε→0D_{i}g_{\varepsilon}\to 0onFiF_{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 ofggfactors through the active tangent projection. Combining this factorization with the identity𝖫x​Ri=0\mathsf{L}_{x}R_{i}=0gives the usable oblique informationℓx​(Ri)=0\ell_{x}(R_{i})=0onFiF_{i}, which is exactly what later forcesDi​gε→0D_{i}g_{\varepsilon}\to 0in boundary-measure pairings.

The proof has two ingredients. The algebraic ingredient is the nonsingularity of every active principal reflection block, which gives the explicit projection𝖫x\mathsf{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 ofTheorem3.3, 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.

Forx∈Ex\in E, defineI​(x)={i∈J:xi=0},I(x)=\{i\in J:x_{i}=0\},and put

Gx={w∈ℝd:wi≥0​for​i∈I​(x)},Hx={v∈ℝd:vi=0​for​i∈I​(x)}.\displaystyle G_{x}=\{w\in\mathbb{R}^{d}:w_{i}\geq 0\text{ for }i\in I(x)\},\qquad H_{x}=\{v\in\mathbb{R}^{d}:v_{i}=0\text{ for }i\in I(x)\}.(3.1)

Proposition 3.1(Resolvent regularity and projected boundary derivatives).

Letλ>0\lambda>0and leth∈Cc∞​(ℝd)h\in C_{c}^{\infty}(\mathbb{R}^{d})be regarded as a function onEE. Defineg(x)=Rλh(x):=𝔼[∫0∞e−λ​th(Ztx)dt],g(x)=R_{\lambda}h(x):=\mathbb{E}\left[\int_{0}^{\infty}e^{-\lambda t}h(Z_{t}^{x})\,dt\right],then we have:

1. (i)ggis bounded and globally Lipschitz onEE.
2. (ii)ggis a classical solution of the resolvent equation inE∘E^{\circ}; more precisely,g∈C∞​(E∘)g\in C^{\infty}(E^{\circ})and(λ−L)​g=h(\lambda-L)g=hinE∘E^{\circ}.
3. (iii)At eachx∈Ex\in E, feasible one-sided directional derivatives∂w+g​(x)\partial_{w}^{+}g(x)exist forw∈Gxw\in G_{x}.
4. (iv)IfA=I​(x)A=I(x), then the principal-block projection𝖫x​v=v−RA​RA​A−1​vA\mathsf{L}_{x}v=v-R_{A}R_{AA}^{-1}v_{A}mapsℝd\mathbb{R}^{d}ontoHxH_{x}, and there is a linear functionalΛx:Hx→ℝ\Lambda_{x}:H_{x}\to\mathbb{R}such that ∂w+g​(x)=Λx​(𝖫x​w),w∈Gx.\partial_{w}^{+}g(x)=\Lambda_{x}(\mathsf{L}_{x}w),\qquad w\in G_{x}.
5. (v)Withℓx​(v)=Λx​(𝖫x​v)\ell_{x}(v)=\Lambda_{x}(\mathsf{L}_{x}v), we haveℓx​(Ri)=0\ell_{x}(R_{i})=0, fori∈I​(x).i\in I(x).

3.1Matrix normalization and active-set projections

Normalize the reflection directions by

Δ=diag⁡(R11,…,Rd​d),R^=R​Δ−1,di=R^i=Ri/Ri​i.\Delta=\operatorname{diag}(R_{11},\ldots,R_{dd}),\qquad\widehat{R}=R\Delta^{-1},\qquad d_{i}=\widehat{R}_{i}=R_{i}/R_{ii}.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.\widehat{R}=I-P^{T},\qquad P\geq 0,\qquad\rho(P)<1.(3.2)Every principal submatrixRA​AR_{AA}is a nonsingularMM-matrix andRA​A−1≥0R_{AA}^{-1}\geq 0. In particular, for every nonemptyA⊂JA\subset J, the active directions{di:i∈A}\{d_{i}:i\in A\}are linearly independent. ForA⊂JA\subset J, define

LA​v=v−RA​RA​A−1​vA,L_{A}v=v-R_{A}R_{AA}^{-1}v_{A},(3.3)withL∅L_{\varnothing}equal to the identity. IfA=I​(x)A=I(x), thenLA=𝖫xL_{A}=\mathsf{L}_{x}is the (unique) linear map fromℝd\mathbb{R}^{d}toHxH_{x}such that𝖫x​v−v∈span⁡{Ri:i∈A}\mathsf{L}_{x}v-v\in\operatorname{span}\{R_{i}:i\in A\}. Moreover,

LA​Ri=0,i∈A,L_{A}R_{i}=0,\qquad i\in A,(3.4)and

C𝖫:=maxA⊂J⁡‖I−RA​RA​A−1​πA‖<∞,C_{\mathsf{L}}:=\max_{A\subset J}\left\|I-R_{A}R_{AA}^{-1}\pi_{A}\right\|<\infty,(3.5)whereπA​v=vA\pi_{A}v=v_{A}and the expression forA=∅A=\varnothingis the identity.

Proof.

The diagonal ofR^\widehat{R}is one and its off-diagonal entries are nonpositive, soPT:=I−R^P^{T}:=I-\widehat{R}is nonnegative. Also

R^−1=Δ​R−1≥0.\widehat{R}^{-1}=\Delta R^{-1}\geq 0.By Perron–Frobenius, in the standard nonnegative-matrix form summarized for example in[1, Chapter 2],PTP^{T}has a nonzero vectorv≥0v\geq 0withPT​v=ρ​(P)​vP^{T}v=\rho(P)v. Ifρ​(P)=1\rho(P)=1, thenR^​v=0\widehat{R}v=0, contradicting invertibility. Ifρ​(P)>1\rho(P)>1, thenR^​v=(1−ρ​(P))​v≤0\widehat{R}v=(1-\rho(P))v\leq 0; multiplying byR^−1≥0\widehat{R}^{-1}\geq 0givesv≤0v\leq 0, again a contradiction. Henceρ​(P)<1\rho(P)<1.

For a principal index setAA, the principal block(PT)A​A(P^{T})_{AA}is nonnegative andρ​((PT)A​A)≤ρ​(PT)<1\rho((P^{T})_{AA})\leq\rho(P^{T})<1. One direct verification of the inequality is that((PT)A​A)n((P^{T})_{AA})^{n}is entrywise bounded by theA​AAAblock of(PT)n(P^{T})^{n}, after which Gelfand’s formula applies. Therefore

(IA−(PT)A​A)−1=∑n=0∞((PT)A​A)n≥0.(I_{A}-(P^{T})_{AA})^{-1}=\sum_{n=0}^{\infty}((P^{T})_{AA})^{n}\geq 0.SinceRA​A=(IA−(PT)A​A)​ΔAR_{AA}=(I_{A}-(P^{T})_{AA})\Delta_{A}, it follows that

RA​A−1=ΔA−1​(IA−(PT)A​A)−1≥0.R_{AA}^{-1}=\Delta_{A}^{-1}(I_{A}-(P^{T})_{AA})^{-1}\geq 0.Linear independence of the active normalized columns follows by restricting a relation∑i∈Aai​di=0\sum_{i\in A}a_{i}d_{i}=0to rows inAA. The maximum in (3.5) is finite because the active-set lattice is finite. For the projection claim, a vector of the formv−RA​av-R_{A}abelongs toHxH_{x}exactly whenvA−RA​A​a=0v_{A}-R_{AA}a=0. The preceding paragraph givesa=RA​A−1​vAa=R_{AA}^{-1}v_{A}. Ifv=Riv=R_{i}withi∈Ai\in A, thenvA=RA​A​eiv_{A}=R_{AA}e_{i}, which proves (3.4). ∎

3.2Regularity of SRBM

This subsection provesTheorem3.3, the stochastic regularity statement used inSection3. More specifically, forg=Rλ​hg=R_{\lambda}h, we will need the reflected semigroup onC0​(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,…,Rd​d),R^=R​Δ−1,di=R^i=RiRi​i.\Delta=\operatorname{diag}(R_{11},\ldots,R_{dd}),\qquad\widehat{R}=R\Delta^{-1},\qquad d_{i}=\widehat{R}_{i}=\frac{R_{i}}{R_{ii}}.Then⟨di,ei⟩=1\langle d_{i},e_{i}\rangle=1. ReplacingRiR_{i}by the positive multipledi=Ri/Ri​id_{i}=R_{i}/R_{ii}only rescales theiith regulator coordinate and leaves the reflected path unchanged. Therefore pathwise statements proved for the normalized matrixR^\widehat{R}apply to the original BAR normalizationRR.

Theorem3.3records the regularity consequences needed for the proof ofSection3. 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.

1. (i)For eachx∈Ex\in Eand each prescribed Brownian motion there is a pathwise unique SRBMZxZ^{x}, andZxZ^{x}is strong Markov.
2. (ii)The semigroup(Pt)(P_{t})mapsC0​(E)C_{0}(E)into itself and is strongly continuous there.
3. (iii)There is a constantKΓ<∞K_{\Gamma}<\infty, depending only on the normalized reflection data, such that synchronous solutions satisfy, for allx,y∈Ex,y\in Eandt≥0t\geq 0, sup0≤s≤t|Zsx−Zsy|≤KΓ​|x−y|almost surely.\sup_{0\leq s\leq t}\left\lvert Z_{s}^{x}-Z_{s}^{y}\right\rvert\leq K_{\Gamma}\left\lvert x-y\right\rvert\qquad\text{almost surely.}(3.6)
4. (iv)For everyx∈Ex\in Ethere is an adapted RCLL derivative process𝖩tx∈Lin⁡(Hx,ℝd)\mathsf{J}_{t}^{x}\in\operatorname{Lin}(H_{x},\mathbb{R}^{d}),t≥0t\geq 0. For each fixedw∈Gxw\in G_{x}, on an event of probability one the derivative ∂wZtx:=limε↓0Ztx+ε​w−Ztxε\partial_{w}Z_{t}^{x}:=\lim_{\varepsilon\downarrow 0}\frac{Z_{t}^{x+\varepsilon w}-Z_{t}^{x}}{\varepsilon}exists for everyt≥0t\geq 0. Moreover, for every fixedt>0t>0, ∂wZtx=𝖩tx​[𝖫x​w]almost surely.\partial_{w}Z_{t}^{x}=\mathsf{J}_{t}^{x}[\mathsf{L}_{x}w]\qquad\text{almost surely.}(3.7)
5. (v)For everyx∈Ex\in Eand every fixedu∈Hxu\in H_{x}, |𝖩tx​[u]|≤KΓ​|u|for​d​t⊗ℙ​-almost every​(t,ω).\left\lvert\mathsf{J}_{t}^{x}[u]\right\rvert\leq K_{\Gamma}\left\lvert u\right\rvert\qquad\text{for }dt\otimes\mathbb{P}\text{-almost every }(t,\omega).

To proveTheorem3.3, 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 columnsdi=Ri/Ri​id_{i}=R_{i}/R_{ii}; 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 parametersα∈U\alpha\in U, where the parameter familyUUis open, and let

G=⋂i∈J{x:⟨x,ni⟩≥ci},J={1,…,d},G=\bigcap_{i\in J}\{x:\langle x,n_{i}\rangle\geq c_{i}\},\qquad J=\{1,\ldots,d\},be a minimally represented simple polyhedron with unit inward normalsnin_{i}, facesFi={x∈G:⟨x,ni⟩=ci}F_{i}=\{x\in G:\langle x,n_{i}\rangle=c_{i}\}, and active setIG​(x)={i:x∈Fi}I_{G}(x)=\{i:x\in F_{i}\}. The normalized reflection directions satisfy⟨di​(α),ni⟩=1\langle d_{i}(\alpha),n_{i}\rangle=1. In the spatially-homogeneous-coefficient specialization we care about, the family of reflected diffusions parameterized byα\alphais written as

Ztα,x=x+b​(α)​t+σ​(α)​Wt+∑i∈Jdi​(α)​Yiα,x​(t),Z_{t}^{\alpha,x}=x+b(\alpha)t+\sigma(\alpha)W_{t}+\sum_{i\in J}d_{i}(\alpha)Y_{i}^{\alpha,x}(t),where eachYiα,xY_{i}^{\alpha,x}is continuous, nondecreasing, starts from zero, and increases only whenZα,x∈FiZ^{\alpha,x}\in F_{i}. Puta​(α)=σ​(α)​σ​(α)Ta(\alpha)=\sigma(\alpha)\sigma(\alpha)^{T},N=(n1,…,nd)N=(n_{1},\ldots,n_{d}), and𝒟​(α)=(d1​(α),…,dd​(α))\mathcal{D}(\alpha)=(d_{1}(\alpha),\ldots,d_{d}(\alpha)). Forx∈Gx\in G, defineCG​(x)={w:⟨w,ni⟩≥0,i∈IG​(x)}C_{G}(x)=\{w:\langle w,n_{i}\rangle\geq 0,\ i\in I_{G}(x)\}andHG​(x)={v:⟨v,ni⟩=0,i∈IG​(x)}H_{G}(x)=\{v:\langle v,n_{i}\rangle=0,\ i\in I_{G}(x)\}. In the orthant specialization,G=EG=E,ni=ein_{i}=e_{i},CG​(x)=GxC_{G}(x)=G_{x}, andHG​(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, fixα∈U\alpha\in U. Assume the following hypotheses.

1. (A1)GGis minimally represented and simple,UUis open, andα↦di​(α)\alpha\mapsto d_{i}(\alpha),b​(α)b(\alpha), andσ​(α)\sigma(\alpha)areC1C^{1}with bounded first derivatives and local Hölder regularity.
2. (A2)a​(α)a(\alpha)is uniformly elliptic:vT​a​(α)​v≥θ​|v|2v^{T}a(\alpha)v\geq\theta|v|^{2}for someθ>0\theta>0and allv∈ℝdv\in\mathbb{R}^{d}.
3. (A3)NT​𝒟​(α)N^{T}\mathcal{D}(\alpha)is a nonsingularMM-matrix.
4. (A4)The reflection matrix is constant in the parameter, or more generally∂α𝒟​(α)\partial_{\alpha}\mathcal{D}(\alpha)is bounded.

Then the following conclusions are available under the assumptions indicated.

1. (C1)*(Well posedness; uses (A1) and (A3),[30, Theorem 2.8].)*For eachx∈Gx\in Gand each prescribed Brownian motion, there is a pathwise unique reflected diffusionZα,xZ^{\alpha,x}, and it is strong Markov.
2. (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}(\alpha))is well posed, and its extended Skorokhod mapΓ¯α\bar{\Gamma}^{\alpha}is Lipschitz on compact time intervals:sups≤t|Γ¯α​(f)​(s)−Γ¯α​(g)​(s)|≤KΓ​sups≤t|f​(s)−g​(s)|\sup_{s\leq t}\left\lvert\bar{\Gamma}^{\alpha}(f)(s)-\bar{\Gamma}^{\alpha}(g)(s)\right\rvert\leq K_{\Gamma}\sup_{s\leq t}\left\lvert f(s)-g(s)\right\rvertfor someKΓ<∞K_{\Gamma}<\infty, all continuous inputsf,gf,g, and allt≥0t\geq 0.
3. (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.
4. (C4)*(Derivative projection; uses (A1) and (A3),[29, Lemma 3.11].)*For eachx∈Gx\in Gthere is a unique linear projectionℒxα:ℝd→HG​(x)\mathcal{L}_{x}^{\alpha}:\mathbb{R}^{d}\to H_{G}(x)such thatℒxα​v−v∈span⁡{di​(α):i∈IG​(x)}\mathcal{L}_{x}^{\alpha}v-v\in\operatorname{span}\{d_{i}(\alpha):i\in I_{G}(x)\}for everyv∈ℝdv\in\mathbb{R}^{d}.
5. (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]: ifA⊂JA\subset Jand∑i∈Aci​di​(α)=0\sum_{i\in A}c_{i}d_{i}(\alpha)=0, then(NT​𝒟​(α))A​A​cA=0(N^{T}\mathcal{D}(\alpha))_{AA}c_{A}=0, and every principal submatrix of a nonsingularMM-matrix is nonsingular, socA=0c_{A}=0. Hence[29, Lemma 3.9]gives the exceptional set of this theorem𝒲α=∅\mathcal{W}^{\alpha}=\varnothing. Therefore, for eachx∈G\𝒲α=Gx\in G\backslash\mathcal{W}^{\alpha}=Gthere is an adapted RCLL derivative processJtα,x∈Lin⁡(HG​(x),ℝd)J_{t}^{\alpha,x}\in\operatorname{Lin}(H_{G}(x),\mathbb{R}^{d}). For every fixedw∈CG​(x)w\in C_{G}(x), we have almost surely,∂wZtα,x:=limε↓0ε−1​(Ztα,x+ε​w−Ztα,x)\partial_{w}Z_{t}^{\alpha,x}:=\lim_{\varepsilon\downarrow 0}\varepsilon^{-1}(Z_{t}^{\alpha,x+\varepsilon w}-Z_{t}^{\alpha,x})exists for everyt≥0t\geq 0, is continuous at everyt>0t>0such thatZtα,x∈G∘Z_{t}^{\alpha,x}\in G^{\circ}, and its right-continuous regularization satisfieslims↓t∂wZsα,x=Jtα,x​[ℒxα​w]\lim_{s\downarrow t}\partial_{w}Z_{s}^{\alpha,x}=J_{t}^{\alpha,x}[\mathcal{L}_{x}^{\alpha}w]for allt≥0t\geq 0.
6. (C6)*(Fixed-time interior statement; uses (A1)–(A4) and (C3)–(C4),[29, Lemma 4.13].)*For every fixedt>0t>0,ℙ​(Ztα,x∈G∘)=1\mathbb{P}(Z_{t}^{\alpha,x}\in G^{\circ})=1.

Proof ofTheorem3.3.

We applySection3.2to the constant parameter family

G=E,ni=ei,ci=0,di​(α)=di=RiRi​i,b​(α)=μ,σ​(α)=Σ1/2,α∈U=(−1,1).G=E,\quad n_{i}=e_{i},\quad c_{i}=0,\quad d_{i}(\alpha)=d_{i}=\frac{R_{i}}{R_{ii}},\quad b(\alpha)=\mu,\quad\sigma(\alpha)=\Sigma^{1/2},\quad\alpha\in U=(-1,1).Thus𝒟​(α)=R^\mathcal{D}(\alpha)=\widehat{R}andN=IN=I. Here constant parameter family means that the domain, reflection directions, drift, and dispersion do not depend onα\alpha. Now we verify that the Harrison-Reiman Class satisfies all assumptions (A1)-(A4).

*Verification of(A1).*The orthant is the minimally represented simple coneE=⋂i{x:⟨x,ei⟩≥0}E=\bigcap_{i}\{x:\langle x,e_{i}\rangle\geq 0\}. Simplicity follows because the coordinate normals are linearly independent on every active set. Minimality follows because, if theiith half-space is removed, then the point−ei-e_{i}satisfies all remaining half-space inequalities but does not belong toEE. The parameter setU=(−1,1)U=(-1,1)is open. The mapsdi​(α)d_{i}(\alpha),b​(α)b(\alpha), andσ​(α)\sigma(\alpha)are constant, henceC1C^{1}; all first derivatives are zero, and therefore bounded and locally Hölder. The normalization⟨di,ei⟩=1\langle d_{i},e_{i}\rangle=1holds by definition. This verifies (A1).

*Verification of(A2).*Herea​(α)=Σa(\alpha)=\Sigma. SinceΣ\Sigmais symmetric positive definite,

vT​Σ​v≥λmin​(Σ)​|v|2,v∈ℝd.v^{T}\Sigma v\geq\lambda_{\min}(\Sigma)|v|^{2},\qquad v\in\mathbb{R}^{d}.Thus (A2) holds withθ=λmin​(Σ)>0\theta=\lambda_{\min}(\Sigma)>0.

*Verification of(A3).*HereN=IN=I,𝒟​(α)=R^=I−PT\mathcal{D}(\alpha)=\widehat{R}=I-P^{T}, and henceNT​𝒟​(α)=R^N^{T}\mathcal{D}(\alpha)=\widehat{R}. BySection3.1,R^\widehat{R}is a nonsingularMM-matrix. This verifies (A3).

*Verification of(A4).*The reflection matrix𝒟​(α)=R^\mathcal{D}(\alpha)=\widehat{R}is constant inα\alpha, so∂α𝒟​(α)=0\partial_{\alpha}\mathcal{D}(\alpha)=0. This verifies (A4).

All assumptions (A1)–(A4) ofSection3.2have now been verified.

Conclusion (C1) gives pathwise existence, uniqueness, and the strong Markov property for the normalized reflected diffusion. SinceRi=Ri​i​diR_{i}=R_{ii}d_{i}withRi​i>0R_{ii}>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 normalizationRR. This proves assertion (i).

For assertion (iii), let the two processes start fromxxandyyand be driven by the same Brownian path. Their free inputs arefx​(s)=x+μ​s+Σ1/2​Wsf_{x}(s)=x+\mu s+\Sigma^{1/2}W_{s}andfy​(s)=y+μ​s+Σ1/2​Wsf_{y}(s)=y+\mu s+\Sigma^{1/2}W_{s}, sosups≤t|fx​(s)−fy​(s)|=|x−y|\sup_{s\leq t}|f_{x}(s)-f_{y}(s)|=|x-y|. Applying conclusion (C2) gives (3.6), proving assertion (iii).

Assertion (ii) follows from (3.6) and Brownian continuity. PutXt=μ​t+Σ1/2​WtX_{t}=\mu t+\Sigma^{1/2}W_{t}. Comparing the inputx+Xx+Xwith the constant inputxxgives

sups≤t|Zsx−x|≤KΓ​sups≤t|Xs|almost surely.\sup_{s\leq t}\left\lvert Z_{s}^{x}-x\right\rvert\leq K_{\Gamma}\sup_{s\leq t}\left\lvert X_{s}\right\rvert\qquad\text{almost surely.}(3.8)Ifh∈C0​(E)h\in C_{0}(E), thenhhis uniformly continuous. Hence (3.6) gives continuity ofx↦Pt​h​(x)x\mapsto P_{t}h(x). Ifhhis supported in the ball of radiusrr, then

|Pt​h​(x)|≤‖h‖∞​ℙ​(KΓ​sups≤t|Xs|≥|x|−r),|P_{t}h(x)|\leq\|h\|_{\infty}\mathbb{P}\!\left(K_{\Gamma}\sup_{s\leq t}|X_{s}|\geq|x|-r\right),which tends to zero as|x|→∞|x|\to\infty; approximation by compactly supported functions givesPt​C0​(E)⊂C0​(E)P_{t}C_{0}(E)\subset C_{0}(E). Finally, ifωh\omega_{h}is the modulus of continuity ofhh, then (3.8) gives

supx∈E|Pt​h​(x)−h​(x)|≤𝔼​ωh​(KΓ​sups≤t|Xs|)⟶0(t↓0)\sup_{x\in E}|P_{t}h(x)-h(x)|\leq\mathbb{E}\,\omega_{h}\!\left(K_{\Gamma}\sup_{s\leq t}|X_{s}|\right)\longrightarrow 0\qquad(t\downarrow 0)by bounded convergence. Thus(Pt)(P_{t})is strongly continuous onC0​(E)C_{0}(E). This proves assertion (ii).

We now prove assertions (iv)–(v). By conclusion (C4), the derivative projection atxxis the unique linear map ontoHxH_{x}whose difference from the identity lies inspan⁡{di:i∈I​(x)}\operatorname{span}\{d_{i}:i\in I(x)\}. Sincespan⁡{di:i∈A}=span⁡{Ri:i∈A}\operatorname{span}\{d_{i}:i\in A\}=\operatorname{span}\{R_{i}:i\in A\}for everyA⊂JA\subset J, uniqueness andSection3.1identify this projection with𝖫x\mathsf{L}_{x}.

Conclusion (C5), applied to the constant parameter family and with parameter direction equal to zero, gives the derivative process𝖩tx\mathsf{J}_{t}^{x}and the directional derivative∂wZtx\partial_{w}Z_{t}^{x}for every fixedw∈Gxw\in G_{x}. It also gives continuity ofs↦∂wZsxs\mapsto\partial_{w}Z_{s}^{x}at everyt>0t>0such thatZtx∈E∘Z_{t}^{x}\in E^{\circ}, together with the projected right-continuous regularization.

Conclusion (C6) givesℙ​(Ztx∈E∘)=1\mathbb{P}(Z_{t}^{x}\in E^{\circ})=1for every fixedt>0t>0. Therefore, for every fixedw∈Gxw\in G_{x}andt>0t>0, on an event of probability one,

∂wZtx=lims↓t∂wZsx=𝖩tx​[𝖫x​w].\partial_{w}Z_{t}^{x}=\lim_{s\downarrow t}\partial_{w}Z_{s}^{x}=\mathsf{J}_{t}^{x}[\mathsf{L}_{x}w].This proves assertion (iv).

For assertion (v), fixu∈Hxu\in H_{x}. Thenu∈Gxu\in G_{x}and𝖫x​u=u\mathsf{L}_{x}u=u. For smallε>0\varepsilon>0,x+ε​u∈Ex+\varepsilon u\in E. Applying (3.6) withy=x+ε​uy=x+\varepsilon u, dividing byε\varepsilon, and lettingε↓0\varepsilon\downarrow 0gives|∂uZtx|≤KΓ​|u|\left\lvert\partial_{u}Z_{t}^{x}\right\rvert\leq K_{\Gamma}\left\lvert u\right\rverton the event where the directional derivative exists for allt≥0t\geq 0. Combining this bound with (3.7) gives|𝖩tx​[u]|≤KΓ​|u|\left\lvert\mathsf{J}_{t}^{x}[u]\right\rvert\leq K_{\Gamma}\left\lvert u\right\rvertfor every fixedt>0t>0, almost surely. Since𝖩x\mathsf{J}^{x}is RCLL, Fubini gives the same bound ford​t⊗ℙdt\otimes\mathbb{P}-almost every(t,ω)(t,\omega). This proves assertion (v) and completes the proof.

∎

3.3The 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.

Fixλ>0\lambda>0andh∈Cc∞​(ℝd)h\in C_{c}^{\infty}(\mathbb{R}^{d}). We regardhhas a function onEEand define

g​(x)=Rλ​h​(x):=𝔼​[∫0∞e−λ​t​h​(Ztx)​𝑑t].g(x)=R_{\lambda}h(x):=\mathbb{E}\left[\int_{0}^{\infty}e^{-\lambda t}h(Z_{t}^{x})\,dt\right].(3.9)

Lemma 3.5(Boundedness and Lipschitz continuity).

The functionggis bounded and globally Lipschitz onEE, with

‖g‖∞≤‖h‖∞λ,Lip⁡(g)≤KΓ​Lip⁡(h)λ.\left\lVert g\right\rVert_{\infty}\leq\frac{\left\lVert h\right\rVert_{\infty}}{\lambda},\qquad\operatorname{Lip}(g)\leq\frac{K_{\Gamma}\operatorname{Lip}(h)}{\lambda}.(3.10)

Proof.

The first estimate follows immediately from (3.9). For the second, coupleZxZ^{x}andZyZ^{y}with the same Brownian path. By (3.6),

|h​(Ztx)−h​(Zty)|≤Lip⁡(h)​KΓ​|x−y|.\left\lvert h(Z_{t}^{x})-h(Z_{t}^{y})\right\rvert\leq\operatorname{Lip}(h)K_{\Gamma}\left\lvert x-y\right\rvert.Integrating againste−λ​t​d​te^{-\lambda t}\,dtproves the claim. ∎

Lemma 3.6(Interior classical solution of the resolvent equation).

The functionggis a classical solution of the resolvent equation inE∘E^{\circ}; more precisely,g∈C∞​(E∘)g\in C^{\infty}(E^{\circ})and

(λ−L)​g=hin​E∘.(\lambda-L)g=h\qquad\text{in }E^{\circ}.(3.11)

Proof.

Fix concentric ballsB′⋐B⋐E∘B^{\prime}\Subset B\Subset E^{\circ}and letτB\tau_{B}be the first exit time fromBB. BeforeτB\tau_{B}, the reflected process is the unconstrained diffusion with generatorLL. The strong Markov property gives, forx∈Bx\in B,

g​(x)=𝔼x​[∫0τBe−λ​t​h​(Zt)​𝑑t+e−λ​τB​g​(ZτB)].g(x)=\mathbb{E}_{x}\left[\int_{0}^{\tau_{B}}e^{-\lambda t}h(Z_{t})\,dt+e^{-\lambda\tau_{B}}g(Z_{\tau_{B}})\right].(3.12)Sinceg∈C​(B¯)g\in C(\overline{B})and∂B\partial Bis compact, Stone–Weierstrass applied to the restrictions of polynomials onℝd\mathbb{R}^{d}to∂B\partial Bgives polynomialspnp_{n}with‖pn−g‖L∞​(∂B)→0\|p_{n}-g\|_{L^{\infty}(\partial B)}\to 0. Settingφn=pn|B¯\varphi_{n}=p_{n}|_{\overline{B}}, we haveφn∈C∞​(B¯)⊂C2,α​(B¯)\varphi_{n}\in C^{\infty}(\overline{B})\subset C^{2,\alpha}(\overline{B})andφn→g|∂B\varphi_{n}\to g|_{\partial 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]).

LetΩ⊂ℝd\Omega\subset\mathbb{R}^{d}be a boundedC2,αC^{2,\alpha}domain and let𝒜=ai​j​Di​j+bi​Di+c\mathcal{A}=a^{ij}D_{ij}+b^{i}D_{i}+cbe strictly elliptic onΩ\Omega, that is,ai​j​(y)​ξi​ξj≥θ0​|ξ|2a^{ij}(y)\xi_{i}\xi_{j}\geq\theta_{0}\left\lvert\xi\right\rvert^{2}for someθ0>0\theta_{0}>0and ally∈Ωy\in\Omega,ξ∈ℝd\xi\in\mathbb{R}^{d}, with coefficientsai​j,bi,c∈Cα​(Ω¯)a^{ij},b^{i},c\in C^{\alpha}(\overline{\Omega})andc≤0c\leq 0onΩ\Omega. Then for everyf∈Cα​(Ω¯)f\in C^{\alpha}(\overline{\Omega})and everyφ∈C2,α​(Ω¯)\varphi\in C^{2,\alpha}(\overline{\Omega})the Dirichlet problem𝒜​u=f\mathcal{A}u=finΩ\Omega,u=φu=\varphion∂Ω\partial\Omega, has a unique solutionu∈C2,α​(Ω¯)u\in C^{2,\alpha}(\overline{\Omega}).

We apply this withΩ=B\Omega=B,𝒜=L−λ\mathcal{A}=L-\lambda, so that in coordinatesai​j=12​Σi​ja^{ij}=\tfrac{1}{2}\Sigma_{ij},bi=μib^{i}=\mu_{i},c=−λc=-\lambda, together withf=−hf=-handφ=ϕn\varphi=\phi_{n}. The four hypotheses hold in the present setting:

1. (a)BBis an open Euclidean ball, hence aC∞C^{\infty}and a fortioriC2,αC^{2,\alpha}domain.
2. (b)For allξ∈ℝd\xi\in\mathbb{R}^{d},ai​j​ξi​ξj=12​ξ⊤​Σ​ξ≥12​λmin​(Σ)​|ξ|2a^{ij}\xi_{i}\xi_{j}=\tfrac{1}{2}\xi^{\top}\Sigma\xi\geq\tfrac{1}{2}\lambda_{\min}(\Sigma)\left\lvert\xi\right\rvert^{2}, andλmin​(Σ)>0\lambda_{\min}(\Sigma)>0becauseΣ\Sigmais symmetric positive definite; thus𝒜\mathcal{A}is strictly elliptic withθ0=12​λmin​(Σ)\theta_{0}=\tfrac{1}{2}\lambda_{\min}(\Sigma).
3. (c)The coefficients12​Σi​j\tfrac{1}{2}\Sigma_{ij},μi\mu_{i},−λ-\lambdaare constants, hence lie inCα​(B¯)C^{\alpha}(\overline{B})with vanishing Hölder seminorm; andc=−λ<0≤0c=-\lambda<0\leq 0sinceλ>0\lambda>0.
4. (d)The sourcef=−h∈Cc∞​(ℝd)⊂Cα​(B¯)f=-h\in C_{c}^{\infty}(\mathbb{R}^{d})\subset C^{\alpha}(\overline{B}), and eachφ=ϕn∈C∞​(B¯)⊂C2,α​(B¯)\varphi=\phi_{n}\in C^{\infty}(\overline{B})\subset C^{2,\alpha}(\overline{B}).

Therefore[20, Theorem 6.14]yields a uniqueun∈C2,α​(B¯)u_{n}\in C^{2,\alpha}(\overline{B})satisfying

(λ−L)​un=hin​B,un=ϕnon​∂B.(\lambda-L)u_{n}=h\quad\text{in }B,\qquad u_{n}=\phi_{n}\quad\text{on }\partial B.Apply Itô’s formula toe−λ​(t∧τB)​un​(Zt∧τB)e^{-\lambda(t\wedge\tau_{B})}u_{n}(Z_{t\wedge\tau_{B}}). Sinceunu_{n}and its first derivatives are bounded onB¯\overline{B}, the stopped stochastic integral has mean zero. Lettingt→∞t\to\inftyis justified by bounded convergence. Indeed, beforeτB\tau_{B}the process isx+μ​t+Σ1/2​Wtx+\mu t+\Sigma^{1/2}W_{t}; a nonzero one-dimensional projection is a Brownian motion with drift and exits the bounded projection ofBBalmost surely, soτB<∞\tau_{B}<\inftyalmost surely. This gives the Feynman–Kac representation

un​(x)=𝔼x​[∫0τBe−λ​t​h​(Zt)​𝑑t+e−λ​τB​ϕn​(ZτB)].u_{n}(x)=\mathbb{E}_{x}\left[\int_{0}^{\tau_{B}}e^{-\lambda t}h(Z_{t})\,dt+e^{-\lambda\tau_{B}}\phi_{n}(Z_{\tau_{B}})\right].Comparing it with (3.12) yields

‖un−g‖L∞​(B)≤‖ϕn−g‖L∞​(∂B)⟶0.\|u_{n}-g\|_{L^{\infty}(B)}\leq\|\phi_{n}-g\|_{L^{\infty}(\partial B)}\longrightarrow 0.Forn,mn,m, the differencew=un−um∈C2,α​(B¯)w=u_{n}-u_{m}\in C^{2,\alpha}(\overline{B})solves the homogeneous equation(λ−L)​w=h−h=0(\lambda-L)w=h-h=0inBB. To pass to the limit we use the interior Schauder estimate, recalled in the form used.

Next, we use interior Schauder estiamte to proveunu_{n}is Cauchy inC2,α​(B′)C^{2,\alpha}(B^{\prime}):

Theorem 3.8(Interior Schauder estimate[20, Theorem 6.2]).

Let𝒜=ai​j​Di​j+bi​Di+c\mathcal{A}=a^{ij}D_{ij}+b^{i}D_{i}+cbe strictly elliptic on a domainΩ⊂ℝd\Omega\subset\mathbb{R}^{d}with ellipticity constantθ0>0\theta_{0}>0and coefficients bounded inCα​(Ω)C^{\alpha}(\Omega)by a constantΘ\Theta. Ifu∈C2,α​(Ω)u\in C^{2,\alpha}(\Omega)satisfies𝒜​u=f\mathcal{A}u=fwithf∈Cα​(Ω)f\in C^{\alpha}(\Omega), then for every subdomainΩ′⋐Ω\Omega^{\prime}\Subset\Omega,

‖u‖C2,α​(Ω′¯)≤C​(‖u‖L∞​(Ω)+‖f‖Cα​(Ω)),C=C​(d,α,θ0,Θ,dist⁡(Ω′,∂Ω)).\left\lVert u\right\rVert_{C^{2,\alpha}(\overline{\Omega^{\prime}})}\leq C\bigl(\left\lVert u\right\rVert_{L^{\infty}(\Omega)}+\left\lVert f\right\rVert_{C^{\alpha}(\Omega)}\bigr),\qquad C=C\bigl(d,\alpha,\theta_{0},\Theta,\operatorname{dist}(\Omega^{\prime},\partial\Omega)\bigr).

The operator𝒜=L−λ\mathcal{A}=L-\lambdasatisfies these hypotheses with the ellipticity constantθ0=12​λmin​(Σ)\theta_{0}=\tfrac{1}{2}\lambda_{\min}(\Sigma)of(b)and the coefficient boundΘ=max⁡{12​‖Σ‖,|μ|,λ}\Theta=\max\{\tfrac{1}{2}\left\lVert\Sigma\right\rVert,\ \left\lvert\mu\right\rvert,\ \lambda\}, both independent ofn,mn,m. Applying it tow=un−umw=u_{n}-u_{m}, withΩ=B\Omega=B,Ω′=B′\Omega^{\prime}=B^{\prime}, andf≡0f\equiv 0, gives

‖un−um‖C2,α​(B′¯)≤CB′,B​‖un−um‖L∞​(B),CB′,B=C​(d,α,12​λmin​(Σ),Θ,dist⁡(B′,∂B)),\left\lVert u_{n}-u_{m}\right\rVert_{C^{2,\alpha}(\overline{B^{\prime}})}\leq C_{B^{\prime},B}\,\left\lVert u_{n}-u_{m}\right\rVert_{L^{\infty}(B)},\qquad C_{B^{\prime},B}=C\bigl(d,\alpha,\tfrac{1}{2}\lambda_{\min}(\Sigma),\Theta,\operatorname{dist}(B^{\prime},\partial B)\bigr),where the constantCB′,BC_{B^{\prime},B}does not depend onn,mn,m. Becauseun→gu_{n}\to guniformly onBB, the right-hand side tends to zero asn,m→∞n,m\to\infty. Thus(un)(u_{n})is Cauchy inC2,α​(B′)C^{2,\alpha}(B^{\prime})and converges there to someu∈C2,α​(B′)u\in C^{2,\alpha}(B^{\prime}). The same sequence converges uniformly toggonBB, so theC2,α​(B′)C^{2,\alpha}(B^{\prime})limit must beu=gu=g. Passing to the limit in the equations satisfied by the classical solutionsunu_{n}givesg∈C2,α​(B′)g\in C^{2,\alpha}(B^{\prime})and (3.11) onB′B^{\prime}. Since the coefficients ofLLare constant andhhis smooth, standard interior elliptic regularity, equivalently the usual bootstrapping by interior estimates, givesg∈C∞​(B′)g\in C^{\infty}(B^{\prime}). SinceB′⋐E∘B^{\prime}\Subset E^{\circ}was arbitrary, the conclusion follows. ∎

Proposition 3.9(Directional factorization of the resolvent).

Letx∈Ex\in E. There is a bounded linear functionalΛx:Hx→ℝ\Lambda_{x}:H_{x}\to\mathbb{R}such that, for everyw∈Gxw\in G_{x}, the one sided directional derivative exists and satisfies

∂w+g​(x):=limε↓0g​(x+ε​w)−g​(x)ε=Λx​(𝖫x​w).\partial_{w}^{+}g(x):=\lim_{\varepsilon\downarrow 0}\frac{g(x+\varepsilon w)-g(x)}{\varepsilon}=\Lambda_{x}(\mathsf{L}_{x}w).(3.13)The linear functionalℓx​(v):=Λx​(𝖫x​v)\ell_{x}(v):=\Lambda_{x}(\mathsf{L}_{x}v)is bounded by

|ℓx​(v)|≤KΓ​C𝖫​‖∇h‖∞λ​|v||\ell_{x}(v)|\leq\frac{K_{\Gamma}C_{\mathsf{L}}\|\nabla h\|_{\infty}}{\lambda}|v|(3.14)for allv∈ℝdv\in\mathbb{R}^{d}. Ifi∈I​(x)i\in I(x), thenℓx​(Ri)=0.\ell_{x}(R_{i})=0.Note that no continuity or measurability of the mapx↦ℓxx\mapsto\ell_{x}is asserted.

Proof.

Fixx∈Ex\in E. Foru∈Hxu\in H_{x}, defineΛx​(u):=𝔼​∫0∞e−λ​t​∇h​(Ztx)⋅𝖩tx​[u]​𝑑t.\Lambda_{x}(u):=\mathbb{E}\int_{0}^{\infty}e^{-\lambda t}\nabla h(Z_{t}^{x})\cdot\mathsf{J}_{t}^{x}[u]\,dt.We first record the measurability and integrability facts needed to defineΛx\Lambda_{x}. Since𝖩x\mathsf{J}^{x}is adapted and RCLL with values in the finite dimensional spaceLin⁡(Hx,ℝd)\operatorname{Lin}(H_{x},\mathbb{R}^{d}), for every fixedu∈Hxu\in H_{x}the process(t,ω)↦𝖩tx​[u]​(ω)(t,\omega)\mapsto\mathsf{J}_{t}^{x}[u](\omega)is progressively measurable, henceℬ​([0,∞))⊗ℱ\mathcal{B}([0,\infty))\otimes\mathcal{F}-measurable.

By Theorem3.3, for each fixedu∈Hxu\in H_{x}, we have|𝖩tx​[u]|≤KΓ​|u||\mathsf{J}_{t}^{x}[u]|\leq K_{\Gamma}|u|holds ford​t⊗ℙdt\otimes\mathbb{P}-almost every(t,ω)(t,\omega). Hence𝔼​∫0∞e−λ​t​|∇h​(Ztx)⋅𝖩tx​[u]|​𝑑t≤KΓ​‖∇h‖∞λ​|u|<∞\mathbb{E}\int_{0}^{\infty}e^{-\lambda t}\left|\nabla h(Z_{t}^{x})\cdot\mathsf{J}_{t}^{x}[u]\right|\,dt\leq\frac{K_{\Gamma}\|\nabla h\|_{\infty}}{\lambda}|u|<\infty,*i.e.*the integral definingΛx​(u)\Lambda_{x}(u)is absolutely convergent with respect toe−λ​t​d​t⊗ℙe^{-\lambda t}dt\otimes\mathbb{P}, and

|Λx​(u)|≤KΓ​‖∇h‖∞λ​|u|,u∈Hx.|\Lambda_{x}(u)|\leq\frac{K_{\Gamma}\|\nabla h\|_{\infty}}{\lambda}|u|,\qquad u\in H_{x}.Since𝖩tx∈Lin⁡(Hx,ℝd)\mathsf{J}_{t}^{x}\in\operatorname{Lin}(H_{x},\mathbb{R}^{d}), linearity ofΛx\Lambda_{x}follows from linearity of𝖩tx\mathsf{J}_{t}^{x}and the preceding bound. ThusΛx\Lambda_{x}is a bounded linear functional onHxH_{x}.

Now fixw∈Gxw\in G_{x}. For all sufficiently smallε>0\varepsilon>0,x+ε​w∈Ex+\varepsilon w\in E. By the definition ofgg,

g​(x+ε​w)−g​(x)ε=𝔼​∫0∞e−λ​t​h​(Ztx+ε​w)−h​(Ztx)ε​𝑑t.\frac{g(x+\varepsilon w)-g(x)}{\varepsilon}=\mathbb{E}\int_{0}^{\infty}e^{-\lambda t}\frac{h(Z_{t}^{x+\varepsilon w})-h(Z_{t}^{x})}{\varepsilon}\,dt.(3.15)The synchronous Lipschitz estimate gives the deterministic domination

|h​(Ztx+ε​w)−h​(Ztx)ε|≤KΓ​‖∇h‖∞​|w|,\left|\frac{h(Z_{t}^{x+\varepsilon w})-h(Z_{t}^{x})}{\varepsilon}\right|\leq K_{\Gamma}\|\nabla h\|_{\infty}|w|,(3.16)uniformly for all sufficiently smallε>0\varepsilon>0, allt≥0t\geq 0, and all sample paths.

On the probability one event inTheorem3.3corresponding to this fixed pair(x,w)(x,w), the pathwise directional derivative∂wZtx=limε↓0Ztx+ε​w−Ztxε\partial_{w}Z_{t}^{x}=\lim_{\varepsilon\downarrow 0}\frac{Z_{t}^{x+\varepsilon w}-Z_{t}^{x}}{\varepsilon}exists for everyt≥0t\geq 0. Sinceh∈Cc∞​(ℝd)h\in C_{c}^{\infty}(\mathbb{R}^{d}), the mean value formula gives

h​(Ztx+ε​w)−h​(Ztx)ε⟶∇h​(Ztx)⋅∂wZtx\frac{h(Z_{t}^{x+\varepsilon w})-h(Z_{t}^{x})}{\varepsilon}\longrightarrow\nabla h(Z_{t}^{x})\cdot\partial_{w}Z_{t}^{x}ford​t⊗ℙdt\otimes\mathbb{P}-almost every(t,ω)(t,\omega). Dominated convergence with respect toe−λ​t​d​t⊗ℙe^{-\lambda t}\,dt\otimes\mathbb{P}therefore yields

∂w+g​(x)=𝔼​∫0∞e−λ​t​∇h​(Ztx)⋅∂wZtx​d​t.\partial_{w}^{+}g(x)=\mathbb{E}\int_{0}^{\infty}e^{-\lambda t}\nabla h(Z_{t}^{x})\cdot\partial_{w}Z_{t}^{x}\,dt.(3.17) For every fixedt>0t>0, Theorem3.3gives∂wZtx=𝖩tx​[𝖫x​w]\partial_{w}Z_{t}^{x}=\mathsf{J}_{t}^{x}[\mathsf{L}_{x}w]almost surely. For fixedxxandww, the map(t,ω)↦∂wZtx​(ω)(t,\omega)\mapsto\partial_{w}Z_{t}^{x}(\omega)isℬ​([0,∞))⊗ℱ\mathcal{B}([0,\infty))\otimes\mathcal{F}-measurable, since it is the pointwise limit of the continuous-in-ttdifference quotientsε−1​(Ztx+ε​w−Ztx).\varepsilon^{-1}\bigl(Z_{t}^{x+\varepsilon w}-Z_{t}^{x}\bigr).The processt↦𝖩tx​[𝖫x​w]t\mapsto\mathsf{J}_{t}^{x}[\mathsf{L}_{x}w]is measurable because𝖩x\mathsf{J}^{x}is RCLL. Hence the fixed-time almost sure identity can be integrated intt, giving the identityd​t⊗ℙdt\otimes\mathbb{P}-almost everywhere. By Fubini, this identity holds ford​t⊗ℙdt\otimes\mathbb{P}-almost every(t,ω)(t,\omega). The value att=0t=0is irrelevant for the time integral. Substituting into the preceding display gives

∂w+g​(x)=𝔼​∫0∞e−λ​t​∇h​(Ztx)⋅𝖩tx​[𝖫x​w]​𝑑t=Λx​(𝖫x​w),\partial_{w}^{+}g(x)=\mathbb{E}\int_{0}^{\infty}e^{-\lambda t}\nabla h(Z_{t}^{x})\cdot\mathsf{J}_{t}^{x}[\mathsf{L}_{x}w]\,dt=\Lambda_{x}(\mathsf{L}_{x}w),(3.18)which proves the factorization.

Finally, by Lemma3.1, we have|𝖫x​v|≤C𝖫​|v|,|\mathsf{L}_{x}v|\leq C_{\mathsf{L}}|v|,holds for allv∈ℝd.v\in\mathbb{R}^{d}.Therefore

|ℓx​(v)|=|Λx​(𝖫x​v)|≤KΓ​‖∇h‖∞λ​|𝖫x​v|≤KΓ​C𝖫​‖∇h‖∞λ​|v|.|\ell_{x}(v)|=|\Lambda_{x}(\mathsf{L}_{x}v)|\leq\frac{K_{\Gamma}\|\nabla h\|_{\infty}}{\lambda}|\mathsf{L}_{x}v|\leq\frac{K_{\Gamma}C_{\mathsf{L}}\|\nabla h\|_{\infty}}{\lambda}|v|.Ifi∈I​(x)i\in I(x), then Lemma3.1gives𝖫x​Ri=0\mathsf{L}_{x}R_{i}=0, and henceℓx​(Ri)=Λx​(𝖫x​Ri)=0.\ell_{x}(R_{i})=\Lambda_{x}(\mathsf{L}_{x}R_{i})=0.This completes the proof. ∎

Proof ofSection3.

Boundedness and Lipschitz continuity areSection3.3. The statement thatggis a classical solution of the resolvent equation inE∘E^{\circ}isSection3.3. The projection formula and the identity𝖫x​Ri=0\mathsf{L}_{x}R_{i}=0fori∈I​(x)i\in I(x)areSection3.1. Finally,Section3.3gives the feasible one-sided derivatives, the factorization through𝖫x\mathsf{L}_{x}, and the linear extensionℓx\ell_{x}satisfyingℓx​(Ri)=0\ell_{x}(R_{i})=0on active faces. ∎

4One-sided smoothing and the measure–Neumann approximation

In this section we prove the resolvent insertion theorem,Theorem4.1. We first prove the identity forh∈Cc∞​(ℝd)h\in C_{c}^{\infty}(\mathbb{R}^{d}), regarded as a function onEE, by a one-sided smoothing argument. At the end of the proof, a density argument extends the identity to allh∈C0​(E)h\in C_{0}(E). Thus, until this final density step throughout this whole section, we fixλ>0\lambda>0,h∈Cc∞​(ℝd)h\in C_{c}^{\infty}(\mathbb{R}^{d}), and writeg=Rλ​hg=R_{\lambda}has in (3.9).

Theorem 4.1(Resolvent insertion theorem).

For every finite signed BAR tuple(π¯,ν¯1,…,ν¯d)(\bar{\pi},\bar{\nu}_{1},\ldots,\bar{\nu}_{d}), everyh∈C0​(E)h\in C_{0}(E), and everyλ>0\lambda>0,

∫E(λ​Rλ​h−h)​𝑑π¯=0.\int_{E}(\lambda R_{\lambda}h-h)\,d\bar{\pi}=0.(RI)

First we prove the insertion for smooth compactly supportedhh, then extend it toC0​(E)C_{0}(E)by uniform approximation. A convolution supported strictly inside the orthant produces boundedC2C^{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 onCb2​(E)C_{b}^{2}(E), no spatial cutoff is needed in the resolvent insertion.

Now we one-sided smoothing the functiongg. We seek a mollifierρ∈Cc∞​((1,2)d)\rho\in C_{c}^{\infty}((1,2)^{d})that satisfiesρ≥0\rho\geq 0and∫ℝdρ​(w)​𝑑w=1.\int_{\mathbb{R}^{d}}\rho(w)\,dw=1.Forε>0\varepsilon>0, define

gε​(x)=∫ℝdρ​(w)​g​(x+ε​w)​𝑑w,hε​(x)=∫ℝdρ​(w)​h​(x+ε​w)​𝑑w,x∈E.g_{\varepsilon}(x)=\int_{\mathbb{R}^{d}}\rho(w)g(x+\varepsilon w)\,dw,\qquad h_{\varepsilon}(x)=\int_{\mathbb{R}^{d}}\rho(w)h(x+\varepsilon w)\,dw,\qquad x\in E.(4.1)Because the support ofρ\rholies strictly inside the positive orthant,gεg_{\varepsilon}is defined and smooth on an open neighborhood ofEE.

Lemma 4.2(One-sided smoothing).

gε∈Cb2​(E)g_{\varepsilon}\in C_{b}^{2}(E)for each fixedε>0\varepsilon>0,gεg_{\varepsilon}is smooth on an open neighborhood ofEE,gε→gg_{\varepsilon}\to guniformly,gεg_{\varepsilon}is uniformly bounded and Lipschitz, and

(λ−L)​gε=hεon​E.(\lambda-L)g_{\varepsilon}=h_{\varepsilon}\qquad\text{on }E.(4.2)Moreover,

‖gε−g‖∞\displaystyle\left\lVert g_{\varepsilon}-g\right\rVert_{\infty}≤Cρ​ε​Lip⁡(g),\displaystyle\leq C_{\rho}\varepsilon\operatorname{Lip}(g),(4.3)‖hε−h‖∞\displaystyle\left\lVert h_{\varepsilon}-h\right\rVert_{\infty}⟶0,\displaystyle\longrightarrow 0,(4.4)‖gε‖∞\displaystyle\left\lVert g_{\varepsilon}\right\rVert_{\infty}≤‖g‖∞,\displaystyle\leq\left\lVert g\right\rVert_{\infty},(4.5)‖∇gε‖∞\displaystyle\left\lVert\nabla g_{\varepsilon}\right\rVert_{\infty}≤Cρ​Lip⁡(g),\displaystyle\leq C_{\rho}\operatorname{Lip}(g),(4.6)‖D2​gε‖∞\displaystyle\left\lVert D^{2}g_{\varepsilon}\right\rVert_{\infty}≤Cρ​ε−1​Lip⁡(g).\displaystyle\leq C_{\rho}\varepsilon^{-1}\operatorname{Lip}(g).(4.7)

Proof.

We first justify the smoothness ofgεg_{\varepsilon}. Letδρ:=dist⁡(supp⁡ρ,∂ℝ+d)>0.\delta_{\rho}:=\operatorname{dist}(\operatorname{supp}\rho,\partial\mathbb{R}_{+}^{d})>0.For each fixedε>0\varepsilon>0, defineUε:={x∈ℝd:x+ε​w∈E∘​for every​w∈supp⁡ρ}.U_{\varepsilon}:=\{x\in\mathbb{R}^{d}:x+\varepsilon w\in E^{\circ}\text{ for every }w\in\operatorname{supp}\rho\}.ThenUεU_{\varepsilon}is an open neighborhood ofEE, becausesupp⁡ρ⋐(0,∞)d\operatorname{supp}\rho\Subset(0,\infty)^{d}. Thusgεg_{\varepsilon}is well defined onUεU_{\varepsilon}. Althoughggis only known to be globally Lipschitz onEE, the derivatives ofgεg_{\varepsilon}may be computed by integration by parts in the convolution variable. For every multiindexα\alpha,

∂xαgε​(x)=(−1)|α|​ε−|α|​∫ℝd∂wαρ​(w)​g​(x+ε​w)​d​w,x∈Uε.\partial_{x}^{\alpha}g_{\varepsilon}(x)=(-1)^{|\alpha|}\varepsilon^{-|\alpha|}\int_{\mathbb{R}^{d}}\partial_{w}^{\alpha}\rho(w)\,g(x+\varepsilon w)\,dw,\qquad x\in U_{\varepsilon}.This identity is first obtained in the sense of distributions onUεU_{\varepsilon}. Since the right hand side is continuous inxx, it is the classical derivative. Iterating the same argument gives derivatives of all orders; hencegε∈C∞​(Uε)g_{\varepsilon}\in C^{\infty}(U_{\varepsilon}). In particular,gε∈C2​(E)g_{\varepsilon}\in C^{2}(E)in the closed domain sense.

For fixedx∈Ex\in E, the compact setx+ε​supp⁡ρx+\varepsilon\operatorname{supp}\rholies inE∘E^{\circ}. Same as the proof of Lemma3.3,ggis a classical solution of(λ−L)​g=h(\lambda-L)g=hon this compact subset of the interior. SinceLLhas constant coefficients, differentiating under the integral on this interior compact set gives(λ−L)​gε​(x)=∫ℝdρ​(w)​(λ−L)​g​(x+ε​w)​𝑑w=∫ℝdρ​(w)​h​(x+ε​w)​𝑑w=hε​(x),(\lambda-L)g_{\varepsilon}(x)=\int_{\mathbb{R}^{d}}\rho(w)(\lambda-L)g(x+\varepsilon w)\,dw=\int_{\mathbb{R}^{d}}\rho(w)h(x+\varepsilon w)\,dw=h_{\varepsilon}(x),which proves (4.2).

Next, the global Lipschitz continuity ofgggives

|gε​(x)−g​(x)|≤ε​Lip⁡(g)​∫|w|​ρ​(w)​𝑑w,\left\lvert g_{\varepsilon}(x)-g(x)\right\rvert\leq\varepsilon\operatorname{Lip}(g)\int\left\lvert w\right\rvert\rho(w)\,dw,which is (4.3). Sinceh∈Cc∞​(ℝd)h\in C_{c}^{\infty}(\mathbb{R}^{d}), it is uniformly continuous, and therefore‖hε−h‖∞→0\|h_{\varepsilon}-h\|_{\infty}\to 0which proves gives (4.4). The bound (4.5) follows fromρ≥0\rho\geq 0and∫ρ=1\int\rho=1:

|gε​(x)|≤∫ℝdρ​(w)​|g​(x+ε​w)|​𝑑w≤‖g‖∞.|g_{\varepsilon}(x)|\leq\int_{\mathbb{R}^{d}}\rho(w)|g(x+\varepsilon w)|\,dw\leq\|g\|_{\infty}. For the gradient, integration by parts inwwand∫ℝd∂wjρ​(w)​d​w=0\int_{\mathbb{R}^{d}}\partial_{w_{j}}\rho(w)\,dw=0gives

∂xjgε​(x)=−1ε​∫∂wjρ​(w)​(g​(x+ε​w)−g​(x))​d​w.\partial_{x_{j}}g_{\varepsilon}(x)=-\frac{1}{\varepsilon}\int\partial_{w_{j}}\rho(w)\bigl(g(x+\varepsilon w)-g(x)\bigr)\,dw.(4.8)Hence|∂xjgε​(x)|≤Lip⁡(g)​∫ℝd|∂wjρ​(w)|​|w|​𝑑w.|\partial_{x_{j}}g_{\varepsilon}(x)|\leq\operatorname{Lip}(g)\int_{\mathbb{R}^{d}}|\partial_{w_{j}}\rho(w)|\,|w|\,dw.The Lipschitz bound on the difference proves (4.6). Differentiating once more in the same distributional-convolution formula and subtracting the constantg​(x)g(x)gives

∂xj​xk2gε​(x)=1ε2​∫∂wj​wk2ρ​(w)​(g​(x+ε​w)−g​(x))​d​w,\partial_{x_{j}x_{k}}^{2}g_{\varepsilon}(x)=\frac{1}{\varepsilon^{2}}\int\partial_{w_{j}w_{k}}^{2}\rho(w)\bigl(g(x+\varepsilon w)-g(x)\bigr)\,dw,and therefore|∂xj​xk2gε​(x)|≤ε−1​Lip⁡(g)​∫ℝd|∂wj​wk2ρ​(w)|​|w|​𝑑w.|\partial_{x_{j}x_{k}}^{2}g_{\varepsilon}(x)|\leq\varepsilon^{-1}\operatorname{Lip}(g)\int_{\mathbb{R}^{d}}|\partial_{w_{j}w_{k}}^{2}\rho(w)|\,|w|\,dw.Taking the maximum overj,kj,kproves (4.7). Thusgεg_{\varepsilon}, its first derivatives, and its second derivatives are bounded onEE, sogε∈Cb2​(E)g_{\varepsilon}\in C_{b}^{2}(E). ∎

The next proposition is where the projected derivative information forggis used at the boundary. It proves two facts on each faceFiF_{i}: first,Di​gε​(x)→0D_{i}g_{\varepsilon}(x)\to 0for everyx∈Fix\in 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 fromSection3.3: ifx∈Fix\in F_{i}andw∈supp⁡ρ⊂(0,∞)dw\in\operatorname{supp}\rho\subset(0,\infty)^{d}, thenw∈Gxw\in G_{x}and(g​(x+ε​w)−g​(x))/ε→ℓx​(w)(g(x+\varepsilon w)-g(x))/\varepsilon\to\ell_{x}(w). After integration by parts in the smoothing variable, the limit ofDi​gε​(x)D_{i}g_{\varepsilon}(x)becomesℓx​(Ri)\ell_{x}(R_{i}), which is zero because𝖫x​Ri=0\mathsf{L}_{x}R_{i}=0on active faces.

Proposition 4.3(Vanishing oblique derivative after one sided smoothing).

For everyi∈Ji\in Jand everyx∈Fix\in F_{i},Di​gε​(x)⟶0D_{i}g_{\varepsilon}(x)\longrightarrow 0asε↓0.\varepsilon\downarrow 0.At the same time, the convergence is pointwise inxx. There is a constantCρ,R<∞C_{\rho,R}<\infty, independent ofxxandε\varepsilon, such that

|Di​gε​(x)|≤Cρ,R​Lip⁡(g),x∈Fi,0<ε<1.|D_{i}g_{\varepsilon}(x)|\leq C_{\rho,R}\operatorname{Lip}(g),\qquad x\in F_{i},\quad 0<\varepsilon<1.(4.9)Consequently, for every finite signed measureηi\eta_{i}onFiF_{i}and every bounded Borel functiona:Fi→ℝa:F_{i}\to\mathbb{R}fixed independently ofε\varepsilon, we have

∫Fia​(x)​Di​gε​(x)​𝑑ηi​(x)⟶0.\int_{F_{i}}a(x)D_{i}g_{\varepsilon}(x)\,d\eta_{i}(x)\longrightarrow 0.(4.10)

Proof.

Sinceρ∈Cc∞​((1,2)d)\rho\in C_{c}^{\infty}((1,2)^{d}), integration by parts in theww-variable gives, forx∈Ex\in E,

∇gε​(x)=−1ε​∫ℝd∇ρ​(w)​g​(x+ε​w)​𝑑w.\nabla g_{\varepsilon}(x)=-\frac{1}{\varepsilon}\int_{\mathbb{R}^{d}}\nabla\rho(w)\,g(x+\varepsilon w)\,dw.Also,∫ℝdRi⋅∇ρ​(w)​𝑑w=0.\int_{\mathbb{R}^{d}}R_{i}\cdot\nabla\rho(w)\,dw=0.Therefore, forx∈Fix\in F_{i},

Di​gε​(x)=−∫ℝd(Ri⋅∇ρ)​(w)​g​(x+ε​w)−g​(x)ε​𝑑w.D_{i}g_{\varepsilon}(x)=-\int_{\mathbb{R}^{d}}(R_{i}\cdot\nabla\rho)(w)\frac{g(x+\varepsilon w)-g(x)}{\varepsilon}\,dw.(4.11) Fixx∈Fix\in F_{i}. Theni∈I​(x)i\in I(x). Sincesupp⁡ρ⊂(1,2)d\operatorname{supp}\rho\subset(1,2)^{d}, everyw∈supp⁡ρw\in\operatorname{supp}\rhobelongs toGxG_{x}. Proposition3.3gives, for each such fixedww,g​(x+ε​w)−g​(x)ε⟶ℓx​(w).\frac{g(x+\varepsilon w)-g(x)}{\varepsilon}\longrightarrow\ell_{x}(w).Moreover,

|(Ri⋅∇ρ)​(w)​g​(x+ε​w)−g​(x)ε|≤|(Ri⋅∇ρ)​(w)|​Lip⁡(g)​|w|.\left|(R_{i}\cdot\nabla\rho)(w)\frac{g(x+\varepsilon w)-g(x)}{\varepsilon}\right|\leq|(R_{i}\cdot\nabla\rho)(w)|\,\operatorname{Lip}(g)|w|.The right hand side is integrable overℝd\mathbb{R}^{d}, becauseρ\rhois smooth and compactly supported. Dominated convergence in the mollifier variablewwgives

limε↓0Di​gε​(x)=−∫ℝd(Ri⋅∇ρ)​(w)​ℓx​(w)​𝑑w.\lim_{\varepsilon\downarrow 0}D_{i}g_{\varepsilon}(x)=-\int_{\mathbb{R}^{d}}(R_{i}\cdot\nabla\rho)(w)\ell_{x}(w)\,dw.(4.12)Sinceℓx\ell_{x}is linear andρ\rhohas compact support, another integration by parts gives

−∫ℝd(Ri⋅∇ρ)​(w)​ℓx​(w)​𝑑w=∫ℝdρ​(w)​ℓx​(Ri)​𝑑w=ℓx​(Ri).-\int_{\mathbb{R}^{d}}(R_{i}\cdot\nabla\rho)(w)\ell_{x}(w)\,dw=\int_{\mathbb{R}^{d}}\rho(w)\ell_{x}(R_{i})\,dw=\ell_{x}(R_{i}).(4.13)There is no boundary term becauseρ∈Cc∞​((1,2)d)\rho\in C_{c}^{\infty}((1,2)^{d}). Sincei∈I​(x)i\in I(x), Proposition3.3givesℓx​(Ri)=0\ell_{x}(R_{i})=0. This proves the pointwise convergence.

The same representation and the Lipschitz bound give

|Di​gε​(x)|≤Lip⁡(g)​∫ℝd|(Ri⋅∇ρ)​(w)|​|w|​𝑑w.|D_{i}g_{\varepsilon}(x)|\leq\operatorname{Lip}(g)\int_{\mathbb{R}^{d}}|(R_{i}\cdot\nabla\rho)(w)|\,|w|\,dw.Thus the uniform estimate holds withCρ,R:=max1≤k≤d​∫ℝd|(Rk⋅∇ρ)​(w)|​|w|​𝑑w<∞.C_{\rho,R}:=\max_{1\leq k\leq d}\int_{\mathbb{R}^{d}}|(R_{k}\cdot\nabla\rho)(w)|\,|w|\,dw<\infty.

Finally, letηi∈M​(Fi)\eta_{i}\in M(F_{i}), and leta:Fi→ℝa:F_{i}\to\mathbb{R}be bounded Borel and fixed independently ofε\varepsilon. SinceDi​gεD_{i}g_{\varepsilon}is continuous onFiF_{i}, the producta​Di​gεaD_{i}g_{\varepsilon}is Borel. The pointwise convergence just proved and the bound

|a​(x)​Di​gε​(x)|≤‖a‖∞​Cρ,R​Lip⁡(g)|a(x)D_{i}g_{\varepsilon}(x)|\leq\|a\|_{\infty}C_{\rho,R}\operatorname{Lip}(g)allow dominated convergence with respect to|ηi||\eta_{i}|. Hence∫Fia​(x)​Di​gε​(x)​𝑑ηi​(x)⟶0.\int_{F_{i}}a(x)D_{i}g_{\varepsilon}(x)\,d\eta_{i}(x)\longrightarrow 0.This completes the proof. ∎

Proposition 4.4(Measure–Neumann resolvent approximation).

Letmmbe a finite signed measure onEEand letηi\eta_{i}be finite signed measures onFiF_{i}. Then, asε↓0\varepsilon\downarrow 0,

∫E(λ−L)​gε​𝑑m\displaystyle\int_{E}(\lambda-L)g_{\varepsilon}\,dm⟶∫Eh​𝑑m,\displaystyle\longrightarrow\int_{E}h\,dm,(4.14)∫Egε​𝑑m\displaystyle\int_{E}g_{\varepsilon}\,dm⟶∫Eg​𝑑m,\displaystyle\longrightarrow\int_{E}g\,dm,(4.15)∫FiDi​gε​𝑑ηi\displaystyle\int_{F_{i}}D_{i}g_{\varepsilon}\,d\eta_{i}⟶0,i=1,…,d.\displaystyle\longrightarrow 0,\qquad i=1,\ldots,d.(4.16)

Proof.

The first assertion follows from(λ−L)​gε=hε(\lambda-L)g_{\varepsilon}=h_{\varepsilon}and the uniform convergencehε→hh_{\varepsilon}\to h. The second follows from the uniform convergencegε→gg_{\varepsilon}\to g. Sincemmis finite signed, uniform convergence is sufficient in both cases. For the boundary terms, take the bounded Borel multipliera≡1a\equiv 1in (4.10). This gives (4.16) for each finite signed boundary measureηi\eta_{i}. ∎

Proof ofTheorem4.1.

First assumeh∈Cc∞​(ℝd)h\in C_{c}^{\infty}(\mathbb{R}^{d}), regarded as a function onEE, and letg=Rλ​hg=R_{\lambda}h. BySection4,gε∈Cb2​(E)g_{\varepsilon}\in C_{b}^{2}(E), so it is an admissible BAR test. Applying the BAR togεg_{\varepsilon}and rearranging gives

∫E(λ−L)​gε​𝑑π¯=λ​∫Egε​𝑑π¯+∑i=1d∫FiDi​gε​𝑑ν¯i.\int_{E}(\lambda-L)g_{\varepsilon}\,d\bar{\pi}=\lambda\int_{E}g_{\varepsilon}\,d\bar{\pi}+\sum_{i=1}^{d}\int_{F_{i}}D_{i}g_{\varepsilon}\,d\bar{\nu}_{i}.(4.17)BySection4, lettingε↓0\varepsilon\downarrow 0yields

∫Eh​𝑑π¯=λ​∫ERλ​h​𝑑π¯.\int_{E}h\,d\bar{\pi}=\lambda\int_{E}R_{\lambda}h\,d\bar{\pi}.Equivalently,∫E(λ​Rλ​h−h)​𝑑π¯=0\int_{E}(\lambda R_{\lambda}h-h)\,d\bar{\pi}=0holds for every smooth compactly supportedhh.

Now leth∈C0​(E)h\in C_{0}(E). The restrictions toEEof functions inCc∞​(ℝd)C_{c}^{\infty}(\mathbb{R}^{d})are uniformly dense inC0​(E)C_{0}(E): extend a function from the closed setEEtoC0​(ℝd)C_{0}(\mathbb{R}^{d}), cut it off, and mollify onℝd\mathbb{R}^{d}. Choosehn∈Cc∞​(ℝd)h_{n}\in C_{c}^{\infty}(\mathbb{R}^{d})with‖hn−h‖∞→0\|h_{n}-h\|_{\infty}\to 0onEE. Since(Pt)(P_{t})is a contraction on bounded functions,

‖Rλ​(hn−h)‖∞≤λ−1​‖hn−h‖∞.\|R_{\lambda}(h_{n}-h)\|_{\infty}\leq\lambda^{-1}\|h_{n}-h\|_{\infty}.The finiteness ofπ¯\bar{\pi}therefore permits passage to the limit in the smooth identity, proving (RI) forh∈C0​(E)h\in C_{0}(E). ∎

5Proof ofSection2.3: From the resolvent identity to signed BAR uniqueness

We now complete the proof of resolvent identity criterion (Section2.3) and then complete the proof of the main Theorem (Theorem2.2). 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 II]).

Leta:[0,∞)→ℝa:[0,\infty)\to\mathbb{R}be locally integrable and of at most exponential growth. Suppose that its Laplace transforma^​(λ)=∫0∞e−λ​t​a​(t)​𝑑t\widehat{a}(\lambda)=\int_{0}^{\infty}e^{-\lambda t}a(t)\,dtvanishes for everyλ\lambdain some interval(λ∗,∞)(\lambda_{\ast},\infty). Thena​(t)=0​for Lebesgue almost every​t≥0.a(t)=0\text{ for Lebesgue almost every }t\geq 0.

Proposition 5.2(Resolvent identity implies semigroup invariance).

Assume that a finite signed measureπ¯\bar{\pi}satisfies (RI) for everyh∈C0​(E)h\in C_{0}(E)and everyλ>0\lambda>0. Thenπ¯\bar{\pi}is invariant for(Pt)(P_{t}):

∫EPt​φ​𝑑π¯=∫Eφ​𝑑π¯,t≥0,φ∈C0​(E).\int_{E}P_{t}\varphi\,d\bar{\pi}=\int_{E}\varphi\,d\bar{\pi},\qquad t\geq 0,\quad\varphi\in C_{0}(E).(5.1)

Proof.

From (RI), for everyh∈C0​(E)h\in C_{0}(E)and everyλ>0\lambda>0,

∫ERλ​h​𝑑π¯=λ−1​∫Eh​𝑑π¯.\int_{E}R_{\lambda}h\,d\bar{\pi}=\lambda^{-1}\int_{E}h\,d\bar{\pi}.(5.2)Fixh∈C0​(E)h\in C_{0}(E)and defineFh​(t)=∫EPt​h​𝑑π¯​​(t>0).F_{h}(t)=\int_{E}P_{t}h\,d\bar{\pi}\text{ }(t>0).We first record the elementary regularity ofFhF_{h}. By the Feller statement (ii) inTheorem3.3,Pt​C0​(E)⊂C0​(E)P_{t}C_{0}(E)\subset C_{0}(E), and‖Pt​h−h‖∞→t↓00.\|P_{t}h-h\|_{\infty}\xrightarrow{t\downarrow 0}0.The semigroup property and the contraction property imply norm continuity oft↦Pt​ht\mapsto P_{t}hon all of[0,∞)[0,\infty). Indeed,‖Ps​h−Pt​h‖∞=‖Pmin⁡{s,t}​(P|s−t|​h−h)‖∞≤‖P|s−t|​h−h‖∞⟶0\|P_{s}h-P_{t}h\|_{\infty}=\|P_{\min\{s,t\}}(P_{|s-t|}h-h)\|_{\infty}\leq\|P_{|s-t|}h-h\|_{\infty}\longrightarrow 0ass→ts\to t.

Sinceπ¯\bar{\pi}is finite signed, it follows thatFhF_{h}is continuous:

|Fh​(s)−Fh​(t)|≤|π¯|​(E)​‖Ps​h−Pt​h‖∞.|F_{h}(s)-F_{h}(t)|\leq|\bar{\pi}|(E)\,\|P_{s}h-P_{t}h\|_{\infty}.Moreover, we have|Fh​(t)|≤‖h‖∞​|π¯|​(E),|F_{h}(t)|\leq\|h\|_{\infty}|\bar{\pi}|(E),for|Pt​h|≤‖h‖∞|P_{t}h|\leq\|h\|_{\infty}. We now pass from the resolvent identity to a Laplace transform identity. Since|e−λ​t​Pt​h​(x)|≤e−λ​t​‖h‖∞|e^{-\lambda t}P_{t}h(x)|\leq e^{-\lambda t}\|h\|_{\infty}and∫E∫0∞e−λ​t​‖h‖∞​𝑑t​d​|π¯|​(x)=λ−1​‖h‖∞​|π¯|​(E)<∞,\int_{E}\int_{0}^{\infty}e^{-\lambda t}\|h\|_{\infty}\,dt\,d|\bar{\pi}|(x)=\lambda^{-1}\|h\|_{\infty}|\bar{\pi}|(E)<\infty,Fubini’s theorem gives

∫ERλ​h​𝑑π¯\displaystyle\int_{E}R_{\lambda}h\,d\bar{\pi}=∫E∫0∞e−λ​t​Pt​h​(x)​𝑑t​𝑑π¯​(x)\displaystyle=\int_{E}\int_{0}^{\infty}e^{-\lambda t}P_{t}h(x)\,dt\,d\bar{\pi}(x)=∫0∞e−λ​t​(∫EPt​h​𝑑π¯)​𝑑t=∫0∞e−λ​t​Fh​(t)​𝑑t.\displaystyle=\int_{0}^{\infty}e^{-\lambda t}\left(\int_{E}P_{t}h\,d\bar{\pi}\right)dt=\int_{0}^{\infty}e^{-\lambda t}F_{h}(t)\,dt.At the same time, sinceλ−1​∫Eh​𝑑π¯=∫0∞e−λ​t​Fh​(0)​𝑑t,\lambda^{-1}\int_{E}h\,d\bar{\pi}=\int_{0}^{\infty}e^{-\lambda t}F_{h}(0)\,dt,therefore (5.2) is equivalent to

∫0∞e−λ​t​(Fh​(t)−Fh​(0))​𝑑t=0,λ>0.\int_{0}^{\infty}e^{-\lambda t}\bigl(F_{h}(t)-F_{h}(0)\bigr)\,dt=0,\qquad\lambda>0.Putah​(t)=Fh​(t)−Fh​(0).a_{h}(t)=F_{h}(t)-F_{h}(0).Thenaha_{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)|\leq|F_{h}(t)|+|F_{h}(0)|\leq 2\|h\|_{\infty}|\bar{\pi}|(E),\qquad t\geq 0.Equation(5.3)(5.3)says precisely that the Laplace transform ofaha_{h}vanishes for everyλ>0\lambda>0. BySection5,ah​(t)=0​for Lebesgue almost every​t≥0.a_{h}(t)=0\text{ for Lebesgue almost every }t\geq 0.Sinceaha_{h}is continuous, this almost everywhere equality upgrades to equality for everyt≥0t\geq 0. HenceFh​(t)=Fh​(0)F_{h}(t)=F_{h}(0)fort≥0t\geq 0,i.e.∫EPt​h​𝑑π¯=∫Eh​𝑑π¯,\int_{E}P_{t}h\,d\bar{\pi}=\int_{E}h\,d\bar{\pi},holds for allh∈C0​(E)h\in C_{0}(E).

For a finite signed measureα\alpha, we write(α​Pt)​(B)=∫EPt​(x,B)​α​(d​x)(\alpha P_{t})(B)=\int_{E}P_{t}(x,B)\,\alpha(dx)forB∈ℬ​(E).B\in\mathcal{B}(E).Thenπ¯​Pt\bar{\pi}P_{t}is a finite signed Radon measure, and for everyφ∈C0​(E)\varphi\in C_{0}(E),

∫Eφ​d​(π¯​Pt)=∫EPt​φ​𝑑π¯=∫Eφ​𝑑π¯.\int_{E}\varphi\,d(\bar{\pi}P_{t})=\int_{E}P_{t}\varphi\,d\bar{\pi}=\int_{E}\varphi\,d\bar{\pi}.SinceC0​(E)C_{0}(E)separates finite Radon measures on the locally compact spaceEE, this impliesπ¯​Pt=π¯.\bar{\pi}P_{t}=\bar{\pi}.∎

Section5indicates that the signed-BAR interior solutionπ¯\bar{\pi}is invariant as a signed measure for the SRBM semigroup, i.e.π¯​Pt=π¯\bar{\pi}P_{t}=\bar{\pi}. Then we follows the Dai-Dieker[5], using Jordan decomposition to show thatπ¯\bar{\pi}uniquely characterizes the stationary probability distributionπ0\pi_{0}of the diffusion process in the sense thatπ¯=π¯​(E)​π0\bar{\pi}=\bar{\pi}(E)\pi_{0}.

Proposition 5.3(Identification of the interior measure).

Ifπ¯\bar{\pi}is a finite signed invariant measure for the SRBM semigroup, thenπ¯=c​π0\bar{\pi}=c\pi_{0}wherec=π¯​(E)c=\bar{\pi}(E).

Proof.

For a Markov kernelPPand a finite signed measureπ¯\bar{\pi}, positivity gives the measure inequality

|π¯​P|≤|π¯|​P.|\bar{\pi}P|\leq|\bar{\pi}|P.(5.2)Indeed, for every Borel setBB,|∫P​(x,B)​𝑑π¯​(x)|≤∫P​(x,B)​d​|π¯|​(x)|\int P(x,B)\,d\bar{\pi}(x)|\leq\int P(x,B)\,d|\bar{\pi}|(x), and the same domination holds for finite measurable partitions, hence for total variation. Ifπ¯​Pt=π¯\bar{\pi}P_{t}=\bar{\pi}, then (5.2) gives|π¯|≤|π¯|​Pt|\bar{\pi}|\leq|\bar{\pi}|P_{t}. Both positive measures have total mass|π¯|​(E)|\bar{\pi}|(E), becausePt​(x,E)=1P_{t}(x,E)=1. Thus the domination is actually equality: if finite positive measuresα≤β\alpha\leq\betahaveα​(E)=β​(E)\alpha(E)=\beta(E), thenβ−α\beta-\alphais a positive measure with total mass zero, hence vanishes. Applying this withα=|π¯|\alpha=|\bar{\pi}|andβ=|π¯|​Pt\beta=|\bar{\pi}|P_{t}gives|π¯|​Pt=|π¯|.|\bar{\pi}|P_{t}=|\bar{\pi}|.

Consequently the Jordan componentsπ¯+=12​(|π¯|+π¯)\bar{\pi}^{+}=\tfrac{1}{2}(|\bar{\pi}|+\bar{\pi})andπ¯−=12​(|π¯|−π¯)\bar{\pi}^{-}=\tfrac{1}{2}(|\bar{\pi}|-\bar{\pi})are invariant positive finite measures. Each nonzero component, after normalization by its total mass, is an invariant probability and therefore equalsπ0\pi_{0}, which is unique. Thusπ¯=(π¯+​(E)−π¯−​(E))​π0=π¯​(E)​π0.\bar{\pi}=(\bar{\pi}^{+}(E)-\bar{\pi}^{-}(E))\pi_{0}=\bar{\pi}(E)\pi_{0}.∎

The precedingSections5and5identifies 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(\eta_{i})_{i=1}^{d}, withηi\eta_{i}supported onFiF_{i}, whose boundary pairing vanishes, i.e.∑i=1d∫FiDi​f​𝑑ηi=0\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\eta_{i}=0against every test functionf∈Cb2​(E)f\in 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 nonsingularMM-matrix assumption only through the invertibility of the principal reflection blocksRA​AR_{AA}. On a stratumSAS_{A}, the active oblique derivatives are determined by the active normal jet:(Di​f|SA)i∈A=RA​AT​a(D_{i}f|_{S_{A}})_{i\in A}=R_{AA}^{T}a, wherea=(∂xjf|SA)j∈Aa=(\partial_{x_{j}}f|_{S_{A}})_{j\in A}. SinceRA​AR_{AA}is invertible, we can prescribe these oblique derivatives independently. In particular, choosinga=RA​A−T​ek​ψa=R_{AA}^{-T}e_{k}\,\psigivesDi​f|SA=δi​k​ψD_{i}f|_{S_{A}}=\delta_{ik}\psifori∈Ai\in A, and for any test functionψ∈Cc∞​(SA)\psi\in C_{c}^{\infty}(S_{A}). This isolates thekk-th boundary measure onSAS_{A}. An induction over the codimension|A||A|removes all lower-stratum contributions and forcesηk|SA=0\eta_{k}|_{S_{A}}=0. Since bothAAandk∈Ak\in Aare arbitrary, all boundary measures vanish.

Proposition 5.4(Pure boundary injectivity).

Letηi∈ℳ​(Fi)\eta_{i}\in\mathcal{M}(F_{i})be finite signed measures. If

∑i=1d∫FiDi​f​𝑑ηi=0,f∈Cb2​(E),\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\eta_{i}=0,\qquad f\in C_{b}^{2}(E),(5.3)thenηi=0,\eta_{i}=0,for alli=1​…,d.i=1\ldots,d.

Proof.

ForA⊂JA\subset Jandi∈Ai\in A, letηiA=ηi|SA.\eta_{i}^{A}=\eta_{i}|_{S_{A}}.Thenηi=∑A∋iηiA\eta_{i}=\sum_{A\ni i}\eta_{i}^{A}as a finite sum of mutually singular signed measures. We prove by induction onn=|A|n=|A|that

ηiA=0for every​A⊂J​with​|A|=n​and every​i∈A.\eta_{i}^{A}=0\quad\text{for every }A\subset J\text{ with }|A|=n\text{ and every }i\in A.(5.4) Assume the claim has been proved for all strata of cardinality less thannn, and fixAAwith|A|=n|A|=n. Letk∈Ak\in Aand letψ∈Cc∞​(SA)\psi\in C_{c}^{\infty}(S_{A}). Write points asx=(xA,y)x=(x_{A},y), wherey=xAc∈(0,∞)Acy=x_{A^{c}}\in(0,\infty)^{A^{c}}. We identifyψ\psiwith its extension by zero toℝAc\mathbb{R}^{A^{c}}; this extension is smooth becausesupp⁡ψ\operatorname{supp}\psiis compact in the open orthant. Choose a tangential cutoffϑ∈Cc∞​((0,∞)Ac)\vartheta\in C_{c}^{\infty}((0,\infty)^{A^{c}})that equals one on a neighborhood ofsupp⁡ψ\operatorname{supp}\psi. WhenA=JA=J, interpret the tangential space as a point and putϑ=1\vartheta=1. Choose a normal cutoffζ∈Cc∞​(ℝA)\zeta\in C_{c}^{\infty}(\mathbb{R}^{A})that equals one near the origin. Define theAA-vector

a​(y)=RA​A−T​ek​ψ​(y)a(y)=R_{AA}^{-T}e_{k}\,\psi(y)(5.5)and the test function

f​(xA,y)=ζ​(xA)​ϑ​(y)​∑j∈Axj​aj​(y).f(x_{A},y)=\zeta(x_{A})\vartheta(y)\sum_{j\in A}x_{j}a_{j}(y).(5.6)After shrinking the support ofϑ\varthetaif necessary,ffis supported away from every faceFjF_{j}withj∉Aj\notin A. At a point ofSAS_{A}, the tangential derivatives offfvanish and

∂xjf​(0,y)=aj​(y),j∈A.\partial_{x_{j}}f(0,y)=a_{j}(y),\qquad j\in A.(5.7)Therefore, fori∈Ai\in A,

Di​f|SA=∑j∈ARj​i​aj=(RA​AT​a)i=δi​k​ψ.D_{i}f|_{S_{A}}=\sum_{j\in A}R_{ji}a_{j}=(R_{AA}^{T}a)_{i}=\delta_{ik}\psi.(5.8) The support condition implies that the only boundary strata meetingsupp⁡f\operatorname{supp}fareSCS_{C}with∅≠C⊂A\varnothing\neq C\subset A. The contributions from|C|<n|C|<nvanish by the induction hypothesis. Hence (5.3) and (5.8) give

0=∑i∈A∫SADi​f​𝑑ηiA=∫SAψ​𝑑ηkA.0=\sum_{i\in A}\int_{S_{A}}D_{i}f\,d\eta_{i}^{A}=\int_{S_{A}}\psi\,d\eta_{k}^{A}.Sinceψ\psiis arbitrary,ηkA=0\eta_{k}^{A}=0. Sincek∈Ak\in Awas arbitrary, the induction step is complete. The base casen=1n=1is the same argument with no lower strata. Thus allηiA\eta_{i}^{A}vanish and hence allηi\eta_{i}vanish. ∎

Now, the uniqueness of Signed BAR solution in the Harrison-Reiman Class is the natural conclusion ofSection5-Section5.

Proof ofSection2.3.

Assume (RI) for the finite signed BAR tuple(π¯,ν¯1,…,ν¯d)(\bar{\pi},\bar{\nu}_{1},\ldots,\bar{\nu}_{d}). BySection5, we know the signed measureπ¯\bar{\pi}is invariant under the semigroup,*i.e.*π¯​Pt=π¯\bar{\pi}P_{t}=\bar{\pi}holds fort≥0t\geq 0. Then bySection5, we have the uniquness of the interior measureπ¯=c​π0\bar{\pi}=c\pi_{0}forc=π¯​(E)c=\bar{\pi}(E).

Defineηi=ν¯i−c​νi0.\eta_{i}=\bar{\nu}_{i}-c\nu_{i}^{0}.Subtractcctimes the stationary BAR (2.4) from the signed BAR (2.3). The interior measures cancel, and we obtain∑i=1d∫FiDi​f​𝑑ηi=0,\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\eta_{i}=0,for all test functionf∈Cb2​(E).f\in C_{b}^{2}(E).Section5givesηi=0\eta_{i}=0for everyii, henceν¯i=c​νi0\bar{\nu}_{i}=c\nu_{i}^{0}. This provesSection2.3. ∎

Finally, we finish the proof of the signed-BAR problem.

Proof ofTheorem2.2.

Theorem4.1proves (RI) for every finite signed BAR tuple.Section2.3converts (RI) into full signed BAR uniqueness. ThereforeTheorem2.2follows. ∎

6Failure in the completelySSclass

In this section, we demonstrate a family of counterexamples in the completely𝒮\mathcal{S}class due to the singular proper active block of the relection matrix. The negative construction is summarized inFig.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 matrixAAis anSSmatrix if there exists a vectoru>0u>0such thatA​u>0Au>0, and is completely-𝒮\mathcal{S}if every principal submatrix is anSSmatrix. completely-𝒮\mathcal{S}reflection matrices are the natural existence class for orthant SRBMs, but they need not have invertible principal blocks.

Boundary gauge on a singular stratumExtension into the interior by a zero-potentialSingular active blockChoose∅≠A⊊J\varnothing\neq A\subsetneq Jand0≠v∈ker⁡RA​A.0\neq v\in\ker R_{AA}.WithT=J∖AT=J\setminus A, putw:=RT​A​v≠0.w:=R_{TA}v\neq 0.The nonzero vectorwwis tangent to the stratumSAS_{A}.Boundary gauge onSAS_{A}Fory∈(0,∞)Ty\in(0,\infty)^{T}andιA​(y)∈SA\iota_{A}(y)\in S_{A}, setd​ζi​(ιA​(y))=vi​φ​(y)​d​y,i∈A,d\zeta_{i}(\iota_{A}(y))=v_{i}\varphi(y)\,dy,\qquad i\in A,and setζi=0\zeta_{i}=0fori∉Ai\notin A.Normal components cancelCombining active faces gives∑i∈Avi​Ri=(RA​A​v,RT​A​v)=(0,w).\sum_{i\in A}v_{i}R_{i}=(R_{AA}v,R_{TA}v)=(0,w).Thus the gauge sees only the tangential derivativew⋅∇Tfw\cdot\nabla_{T}fonSAS_{A}.Tangential integration by partsBecauseφ∈Cc∞​((0,∞)T)\varphi\in C_{c}^{\infty}((0,\infty)^{T}),∑i∫FiDi​f​𝑑ζi\displaystyle\sum_{i}\int_{F_{i}}D_{i}f\,d\zeta_{i}=∫Ef​𝑑χ,\displaystyle=\int_{E}f\,d\chi,d​χ​(ιA​(y))\displaystyle d\chi(\iota_{A}(y))=−(w⋅∇Tφ​(y))​d​y.\displaystyle=-(w\cdot\nabla_{T}\varphi(y))\,dy.The source is supported onSAS_{A}, nonzero, and centered:χ​(E)=0\chi(E)=0.Interior zero-potentialSpread the centered source by the reflected semigroup:π¯=∫0∞χ​Pt​𝑑t.\bar{\pi}=\int_{0}^{\infty}\chi P_{t}\,dt.Exponential ergodicity andχ​(E)=0\chi(E)=0make this finite and giveπ¯​(E)=0\bar{\pi}(E)=0.Boundary occupation correctionUse the one-unit regulator kernelsKiK_{i}and setθi=∑n≥0(χ​Pn)​Ki.\theta_{i}=\sum_{n\geq 0}(\chi P_{n})K_{i}.The one-step regulator bound makes eachθi\theta_{i}finite onFiF_{i}.Poisson identity cancels the sourceItô’s formula over integer intervals gives∫EL​f​𝑑π¯+∑i∫FiDi​f​𝑑θi=−∫Ef​𝑑χ.\int_{E}Lf\,d\bar{\pi}+\sum_{i}\int_{F_{i}}D_{i}f\,d\theta_{i}=-\int_{E}f\,d\chi.Adding the boundary-gauge identity from the upper half cancels∫f​𝑑χ\int f\,d\chi.Signed BAR tuple with zero interior massν¯i:=θi+ζi,∫EL​f​𝑑π¯+∑i∫FiDi​f​𝑑ν¯i=0,π¯​(E)=0.\bar{\nu}_{i}:=\theta_{i}+\zeta_{i},\qquad\int_{E}Lf\,d\bar{\pi}+\sum_{i}\int_{F_{i}}D_{i}f\,d\bar{\nu}_{i}=0,\qquad\bar{\pi}(E)=0.Sinceπ¯≠0\bar{\pi}\neq 0butπ¯​(E)=0\bar{\pi}(E)=0whereasπ0​(E)=1\pi_{0}(E)=1, the interior coordinate cannot be a scalar multiple of the stationary probability. Varyingφ\varphion disjoint supports gives the infinite-dimensional failure.+∫f​𝑑χ+\int f\,d\chi

Figure 2:Counterexample construction in the completely-𝒮\mathcal{S}class.The upper group is the boundary algebra ofSection6.1: a gauge supported on a lower-dimensional stratum has its active normal components killed byRA​A​v=0R_{AA}v=0, leaving a tangential derivative and hence a centered sourceχ\chi. The lower group is the zero-potential correction ofSection6.2: the semigroup potentialπ¯=∫0∞χ​Pt​𝑑t\bar{\pi}=\int_{0}^{\infty}\chi P_{t}\,dtand the boundary occupation potentialsθi\theta_{i}cancel the source and produce a nonzero zero-mass signed BAR tuple.### 6.1A 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.

LetJ={1,…,d}J=\{1,\ldots,d\}. For a nonempty proper setA⊊JA\subsetneq J, putT=J∖AT=J\setminus Aand writeRB​CR_{BC}for the submatrix with rows inBBand columns inCC. Define the open stratumSA={x∈E:xi=0​for​i∈A,xj>0​for​j∈T}.S_{A}=\{x\in E:x_{i}=0\text{ for }i\in A,\ x_{j}>0\text{ for }j\in T\}.We identifySAS_{A}with(0,∞)T(0,\infty)^{T}through the embeddingιA:(0,∞)T→E\iota_{A}:(0,\infty)^{T}\to Ethat inserts zeros in the coordinates indexed byAA.

Proposition 6.1(Boundary gauge from a singular principal block).

Assume thatRRis nonsingular and thatRA​AR_{AA}is singular for some nonempty proper setA⊊JA\subsetneq J. Choose

0≠v∈ker⁡RA​A,w=RT​A​v∈ℝT.0\neq v\in\ker R_{AA},\qquad w=R_{TA}v\in\mathbb{R}^{T}.Thenw≠0w\neq 0. Letϕ∈Cc∞​((0,∞)T)\phi\in C_{c}^{\infty}((0,\infty)^{T})satisfyw⋅∇Tϕ≢0w\cdot\nabla_{T}\phi\not\equiv 0. Fori∈Ai\in A, define a finite signed measureζi\zeta_{i}onFiF_{i}by

∫Fig​(x)​𝑑ζi​(x)=vi​∫(0,∞)Tg​(ιA​(y))​ϕ​(y)​𝑑y,\int_{F_{i}}g(x)\,d\zeta_{i}(x)=v_{i}\int_{(0,\infty)^{T}}g(\iota_{A}(y))\phi(y)\,dy,(6.1)and setζi=0\zeta_{i}=0fori∉Ai\notin A. Defineχ∈ℳ​(E)\chi\in\mathcal{M}(E)by

∫Eg​(x)​𝑑χ​(x)=−∫(0,∞)Tg​(ιA​(y))​(w⋅∇Tϕ​(y))​𝑑y.\int_{E}g(x)\,d\chi(x)=-\int_{(0,\infty)^{T}}g(\iota_{A}(y))\bigl(w\cdot\nabla_{T}\phi(y)\bigr)\,dy.(6.2)Thenχ\chiis finite, nonzero, supported onSAS_{A}, and satisfiesχ​(E)=0\chi(E)=0. Moreover,

∑i=1d∫FiDi​f​𝑑ζi=∫Ef​𝑑χ,f∈Cb2​(E).\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\zeta_{i}=\int_{E}f\,d\chi,\qquad f\in C_{b}^{2}(E).(6.3)

Proof.

Ifw=0w=0, extendvvtov~∈ℝd\widetilde{v}\in\mathbb{R}^{d}by setting its coordinates inTTequal to zero. Then

R​v~=(RA​A​vRT​A​v)=0,R\widetilde{v}=\begin{pmatrix}R_{AA}v\\ R_{TA}v\end{pmatrix}=0,contradicting the nonsingularity ofRR. Hencew≠0w\neq 0, and a compactly supported smoothϕ\phiwith nonzero directional derivative alongwwexists.

Forf∈Cb2​(E)f\in C_{b}^{2}(E), combine the face labels before integrating:

∑i=1d∫FiDi​f​𝑑ζi\displaystyle\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\zeta_{i}=∫(0,∞)Tϕ​(y)​(∑i∈Avi​Ri)⋅∇f​(ιA​(y))​𝑑y\displaystyle=\int_{(0,\infty)^{T}}\phi(y)\left(\sum_{i\in A}v_{i}R_{i}\right)\cdot\nabla f(\iota_{A}(y))\,dy=∫(0,∞)Tϕ​(y)​[(RA​A​v)⋅∇Af​(ιA​(y))+(RT​A​v)⋅∇Tf​(ιA​(y))]​𝑑y\displaystyle=\int_{(0,\infty)^{T}}\phi(y)\left[(R_{AA}v)\cdot\nabla_{A}f(\iota_{A}(y))+(R_{TA}v)\cdot\nabla_{T}f(\iota_{A}(y))\right]dy=∫(0,∞)Tϕ​(y)​w⋅∇Tf​(ιA​(y))​𝑑y\displaystyle=\int_{(0,\infty)^{T}}\phi(y)w\cdot\nabla_{T}f(\iota_{A}(y))\,dy=−∫(0,∞)T(w⋅∇Tϕ​(y))​f​(ιA​(y))​𝑑y.\displaystyle=-\int_{(0,\infty)^{T}}\bigl(w\cdot\nabla_{T}\phi(y)\bigr)f(\iota_{A}(y))\,dy.There is no boundary term becauseϕ\phiis compactly supported in the open stratum. This proves (6.3). Taking a test function equal to one on a neighborhood ofsupp⁡ϕ\operatorname{supp}\phigivesχ​(E)=0\chi(E)=0, and the choice ofϕ\phigivesχ≠0\chi\neq 0. ∎

The singular block cancels the components normal to the active faces. The remaining vectorwwis tangent toSAS_{A}, and tangential integration by parts turns the boundary gauge into the centered sourceχ\chi.

6.2Interior 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\geq 0}be the transition semigroup of the SRBM, and letY=(Y1,…,Yd)Y=(Y_{1},\ldots,Y_{d})be its regulator. For a finite signed measureα\alpha, we define the semigroup(α​Pt)​(B)=∫EPt​(x,B)​α​(d​x).(\alpha P_{t})(B)=\int_{E}P_{t}(x,B)\,\alpha(dx).For each face define the boundary occupation kernel

Ki​(x,B)=𝔼x​∫01𝟏B​(Z​(s))​𝑑Yi​(s),K_{i}(x,B)=\mathbb{E}_{x}\int_{0}^{1}\mathbf{1}_{B}(Z(s))\,dY_{i}(s),(6.4)which is supported onFiF_{i}.

Proposition 6.2(Interior correction by a zero-potential).

Assume that the SRBM is a strong Markov process whose transition semigroup is Feller onC0​(E)C_{0}(E), and that it has stationary probabilityπ0\pi_{0}. Suppose that there are a locally bounded functionV:E→[1,∞)V:E\to[1,\infty)and constantsM,κ>0M,\kappa>0such that

‖Pt​(x,⋅)−π0‖TV≤M​V​(x)​e−κ​t,x∈E,t≥0.\|P_{t}(x,\cdot)-\pi_{0}\|_{\mathrm{TV}}\leq MV(x)e^{-\kappa t},\qquad x\in E,\ t\geq 0.(6.5)Suppose also that

ci:=supx∈E𝔼x​Yi​(1)<∞,i=1,…,d.c_{i}:=\sup_{x\in E}\mathbb{E}_{x}Y_{i}(1)<\infty,\qquad i=1,\ldots,d.(6.6)Letχ∈ℳ​(E)\chi\in\mathcal{M}(E)satisfyχ​(E)=0\chi(E)=0and∫EV​d​|χ|<∞,\int_{E}V\,d|\chi|<\infty,and letζi∈ℳ​(Fi)\zeta_{i}\in\mathcal{M}(F_{i})satisfy

∑i=1d∫FiDi​f​𝑑ζi=∫Ef​𝑑χ,f∈Cb2​(E).\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\zeta_{i}=\int_{E}f\,d\chi,\qquad f\in C_{b}^{2}(E).(6.7)Define

π¯\displaystyle\overline{\pi}=∫0∞χ​Pt​𝑑t,\displaystyle=\int_{0}^{\infty}\chi P_{t}\,dt,(6.8)θi\displaystyle\theta_{i}=∑n=0∞(χ​Pn)​Ki,\displaystyle=\sum_{n=0}^{\infty}(\chi P_{n})K_{i},(6.9)ν¯i\displaystyle\overline{\nu}_{i}=θi+ζi.\displaystyle=\theta_{i}+\zeta_{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 allf∈Cb2​(E)f\in C_{b}^{2}(E), hence also for allf∈Cc2​(E)f\in C_{c}^{2}(E), andπ¯​(E)=0.\overline{\pi}(E)=0.Ifχ≠0\chi\neq 0, thenπ¯≠0\overline{\pi}\neq 0.

Proof.

Becauseχ​(E)=0\chi(E)=0,

χ​Pt=∫E(Pt​(x,⋅)−π0)​χ​(d​x).\chi P_{t}=\int_{E}\bigl(P_{t}(x,\cdot)-\pi_{0}\bigr)\,\chi(dx).Therefore

‖χ​Pt‖TV≤M​e−κ​t​∫EV​d​|χ|.\|\chi P_{t}\|_{\mathrm{TV}}\leq Me^{-\kappa t}\int_{E}V\,d|\chi|.(6.11)The integral in (6.8) converges in total variation, andπ¯​(E)=0\overline{\pi}(E)=0becauseχ​Pt​(E)=χ​(E)=0\chi P_{t}(E)=\chi(E)=0.

For a finite signed measureα\alphaand the positive kernelKiK_{i},

‖α​Ki‖TV≤∫EKi​(x,E)​|α|​(d​x)≤ci​‖α‖TV.\|\alpha K_{i}\|_{\mathrm{TV}}\leq\int_{E}K_{i}(x,E)\,|\alpha|(dx)\leq c_{i}\|\alpha\|_{\mathrm{TV}}.(6.12)Combining (6.11) at integer times with (6.12) proves absolute convergence of (6.9). Eachθi\theta_{i}is supported onFiF_{i}.

Fixf∈Cb2​(E)f\in C_{b}^{2}(E). Itô’s formula up to an integer timeNN, followed by integration against the Jordan decomposition ofχ\chi, gives

χ​PN​(f)−χ​(f)=∫0N(χ​Pt)​(L​f)​𝑑t+∑i=1d∫E𝔼x​∫0NDi​f​(Z​(t))​𝑑Yi​(t)​χ​(d​x).\chi P_{N}(f)-\chi(f)=\int_{0}^{N}(\chi P_{t})(Lf)\,dt+\sum_{i=1}^{d}\int_{E}\mathbb{E}_{x}\int_{0}^{N}D_{i}f(Z(t))\,dY_{i}(t)\,\chi(dx).(6.13)Splitting the boundary integral into unit intervals and using the strong Markov property yields

∫E𝔼x​∫0NDi​f​(Z​(t))​𝑑Yi​(t)​χ​(d​x)=∑n=0N−1((χ​Pn)​Ki)​(Di​f).\int_{E}\mathbb{E}_{x}\int_{0}^{N}D_{i}f(Z(t))\,dY_{i}(t)\,\chi(dx)=\sum_{n=0}^{N-1}\bigl((\chi P_{n})K_{i}\bigr)(D_{i}f).(6.14)The estimates above justify passage to the limit. Sinceχ​PN​(f)→0\chi P_{N}(f)\to 0, equations (6.13) and (6.14) give

∫EL​f​𝑑π¯+∑i=1d∫FiDi​f​𝑑θi=−∫Ef​𝑑χ.\int_{E}Lf\,d\overline{\pi}+\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\theta_{i}=-\int_{E}f\,d\chi.Adding (6.7) proves the BAR for(π¯,ν¯1,…,ν¯d)(\overline{\pi},\overline{\nu}_{1},\ldots,\overline{\nu}_{d}).

It remains to prove that the zero potential is injective on this centered class. Let𝒜\mathcal{A}be the generator of the Feller semigroup onC0​(E)C_{0}(E). Forh∈D​(𝒜)h\in D(\mathcal{A}), semigroup differentiation gives

π¯​(𝒜​h)=limT→∞∫0Tχ​Pt​(𝒜​h)​𝑑t=limT→∞χ​(PT​h−h)=−χ​(h),\displaystyle\overline{\pi}(\mathcal{A}h)=\lim_{T\to\infty}\int_{0}^{T}\chi P_{t}(\mathcal{A}h)\,dt=\lim_{T\to\infty}\chi(P_{T}h-h)=-\chi(h),where the last limit follows from (6.11). Ifπ¯=0\overline{\pi}=0, thenχ​(h)=0\chi(h)=0for everyh∈D​(𝒜)h\in D(\mathcal{A}). To conclude thatχ=0\chi=0we use the following density theorem.

Theorem 6.3([17, Chapter 1, Section 2]).

The infinitesimal generator𝒜\mathcal{A}of a strongly continuous contraction semigroup on a Banach space has domainD​(𝒜)D(\mathcal{A})dense in that space. In particular, for a Feller semigroup onC0​(E)C_{0}(E), the domainD​(𝒜)D(\mathcal{A})is dense inC0​(E)C_{0}(E).

Its hypothesis is exactly the standing assumption of the present proposition:(Pt)(P_{t})is Feller onC0​(E)C_{0}(E). HenceD​(𝒜)D(\mathcal{A})is dense inC0​(E)C_{0}(E). Sinceℳ​(E)=C0​(E)∗\mathcal{M}(E)=C_{0}(E)^{*}, the identityχ​(h)=0\chi(h)=0onD​(𝒜)D(\mathcal{A})impliesχ=0\chi=0. Thereforeχ≠0\chi\neq 0impliesπ¯≠0\overline{\pi}\neq 0. ∎

Theorem 6.4(Failure in the completely-𝒮\mathcal{S}class).

Consider an SRBM inE=ℝ+dE=\mathbb{R}_{+}^{d}with positive definite covariance matrix, nonsingular completely-𝒮\mathcal{S}reflection matrixRR, and stationary probabilityπ0\pi_{0}. Assume that the process is strong Markov, is Feller onC0​(E)C_{0}(E), and satisfies the quantitative recurrence conditions (6.5) and (6.6). IfRA​AR_{AA}is singular for some nonempty proper setA⊊JA\subsetneq J, then signed BAR uniqueness fails.

More precisely, there exists a finite signed BAR tuple(π¯,ν¯1,…,ν¯d)(\overline{\pi},\overline{\nu}_{1},\ldots,\overline{\nu}_{d})such that

π¯​(E)=0,π¯≠0.\overline{\pi}(E)=0,\qquad\overline{\pi}\neq 0.(6.15)Consequentlyπ¯\overline{\pi}is not a scalar multiple ofπ0\pi_{0}.

Proof.

ApplySection6.1. Its sourceχ\chiis compactly supported, and the local boundedness ofVVgives∫EV​d​|χ|<∞\int_{E}V\,d|\chi|<\infty.Section6.2then produces the required BAR tuple. Ifπ¯=c​π0\overline{\pi}=c\pi_{0}, total masses givec=0c=0, contradictingπ¯≠0\overline{\pi}\neq 0. ∎

Corollary 6.5(Infinite-dimensional failure).

Under the assumptions ofTheorem6.4, 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 functionsϕm∈Cc∞​((0,∞)T)\phi_{m}\in C_{c}^{\infty}((0,\infty)^{T})with pairwise disjoint supports andw⋅∇Tϕm≢0w\cdot\nabla_{T}\phi_{m}\not\equiv 0. Letχm\chi_{m}andπ¯m\overline{\pi}_{m}be the corresponding sources and zero potentials. If∑m=1Nam​π¯m=0\sum_{m=1}^{N}a_{m}\overline{\pi}_{m}=0, linearity and the injectivity identity in the proof ofSection6.2imply∑m=1Nam​χm=0\sum_{m=1}^{N}a_{m}\chi_{m}=0. The sources are nonzero and have pairwise disjoint supports, so everyama_{m}is zero. ∎

The theorem concerns a singular proper principal block. Singularity of an arbitrary rectangular or nonprincipal submatrix does not yield the cancellationRA​A​v=0R_{AA}v=0needed inSection6.1. Conversely, every principal block of a nonsingularMM-matrix is nonsingular bySection3.1, so this obstruction is absent from the class covered byTheorem2.2.

6.3A 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. AZZmatrix means a matrix with nonpositive off-diagonal entries.

Corollary 6.6(A checkable completely-𝒮\mathcal{S}subclass).

Assume in addition thatRi​i=1R_{ii}=1andRi​j≥0R_{ij}\geq 0for alli,ji,j. Suppose there is a symmetric positive definite matrixHHsuch thatH​RHRis aZZmatrix andH​μ<0H\mu<0componentwise. IfRRis nonsingular, completelySS, and has a singular proper principal block, then signed BAR uniqueness fails and the conclusion ofSection6.2holds.

Proof.

We verify the standing hypotheses ofTheorem6.4: existence, the strong Markov property, theC0C_{0}-Feller property, the recurrence certificate (6.5), and the regulator bound (6.6).

First, for the existence, the strong Markov property, theC0C_{0}-Feller property, as well as (6.6), we have

Proposition 6.7([34]).

For a symmetric positive definite covarianceΣ\Sigma, a driftμ\mu, and a reflection matrixRRwith unit diagonal, the orthant SRBM with data(Σ,μ,R)(\Sigma,\mu,R)exists and is unique in law if and only ifRRis completely-𝒮\mathcal{S}; when it exists it is a Feller continuous strong Markov process, andx↦Pt​h​(x)x\mapsto P_{t}h(x)is continuous for everyh∈Cb​(E)h\in C_{b}(E).

SinceΣ\Sigmais positive definite,RRhas unit diagonal, andRRis completely-SS, the orthant SRBM exists, is unique in law, is strong Markov, andx↦Pt​h​(x)x\mapsto P_{t}h(x)is continuous for everyh∈Cb​(E)h\in C_{b}(E)(Feller continuity).

Now we check theC0C_{0}Feller property, i.e.Pt​C0​(E)⊂C0​(E)P_{t}C_{0}(E)\subset C_{0}(E)and that‖Pt​h−h‖∞→0\|P_{t}h-h\|_{\infty}\to 0forh∈C0​(E)h\in C_{0}(E). WriteZx​(t)=x+μ​t+B​(t)+R​Yx​(t),Z^{x}(t)=x+\mu t+B(t)+RY^{x}(t),whereB​(t)=Σ1/2​W​(t)B(t)=\Sigma^{1/2}W(t)and eachYixY_{i}^{x}is nondecreasing. SinceRi​j≥0R_{ij}\geq 0andYjx≥0Y_{j}^{x}\geq 0, the one-dimensional Skorokhod formula gives, for0≤s≤t0\leq s\leq t, we haveYix​(s)≤|μi|​t+sup0≤u≤t|Bi​(u)|.Y_{i}^{x}(s)\leq|\mu_{i}|t+\sup_{0\leq u\leq t}|B_{i}(u)|.(This also verifies (6.6) since the right side has finite expectation independent of the initial statexx.)

Hence, withMt:=|μ|∞​t+sup0≤u≤t|B​(u)|∞​and​CR:=1+maxi​∑jRi​j,M_{t}:=|\mu|_{\infty}t+\sup_{0\leq u\leq t}|B(u)|_{\infty}\text{ and }C_{R}:=1+\max_{i}\sum_{j}R_{ij},we have the uniform displacement bound

sup0≤s≤t|Zx​(s)−x|∞≤CR​Mt.\sup_{0\leq s\leq t}|Z^{x}(s)-x|_{\infty}\leq C_{R}M_{t}. Leth∈Cc​(E)h\in C_{c}(E)and supposesupp⁡h⊂{|y|∞≤a}\operatorname{supp}h\subset\{|y|_{\infty}\leq a\}. Then

|Pt​h​(x)|≤‖h‖∞​ℙ​{CR​Mt≥|x|∞−a}⟶0as​|x|∞→∞.|P_{t}h(x)|\leq\|h\|_{\infty}\mathbb{P}\{C_{R}M_{t}\geq|x|_{\infty}-a\}\longrightarrow 0\qquad\text{as }|x|_{\infty}\to\infty.ThusPt​h∈C0​(E)P_{t}h\in C_{0}(E)forh∈Cc​(E)h\in C_{c}(E). By contraction and approximation ofC0​(E)C_{0}(E)by compactly supported continuous functions, the same holds for everyh∈C0​(E)h\in C_{0}(E).

Finally, everyh∈C0​(E)h\in C_{0}(E)is uniformly continuous. Ifωh\omega_{h}is its modulus of continuity, then

supx∈E|Pt​h​(x)−h​(x)|≤𝔼​ωh​(CR​Mt).\sup_{x\in E}|P_{t}h(x)-h(x)|\leq\mathbb{E}\,\omega_{h}(C_{R}M_{t}).SinceMt→0M_{t}\to 0almost surely ast↓0t\downarrow 0and0≤ωh≤2​‖h‖∞0\leq\omega_{h}\leq 2\|h\|_{\infty}, dominated convergence gives‖Pt​h−h‖∞→0.\|P_{t}h-h\|_{\infty}\to 0.Therefore(Pt)(P_{t})is a strongly continuous positive contraction semigroup onC0​(E)C_{0}(E).

Then we verify (6.5).

Proposition 6.8([33, Corollary 3.2]).

Let(Σ,μ,R)(\Sigma,\mu,R)be orthant SRBM data withΣ\Sigmasymmetric positive definite andRRcompletely-𝒮\mathcal{S}. If there is a symmetric positive definite matrixHHwithH​RHRaZZmatrix andH​μ<0H\mu<0componentwise, then the SRBM is positive recurrent with a unique stationary probabilityπ0\pi_{0}and isVV-uniformly exponentially ergodic: there exist a locally boundedV:E→[1,∞)V\colon E\to[1,\infty)and constantsM,κ>0M,\kappa>0such that

‖Pt​(x,⋅)−π0‖TV≤M​V​(x)​e−κ​t,x∈E,t≥0.\left\lVert P_{t}(x,\cdot)-\pi_{0}\right\rVert_{\mathrm{TV}}\leq MV(x)e^{-\kappa t},\qquad x\in E,\ t\geq 0.

The matrixHHin the statement of the corollary is exactly such a certificate: it is symmetric positive definite,H​RHRis aZZmatrix, andH​μ<0H\mu<0componentwise, whileΣ\Sigmais positive definite andRRis completely-𝒮\mathcal{S}. Hence Sarantsev’s criterion supplies the stationary probabilityπ0\pi_{0}and the recurrence certificate (6.5).

ThusTheorem6.4applies. ∎

Set

R​(a,b,c,d)=(11a11bcd1),Σ=I3,μ=(−11−7−11),R(a,b,c,d)=\begin{pmatrix}1&1&a\\ 1&1&b\\ c&d&1\end{pmatrix},\qquad\Sigma=I_{3},\qquad\mu=\begin{pmatrix}-11\\ -7\\ -11\end{pmatrix},(6.16)and let𝒫\mathcal{P}be the parameter region

0<b≤1120,56​b+2560≤a≤3​b+35,0<c≤425,23≤d≤3546.\begin{gathered}0<b\leq\frac{11}{20},\qquad\frac{56b+25}{60}\leq a\leq\frac{3b+3}{5},\qquad 0<c\leq\frac{4}{25},\qquad\frac{2}{3}\leq d\leq\frac{35}{46}.\end{gathered}(6.17)The subset obtained by making all inequalities strict is nonempty, so𝒫\mathcal{P}contains a genuine four-dimensional region.

Theorem 6.9(Four-parameter family).

For every(a,b,c,d)∈𝒫(a,b,c,d)\in\mathcal{P}, the SRBM data in (6.16) define a nonsingular completelySS, exponentially ergodic SRBM with a unique stationary probability. Its reflection matrix has the singular proper principal block

R{1,2},{1,2}=(1111).R_{\{1,2\},\{1,2\}}=\begin{pmatrix}1&1\\ 1&1\end{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 ofR​(a,b,c,d)R(a,b,c,d)is positive. Hence every principal submatrix is anSSmatrix, with the all-ones vector as a witness, andRRis completelySS. A direct calculation givesdetR​(a,b,c,d)=(b−a)​(c−d).\det R(a,b,c,d)=(b-a)(c-d).The parameter bounds implya>ba>bandd>cd>c, so the determinant is positive. The block indexed byA={1,2}A=\{1,2\}is singular, with

v=(1−1)∈ker⁡RA​A,w=R{3},A​v=c−d≠0.v=\begin{pmatrix}1\\ -1\end{pmatrix}\in\ker R_{AA},\qquad w=R_{\{3\},A}v=c-d\neq 0. Consider the symmetric matrix

H=(12−310−310−31072518−3101823100).H=\begin{pmatrix}\frac{1}{2}&-\frac{3}{10}&-\frac{3}{10}\\ -\frac{3}{10}&\frac{7}{25}&\frac{1}{8}\\ -\frac{3}{10}&\frac{1}{8}&\frac{23}{100}\end{pmatrix}.(6.18)Its leading principal minors are12,120,7980000,\frac{1}{2},\frac{1}{20},\frac{79}{80000},soHHis positive definite. Multiplication gives

H​R=(15−3​c1015−3​d10a2−3​b10−310c8−150d8−150−3​a10+7​b25+1823​c100−74023​d100−740−3​a10+b8+23100).HR=\begin{pmatrix}\frac{1}{5}-\frac{3c}{10}&\frac{1}{5}-\frac{3d}{10}&\frac{a}{2}-\frac{3b}{10}-\frac{3}{10}\\[5.69054pt] \frac{c}{8}-\frac{1}{50}&\frac{d}{8}-\frac{1}{50}&-\frac{3a}{10}+\frac{7b}{25}+\frac{1}{8}\\[5.69054pt] \frac{23c}{100}-\frac{7}{40}&\frac{23d}{100}-\frac{7}{40}&-\frac{3a}{10}+\frac{b}{8}+\frac{23}{100}\end{pmatrix}.(6.19)The six off-diagonal entries are nonpositive by (6.17). Also

H​μ=(−110−7200−21200)<0.H\mu=\begin{pmatrix}-\frac{1}{10}\\[2.84526pt] -\frac{7}{200}\\[2.84526pt] -\frac{21}{200}\end{pmatrix}<0.(6.20)Section6.3completes the proof. ∎

For this entire family the algebraic source is explicit. LetS={(0,0,y):y>0}S=\{(0,0,y):y>0\}and choose a nonconstantϕ∈Cc∞​((0,∞))\phi\in C_{c}^{\infty}((0,\infty)). Define

∫F1g​𝑑ζ1\displaystyle\int_{F_{1}}g\,d\zeta_{1}=∫0∞g​(0,0,y)​ϕ​(y)​𝑑y,\displaystyle=\int_{0}^{\infty}g(0,0,y)\phi(y)\,dy,(6.21)∫F2g​𝑑ζ2\displaystyle\int_{F_{2}}g\,d\zeta_{2}=−∫0∞g​(0,0,y)​ϕ​(y)​𝑑y,ζ3=0.\displaystyle=-\int_{0}^{\infty}g(0,0,y)\phi(y)\,dy,\qquad\zeta_{3}=0.(6.22)SinceR1−R2=(0,0,c−d)TR_{1}-R_{2}=(0,0,c-d)^{T},

∑i=13∫FiDi​f​𝑑ζi=(d−c)​∫0∞ϕ′​(y)​f​(0,0,y)​𝑑y.\sum_{i=1}^{3}\int_{F_{i}}D_{i}f\,d\zeta_{i}=(d-c)\int_{0}^{\infty}\phi^{\prime}(y)f(0,0,y)\,dy.(6.23)Thus

χ​(d​x)=(d−c)​ϕ′​(x3)​d​x3​δ0​(d​x1)​δ0​(d​x2)\chi(dx)=(d-c)\phi^{\prime}(x_{3})\,dx_{3}\,\delta_{0}(dx_{1})\delta_{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=\begin{pmatrix}1&1&\frac{3}{5}\\ 1&1&\frac{1}{6}\\ \frac{2}{15}&\frac{2}{3}&1\end{pmatrix},\qquad\Sigma=I_{3},\qquad\mu=\begin{pmatrix}-11\\ -7\\ -11\end{pmatrix},(6.25)there is a finite signed BAR tuple(π¯,ν¯1,ν¯2,ν¯3)(\overline{\pi},\overline{\nu}_{1},\overline{\nu}_{2},\overline{\nu}_{3})such thatπ¯​(E)=0\overline{\pi}(E)=0andπ¯≠0\overline{\pi}\neq 0.

Proof.

The parameter choice belongs to𝒫\mathcal{P}. Exact arithmetic gives

detR=52225>0,R−1​μ=(−365104−203104−12013)<0.\det R=\frac{52}{225}>0,\qquad R^{-1}\mu=\begin{pmatrix}-\frac{365}{104}\\[2.84526pt] -\frac{203}{104}\\[2.84526pt] -\frac{120}{13}\end{pmatrix}<0.(6.26)The matrixHHin (6.18) satisfies

H​R=(4250−120−130019300−1120−4333000−1360017240),HR=\begin{pmatrix}\frac{4}{25}&0&-\frac{1}{20}\\[2.84526pt] -\frac{1}{300}&\frac{19}{300}&-\frac{1}{120}\\[2.84526pt] -\frac{433}{3000}&-\frac{13}{600}&\frac{17}{240}\end{pmatrix},(6.27)and (6.20) holds. Thus all assumptions inSection6.3are verified.

For complete explicitness, take the standard bump

ϕ​(y)={exp⁡(−1(y−1)​(2−y)),1<y<2,0,otherwise.\phi(y)=\begin{cases}\displaystyle\exp\!\left(-\frac{1}{(y-1)(2-y)}\right),&1<y<2,\\[5.69054pt] 0,&\text{otherwise}.\end{cases}(6.28)Defineζ1,ζ2,ζ3\zeta_{1},\zeta_{2},\zeta_{3}by (6.21)–(6.22). Here

R1−R2=(00−815),R_{1}-R_{2}=\begin{pmatrix}0\\ 0\\ -\frac{8}{15}\end{pmatrix},so

χ​(d​x)=815​ϕ′​(x3)​d​x3​δ0​(d​x1)​δ0​(d​x2),χ​(E)=0,χ≠0.\chi(dx)=\frac{8}{15}\phi^{\prime}(x_{3})\,dx_{3}\,\delta_{0}(dx_{1})\delta_{0}(dx_{2}),\qquad\chi(E)=0,\qquad\chi\neq 0.(6.29)LetPtP_{t}be the reflected semigroup and letKiK_{i}be the kernels in (6.4). Set

π¯=∫0∞χ​Pt​𝑑t,ν¯i=ζi+∑n=0∞(χ​Pn)​Ki.\overline{\pi}=\int_{0}^{\infty}\chi P_{t}\,dt,\qquad\overline{\nu}_{i}=\zeta_{i}+\sum_{n=0}^{\infty}(\chi P_{n})K_{i}.(6.30)BySection6.2, all measures in (6.30) are finite and satisfy

∫EL​f​𝑑π¯+∑i=13∫FiDi​f​𝑑ν¯i=0,f∈Cb2​(E).\int_{E}Lf\,d\overline{\pi}+\sum_{i=1}^{3}\int_{F_{i}}D_{i}f\,d\overline{\nu}_{i}=0,\qquad f\in C_{b}^{2}(E).Moreoverπ¯​(E)=0\overline{\pi}(E)=0andπ¯≠0\overline{\pi}\neq 0. Thereforeπ¯\overline{\pi}cannot be a scalar multiple of the stationary probabilityπ0\pi_{0}. ∎

7Unified 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\mathcal{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 nonsingularMM-matrix class this quotient vanishes, while in the completely-𝒮\mathcal{S}singular-block regime the boundary source and its zero-potential lift produce a nonzero class in this quotient. We denote

𝖬E=ℳ​(E),𝖬∂=∏i=1dℳ​(Fi),𝒯=Cb2​(E).\mathsf{M}_{E}=\mathcal{M}(E),\qquad\mathsf{M}_{\partial}=\prod_{i=1}^{d}\mathcal{M}(F_{i}),\qquad\mathcal{T}=C_{b}^{2}(E).For(π,ν)∈𝖬E×𝖬∂(\pi,\nu)\in\mathsf{M}_{E}\times\mathsf{M}_{\partial}, define the BAR functional𝒜​(π,ν)​(f)=∫EL​f​𝑑π+∑i=1d∫FiDi​f​𝑑νi,\mathcal{A}(\pi,\nu)(f)=\int_{E}Lf\,d\pi+\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\nu_{i},forf∈𝒯f\in\mathcal{T}. Thus finite signed BAR tuples are preciselyker⁡𝒜\ker\mathcal{A}. Define the pure boundary operator∂Rζ​(f)=∑i=1d∫FiDi​f​𝑑ζi\partial_{R}\zeta(f)=\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\zeta_{i}. Since one can identify a finite signed measureχ∈ℳ​(E)\chi\in\mathcal{M}(E)with the functionalf↦∫Ef​𝑑χf\mapsto\int_{E}f\,d\chion𝒯\mathcal{T}, we define the relation∂Rζ=χ\partial_{R}\zeta=\chiby the condition that∑i=1d∫FiDi​f​𝑑ζi=∫Ef​𝑑χ\sum_{i=1}^{d}\int_{F_{i}}D_{i}f\,d\zeta_{i}=\int_{E}f\,d\chiholds for allf∈𝒯f\in\mathcal{T}.

7.1The RI-defect quotient

LetZBAR:=ker⁡𝒜Z_{\rm BAR}:=\ker\mathcal{A}andΠE​(π,ν)=π\Pi_{E}(\pi,\nu)=\pi, and define the space of interior coordinates of signed BAR tuples by

ℐBAR:=ΠE​ZBAR={π∈ℳ​(E):there exists​ν∈𝖬∂​with​(π,ν)∈ker⁡𝒜}.\mathcal{I}_{\rm BAR}:=\Pi_{E}Z_{\rm BAR}=\{\pi\in\mathcal{M}(E):\text{ there exists }\nu\in\mathsf{M}_{\partial}\text{ with }(\pi,\nu)\in\ker\mathcal{A}\}.(7.1)For a finite signed measurem∈ℳ​(E)m\in\mathcal{M}(E), define its resolvent-insertion defect by

𝔯​(m)​(λ,h):=∫E(λ​Rλ​h−h)​𝑑m,λ>0,h∈C0​(E).\mathfrak{r}(m)(\lambda,h):=\int_{E}(\lambda R_{\lambda}h-h)\,dm,\qquad\lambda>0,\quad h\in C_{0}(E).(7.2)Thusmmsatisfies the resolvent identity precisely when𝔯​(m)=0\mathfrak{r}(m)=0. SetℐRI:=ℐBAR∩ker⁡𝔯,\mathcal{I}_{\rm RI}:=\mathcal{I}_{\rm BAR}\cap\ker\mathfrak{r},and define the RI-defect quotient𝒬RI:=ℐBAR/ℐRI.\mathcal{Q}_{\rm RI}:=\mathcal{I}_{\rm BAR}/\mathcal{I}_{\rm RI}.Equivalently,𝒬RI≃𝔯​(ℐBAR).\mathcal{Q}_{\rm RI}\simeq\mathfrak{r}(\mathcal{I}_{\rm 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 isC0C_{0}-Feller and strongly continuous, that it has a unique invariant probabilityπ0\pi_{0}, and thats0=(π0,ν0)s_{0}=(\pi_{0},\nu^{0})is the stationary BAR tuple. Then there is a canonical exact sequence

0⟶ker​∂R⟶ker⁡𝒜/ℝ​s0⟶𝒬RI⟶0.0\longrightarrow\ker\partial_{R}\longrightarrow\ker\mathcal{A}/\mathbb{R}s_{0}\longrightarrow\mathcal{Q}_{\rm RI}\longrightarrow 0.(7.3)Equivalently,

ker⁡𝒜ℝ​s0⊕({0}×ker​∂R)≃𝒬RI.\frac{\ker\mathcal{A}}{\mathbb{R}s_{0}\oplus(\{0\}\times\ker\partial_{R})}\simeq\mathcal{Q}_{\rm RI}.(7.4)

Proof.

LetHBAR:=ker⁡𝒜/ℝ​s0.H_{\rm BAR}:=\ker\mathcal{A}/\mathbb{R}s_{0}.DefineΓ:HBAR→𝒬RI,\Gamma:H_{\rm BAR}\to\mathcal{Q}_{\rm RI},andΓ​([(π,ν)])=[π].\Gamma([(\pi,\nu)])=[\pi].Replacing(π,ν)(\pi,\nu)by(π,ν)+c​(π0,ν0)(\pi,\nu)+c(\pi_{0},\nu^{0})changes the interior coordinate byc​π0c\pi_{0}. Sinceπ0\pi_{0}is invariant,

∫ERλ​h​𝑑π0=∫0∞e−λ​t​∫EPt​h​𝑑π0​𝑑t=λ−1​∫Eh​𝑑π0,\int_{E}R_{\lambda}h\,d\pi_{0}=\int_{0}^{\infty}e^{-\lambda t}\int_{E}P_{t}h\,d\pi_{0}\,dt=\lambda^{-1}\int_{E}h\,d\pi_{0},so𝔯​(π0)=0\mathfrak{r}(\pi_{0})=0andπ0∈ℐRI\pi_{0}\in\mathcal{I}_{\rm RI}. Hence[π][\pi]is unchanged in𝒬RI\mathcal{Q}_{\rm RI}. The mapΓ\Gammais surjective by the definition of𝒬RI\mathcal{Q}_{\rm RI}.

We compute its kernel. SupposeΓ​([(π,ν)])=0\Gamma([(\pi,\nu)])=0. Thenπ∈ℐRI\pi\in\mathcal{I}_{\rm RI}, so

∫E(λ​Rλ​h−h)​𝑑π=0,h∈C0​(E),λ>0.\int_{E}(\lambda R_{\lambda}h-h)\,d\pi=0,\qquad h\in C_{0}(E),\quad\lambda>0.BySection5,π​Pt=π\pi P_{t}=\pifor allt≥0t\geq 0. BySection5, we haveπ=c​π0,\pi=c\pi_{0},wherec=π​(E).c=\pi(E).Since both(π,ν)(\pi,\nu)andc​(π0,ν0)c(\pi_{0},\nu^{0})are BAR tuples,(0,ν−c​ν0)=(π,ν)−c​(π0,ν0)∈ker⁡𝒜.(0,\nu-c\nu^{0})=(\pi,\nu)-c(\pi_{0},\nu^{0})\in\ker\mathcal{A}.Equivalently,∂R(ν−c​ν0)=0.\partial_{R}(\nu-c\nu^{0})=0.Thus[(π,ν)]=[(0,η)][(\pi,\nu)]=[(0,\eta)]for someη∈ker​∂R\eta\in\ker\partial_{R}.

Conversely, ifη∈ker​∂R\eta\in\ker\partial_{R}, then(0,η)∈ker⁡𝒜(0,\eta)\in\ker\mathcal{A}and its interior coordinate has zero RI defect. Henceker⁡Γ={[(0,η)]:η∈ker​∂R}.\ker\Gamma=\{[(0,\eta)]:\eta\in\ker\partial_{R}\}.

The mapη↦[(0,η)]\eta\mapsto[(0,\eta)]is injective because if[(0,η)]=0[(0,\eta)]=0inker⁡𝒜/ℝ​s0\ker\mathcal{A}/\mathbb{R}s_{0}, then(0,η)=c​(π0,ν0)(0,\eta)=c(\pi_{0},\nu^{0})for somec∈ℝc\in\mathbb{R}; the interior coordinate givesc​π0=0c\pi_{0}=0, hencec=0c=0andη=0\eta=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. ∎

The positive theorem in the Harrison–Reiman Class

In the Harrison–Reiman nonsingularMM-matrix setting,Theorem4.1says that every signed BAR tuple satisfies the resolvent identity. Equivalently,𝔯​(ℐBAR)=0\mathfrak{r}(\mathcal{I}_{\rm BAR})=0and𝒬RI=0.\mathcal{Q}_{\rm RI}=0.ThenSection7.1reduces the quotientker⁡𝒜/ℝ​s0\ker\mathcal{A}/\mathbb{R}s_{0}to the pure boundary kernel. The latter is killed bySection5,*i.e.*ker​∂R={0}.\ker\partial_{R}=\{0\}.Consequentlyker⁡𝒜=ℝ​s0,\ker\mathcal{A}=\mathbb{R}s_{0},which is the signed uniqueness conclusion ofTheorem2.2.

This formulation separates the two uses of active-block invertibility. First,RA​A−1R_{AA}^{-1}defines the active projection

𝖫A​v=v−RA​RA​A−1​vA,𝖫A​Ri=0,i∈A.\mathsf{L}_{A}v=v-R_{A}R_{AA}^{-1}v_{A},\qquad\mathsf{L}_{A}R_{i}=0,\quad i\in A.This is the algebraic input behind the measure–Neumann resolvent insertion. Second,RA​A−TR_{AA}^{-T}prescribes active oblique jets onSAS_{A}. If the active normal gradient isaa, then(Di​f|SA)i∈A=RA​AT​a,(D_{i}f|_{S_{A}})_{i\in A}=R_{AA}^{T}a,so invertibility ofRA​AR_{AA}permits the choicea=RA​A−T​ψa=R_{AA}^{-T}\psi. The induction over strata inSection5uses exactly this prescription.

The completely-𝒮\mathcal{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 into𝒬RI\mathcal{Q}_{\rm RI}. Starting from the singular boundary gauge, one has a boundary source identity∂Rζ=χ.\partial_{R}\zeta=\chi.Thus the boundary gauge alone has BAR defect+χ+\chi. The zero potentialQχ=(Uχ,Θχ),where​Uχ=∫0∞χ​Pt​𝑑t,Q_{\chi}=(U_{\chi},\Theta_{\chi}),\text{ where }U_{\chi}=\int_{0}^{\infty}\chi P_{t}\,dt,is constructed so that its BAR contribution is exactly the opposite defect:A​(Qχ)=−χ.A(Q_{\chi})=-\chi.Therefore we haveQχ+(0,ζ)=(Uχ,Θχ+ζ)∈ker⁡A,Q_{\chi}+(0,\zeta)=(U_{\chi},\Theta_{\chi}+\zeta)\in\ker A,so the zero potential turns the boundary gauge into a genuine signed BAR tuple. However, this cancellation does not make the sourceχ\chidisappear. Instead, we shows thatχ\chireappears as the resolvent insertion defect of the interior measureUχU_{\chi}:r​(Uχ)​(λ,h)=−∫ERλ​h​𝑑χ.r(U_{\chi})(\lambda,h)=-\int_{E}R_{\lambda}h\,d\chi.Hence, ifχ≠0\chi\neq 0, thenUχU_{\chi}cannot satisfy the resolvent insertion identity; otherwise the strong continuity of the semigroup would imply thatχ\chivanishes on all functions inC0​(E)C_{0}(E), forcingχ=0\chi=0. Consequently, the zero potential is the mechanism that converts the singular boundary source into a concrete nonzero class.0≠[Uχ]∈QRI.0\neq[U_{\chi}]\in Q_{\mathrm{RI}}.This is why the completelySScounterexample is best understood as an RI defect: the boundary algebra creates the centered sourceχ\chi, 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 onC0​(E)C_{0}(E)and that the zero-potential construction ofSection6.2is available for a centered sourceχ\chi. Write

Q​χ=(U​χ,Θ​χ),U​χ=∫0∞χ​Pt​𝑑t,Θi​χ=∑n=0∞(χ​Pn)​Ki.Q\chi=(U\chi,\Theta\chi),\qquad U\chi=\int_{0}^{\infty}\chi P_{t}\,dt,\qquad\Theta_{i}\chi=\sum_{n=0}^{\infty}(\chi P_{n})K_{i}.Ifζ∈𝖬∂\zeta\in\mathsf{M}_{\partial}satisfies∂Rζ=χ\partial_{R}\zeta=\chi, then

Q​χ+(0,ζ)=(U​χ,Θ​χ+ζ)∈ker⁡𝒜.Q\chi+(0,\zeta)=(U\chi,\Theta\chi+\zeta)\in\ker\mathcal{A}.(7.5)Its image under the map inSection7.1is the class[U​χ]∈𝒬RI[U\chi]\in\mathcal{Q}_{\rm RI}, and

𝔯​(U​χ)​(λ,h)=−∫ERλ​h​𝑑χ,λ>0,h∈C0​(E).\mathfrak{r}(U\chi)(\lambda,h)=-\int_{E}R_{\lambda}h\,d\chi,\qquad\lambda>0,\quad h\in C_{0}(E).(7.6)In particular, ifχ≠0\chi\neq 0, then[U​χ]≠0[U\chi]\neq 0in𝒬RI\mathcal{Q}_{\rm RI}.

Proof.

BySection6.2and fact that∂Rζ=χ\partial_{R}\zeta=\chi, thus𝒜​(Q​χ)​(f)=−∫Ef​𝑑χ\mathcal{A}(Q\chi)(f)=-\int_{E}f\,d\chiand𝒜​(0,ζ)​(f)=∫Ef​𝑑χ\mathcal{A}(0,\zeta)(f)=\int_{E}f\,d\chiholds for allf∈𝒯f\in\mathcal{T}. Adding the two identities gives (7.5). HenceU​χ∈ℐBARU\chi\in\mathcal{I}_{\rm BAR}and its class in the RI-defect quotient is[U​χ][U\chi].

It remains to identify its defect. Forh∈C0​(E)h\in C_{0}(E)putF​(t)=∫EPt​h​𝑑χ.F(t)=\int_{E}P_{t}h\,d\chi.The total-variation convergence inSection6.2justifies the following Fubini calculation:

𝔯​(U​χ)​(λ,h)\displaystyle\mathfrak{r}(U\chi)(\lambda,h)=∫0∞∫EPt​(λ​Rλ​h−h)​𝑑χ​𝑑t=∫0∞(λ​∫0∞e−λ​s​F​(t+s)​𝑑s−F​(t))​𝑑t\displaystyle=\int_{0}^{\infty}\int_{E}P_{t}(\lambda R_{\lambda}h-h)\,d\chi\,dt=\int_{0}^{\infty}\left(\lambda\int_{0}^{\infty}e^{-\lambda s}F(t+s)\,ds-F(t)\right)dt=∫0∞F​(u)​(1−e−λ​u)​𝑑u−∫0∞F​(u)​𝑑u=−∫0∞e−λ​u​F​(u)​𝑑u=−∫ERλ​h​𝑑χ.\displaystyle=\int_{0}^{\infty}F(u)(1-e^{-\lambda u})\,du-\int_{0}^{\infty}F(u)\,du=-\int_{0}^{\infty}e^{-\lambda u}F(u)\,du=-\int_{E}R_{\lambda}h\,d\chi.If[U​χ]=0[U\chi]=0in𝒬RI\mathcal{Q}_{\rm RI}, then𝔯​(U​χ)=0\mathfrak{r}(U\chi)=0, henceχ​(Rλ​h)=0\chi(R_{\lambda}h)=0for allλ>0\lambda>0andh∈C0​(E)h\in C_{0}(E). Since the semigroup is strongly continuous onC0​(E)C_{0}(E),

λ​Rλ​h=∫0∞e−s​Ps/λ​h​𝑑s⟶hin​C0​(E)\lambda R_{\lambda}h=\int_{0}^{\infty}e^{-s}P_{s/\lambda}h\,ds\longrightarrow h\quad\text{in }C_{0}(E)asλ→∞\lambda\to\infty. Thereforeχ​(h)=0\chi(h)=0for everyh∈C0​(E)h\in C_{0}(E), and the finite Radon measureχ\chiis zero. Thusχ≠0\chi\neq 0implies[U​χ]≠0[U\chi]\neq 0. ∎

Suppose thatRRis nonsingular and thatRA​AR_{AA}is singular for some nonempty proper subsetA⊊JA\subsetneq J. PutT=J∖AT=J\setminus Aand choose0≠v∈ker⁡RA​A,0\neq v\in\ker R_{AA},wherew=RT​A​v.w=R_{TA}v.SinceRRis nonsingular,w≠0w\neq 0. OnSAS_{A}, the local boundary symbol is

∑i∈Avi​Di​f=(RA​A​v)⋅∇Af+(RT​A​v)⋅∇Tf.\sum_{i\in A}v_{i}D_{i}f=(R_{AA}v)\cdot\nabla_{A}f+(R_{TA}v)\cdot\nabla_{T}f.(7.7)BecauseRA​A​v=0R_{AA}v=0, only the tangential derivativew⋅∇Tfw\cdot\nabla_{T}fremains.Section6.1turns this symbol calculation into a boundary source: for a compactly supported smooth densityϕ\phion the open stratum withw⋅∇Tϕ≢0w\cdot\nabla_{T}\phi\not\equiv 0, it constructs boundary measuresζi\zeta_{i}and a finite nonzero centered measureχ\chisupported onSAS_{A}such that∂Rζ=χ.\partial_{R}\zeta=\chi.Under the recurrence and regulator hypotheses ofSection6.2,Section7.1gives(U​χ,Θ​χ+ζ)∈ker⁡𝒜(U\chi,\Theta\chi+\zeta)\in\ker\mathcal{A}and places its interior coordinate into the RI-defect quotient as0≠[U​χ]∈𝒬RI.0\neq[U\chi]\in\mathcal{Q}_{\rm 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,𝔯​(U​χ)​(λ,h)=−χ​(Rλ​h).\mathfrak{r}(U\chi)(\lambda,h)=-\chi(R_{\lambda}h).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 AA stable example withg=Rλ​h∉C2​(E)g=R_{\lambda}h\notin C^{2}(E)

This appendix gives a concrete example in which the probabilistic resolventg=Rλ​hg=R_{\lambda}his not inC2​(E)C^{2}(E). The point is that interior smoothness and one-sided oblique flatness on open faces do not guarantee closed-domainC2C^{2}regularity at a corner. We choose a smooth nonnegative sourceh∈Cc∞​(E∘)h\in C_{c}^{\infty}(E^{\circ})such thath≢0h\not\equiv 0buth​(0)=0h(0)=0.

We first justify thatg​(0)>0g(0)>0. Sinceh≥0h\geq 0,h≢0h\not\equiv 0, andsupp⁡h⊂E∘\operatorname{supp}h\subset E^{\circ}, choosez∗∈E∘z_{\ast}\in E^{\circ},r>0r>0, andch>0c_{h}>0such thatB​(z∗,r)¯⊂E∘\overline{B(z_{\ast},r)}\subset E^{\circ}andh≥chh\geq c_{h}onB​(z∗,r)¯\overline{B(z_{\ast},r)}. LetΓ\Gammabe the Harrison–Reiman Skorokhod map. On[0,2][0,2],Γ\Gammais Lipschitz with constantKΓK_{\Gamma}. Define

γ​(t)=t​z∗,0≤t≤1,γ​(t)=z∗,1≤t≤2.\gamma(t)=tz_{\ast},\quad 0\leq t\leq 1,\qquad\gamma(t)=z_{\ast},\quad 1\leq t\leq 2.Sinceγ\gammastays inEE,Γ​(γ)=γ\Gamma(\gamma)=\gamma. For the SRBM started from zero, the free input isX​(t)=μ​t+W​(t)X(t)=\mu t+W(t). Putb​(t)=γ​(t)−μ​tb(t)=\gamma(t)-\mu t. Since Brownian motion has full support inC0​([0,2],ℝ3)C_{0}([0,2],\mathbb{R}^{3}), the eventA={sup0≤t≤2|W​(t)−b​(t)|<r/KΓ}A=\left\{\sup_{0\leq t\leq 2}|W(t)-b(t)|<r/K_{\Gamma}\right\}has positive probability. OnAA, the free inputXXis withinr/KΓr/K_{\Gamma}ofγ\gamma, and therefore the reflected pathZ0=Γ​(X)Z^{0}=\Gamma(X)is withinrrofγ\gamma. Sinceγ​(t)=z∗\gamma(t)=z_{\ast}for1≤t≤21\leq t\leq 2, we haveZ0​(t)∈B​(z∗,r)Z^{0}(t)\in B(z_{\ast},r)throughout[1,2][1,2]onAA. Henceg​(0)≥ℙ​(A)​ch​∫12e−λ​t​𝑑t>0.g(0)\geq\mathbb{P}(A)c_{h}\int_{1}^{2}e^{-\lambda t}\,dt>0.

On the other hand, ifggwereC2C^{2}up to the corner and satisfied the exact face conditions, those conditions would force∇g​(0)=0\nabla g(0)=0andD2​g​(0)=0D^{2}g(0)=0. The resolvent equation at the corner would then give0=h​(0)=(λ−L)​g​(0)=λ​g​(0)0=h(0)=(\lambda-L)g(0)=\lambda g(0), contradictingg​(0)>0g(0)>0.

Considerd=3d=3,Σ=I3\Sigma=I_{3}, and

R=(10−12−12100−121),R−1=17​(824482248)≥0.R=\begin{pmatrix}1&0&-\frac{1}{2}\\ -\frac{1}{2}&1&0\\ 0&-\frac{1}{2}&1\end{pmatrix},\qquad R^{-1}=\frac{1}{7}\begin{pmatrix}8&2&4\\ 4&8&2\\ 2&4&8\end{pmatrix}\geq 0.ThusRRis a nonsingularMM-matrix. Letμ=−R​𝟏\mu=-R\mathbf{1}; thenR−1​μ=−𝟏<0R^{-1}\mu=-\mathbf{1}<0, so the data are stable in the sense of (2.2).

Chooseh∈Cc∞​(E∘)h\in C_{c}^{\infty}(E^{\circ})withh≥0h\geq 0andh≢0h\not\equiv 0, and setg=Rλ​hg=R_{\lambda}hfor someλ>0\lambda>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 whichh>0h>0and then remaining there for a nonzero time interval. Henceg​(0)=𝔼0​∫0∞e−λ​t​h​(Zt)​𝑑t>0g(0)=\mathbb{E}_{0}\int_{0}^{\infty}e^{-\lambda t}h(Z_{t})\,dt>0. Alsoh​(0)=0h(0)=0, becausehhis supported in the interior.

Assume, for contradiction, thatg∈C2​(E)g\in C^{2}(E)in the closed-domain sense. On each open faceFi∘F_{i}^{\circ}, the one-sided derivative identity ofSection3.3applies in the feasible directionRiR_{i}. Since𝖫x​Ri=0\mathsf{L}_{x}R_{i}=0forx∈Fi∘x\in F_{i}^{\circ}, it givesDi​g=0D_{i}g=0onFi∘F_{i}^{\circ}. By continuity of the first derivatives, these identities extend to the origin asRT​∇g​(0)=0R^{T}\nabla g(0)=0. SinceRRis invertible,∇g​(0)=0\nabla g(0)=0.

LetH=D2​g​(0)H=D^{2}g(0). Forj≠ij\neq i, differentiatingDi​g=Ri⋅∇g=0D_{i}g=R_{i}\cdot\nabla g=0in the tangential directioneje_{j}alongFi∘F_{i}^{\circ}and then letting the tangential point tend to the origin givesRiT​H​ej=0R_{i}^{T}He_{j}=0. Equivalently,offdiag⁡(RT​H)=0\operatorname{offdiag}(R^{T}H)=0. ThusRT​H=diag⁡(d1,d2,d3)R^{T}H=\operatorname{diag}(d_{1},d_{2},d_{3})for some real numbersd1,d2,d3d_{1},d_{2},d_{3}, and therefore

H=R−T​diag⁡(d1,d2,d3)=17​(8​d14​d22​d32​d18​d24​d34​d12​d28​d3).H=R^{-T}\operatorname{diag}(d_{1},d_{2},d_{3})=\frac{1}{7}\begin{pmatrix}8d_{1}&4d_{2}&2d_{3}\\ 2d_{1}&8d_{2}&4d_{3}\\ 4d_{1}&2d_{2}&8d_{3}\end{pmatrix}.SinceHHis symmetric, comparison of the(1,2)(1,2),(2,3)(2,3), and(1,3)(1,3)entries givesd1=2​d2d_{1}=2d_{2},d2=2​d3d_{2}=2d_{3}, andd3=2​d1d_{3}=2d_{1}. Henced1=d2=d3=0d_{1}=d_{2}=d_{3}=0, soH=0H=0.

The interior resolvent equation gives(λ−L)​g=h(\lambda-L)g=honE∘E^{\circ}. Ifg∈C2​(E)g\in C^{2}(E), the left-hand side extends continuously to the origin. Sinceh​(0)=0h(0)=0,∇g​(0)=0\nabla g(0)=0,D2​g​(0)=0D^{2}g(0)=0, andΣ=I3\Sigma=I_{3}, this gives0=h​(0)=(λ−L)​g​(0)=λ​g​(0)0=h(0)=(\lambda-L)g(0)=\lambda g(0), contradictingg​(0)>0g(0)>0. Consequently, for this stable Harrison–Reiman SRBM and this smooth compactly supported interior source,Rλ​h∉C2​(E)R_{\lambda}h\notin C^{2}(E).

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