Preliminary work
An AI-Assisted Solution to the Signed BAR Conjecture: Uniqueness in the Harrison–Reiman Class and a Completely-S Class Obstruction Yiping Lu1 and Youheng Zhu1
arXiv:2607.03639v1 [math.PR] 3 Jul 2026
1 Department of Industrial Engineering and Management Sciences, McCormick School of Engineering, Northwestern University.
Abstract. For a multidimensional reflected diffusion, determining whether the associated basic adjoint relationship (BAR) uniquely characterizes the stationary distribution is a basic uniqueness problem in the BAR approach. The problem has remained unresolved for more than 35 years since the introduction of the BAR approach. In this paper, we resolve the finite-signed uniqueness problem for stable Harrison–Reiman data with a nonsingular M -matrix reflection matrix. The proof uses pathwise differentiability of the reflected diffusion implies feasible directional differentiability of the probabilistic resolvent to show that, at boundary points, its one-sided initial-state derivative factors through the tangent projection and vanishes along active reflection directions. An interior one-sided convolution then yields smooth test functions whose oblique derivatives are uniformly bounded and converge pointwise to zero on each closed face. The interior signed measure is consequently invariant for the reflected semigroup. A Jordan-decomposition argument identifies it as a scalar multiple of the unique invariant probability, and an induction over boundary strata, using invertibility of the principal reflection blocks, identifies the boundary measures. The proof was discovered with the assistance of ChatGPT 5.5 Pro and subsequently verified by the authors. We also show that the nonsingular M -matrix assumption is structural. In the larger completely-S class, a nonsingular reflection matrix with a singular proper principal block admits boundary gauges supported on lower-dimensional strata. Under standard exponential ergodicity and a mild one-step regulator bound, these gauges produce nonzero zero-mass signed BAR tuples; indeed the zero-mass interior BAR coordinates contain an infinite-dimensional subspace. A four-parameter three-dimensional family, including an explicit rational example, verifies the obstruction. Thus the finite signed version of the Dai–Dieker question has a positive answer in the Harrison–Reiman M -matrix class and a negative answer in a natural completely-S extension. MSC2020 subject classifications: Primary 60J60; 60J55; secondary 35J25; 46A20; 60K25 Keywords: semimartingale reflected Brownian motion; basic adjoint relationship; signed measure; Skorokhod map; pathwise derivative; resolvent; completely S matrix
1. Introduction Semimartingale reflected Brownian motions (SRBMs) in the nonnegative orthant are diffusion approximations for stochastic networks in heavy traffic. In the interior of the orthant the process behaves as a Brownian motion with drift and covariance matrix; when it reaches a face, it is pushed back into the state space in an oblique direction prescribed by the corresponding column of a reflection matrix. The Harrison–Reiman construction [23, 24] is the canonical orthant model behind open queueing networks in heavy traffic [21, 22, 25, 32, 35]; it is the main positive setting of this paper. 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 [24–26], underlies numerical methods for orthant SRBMs [6, 7], has been used in steady-state heavy-traffic approximation through the BAR approach [3, 4], and is one of the standard weak formulations used to characterize stationary distributions of reflected diffusions [5, 27]. If π is an interior measure and νi is a boundary measure on the face Fi = {xi = 0}, the BAR has the form Z Lf dπ +
(1.1) E
d Z X i=1
Di f dνi = 0,
f ∈ Cb2 (E),
Fi
where L is the interior diffusion generator and Di is the directional derivative in the ith reflection direction. The stationary distribution π0 , together with its stationary boundary occupation measures νi0 , 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? 1
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The issue has persisted for more than three decades, remaining an open problem since the inception of the BAR approach. The open problem was first stated as a conjecture in [6] for SRBMs in a two dimensional rectangle and in [7] for SRBMs in a d-dimensional orthant. Dai and Dieker [5] describe the fundamental open problem concerning the Basic Adjoint Relationship (BAR) for multidimensional diffusion processes. Specifically, for both Semimartingale Reflecting Brownian Motions (SRBMs) and piecewise Ornstein–Uhlenbeck (OU) processes. Dai and Dieker [5, Proposition 1 and Open Problem 1] formulated the BAR characterization with bounded C 2 tests, proved the corresponding characterization in the positive-measure setting, and asked for the signed analogue. The compactly supported C 2 formulation leads to the same finite-signed uniqueness problem. The bounded-test identity immediately implies the compactly supported one. Conversely, let f ∈ Cb2 (E) and choose χn ∈ Cc∞ (Rd ) with 0 ≤ χn ≤ 1, χn = 1 on {|x| ≤ n}, and ∥∇χn ∥∞ + ∥D2 χn ∥∞ → 0. Applying the compactly supported identity to χn f and expanding L(χn f ) and Di (χn f ) gives the bounded-test identity after passage to the limit, because χn → 1 pointwise and all error terms are uniformly bounded by constants times ∥∇χn ∥∞ + ∥D2 χn ∥∞ against finite signed measures. Throughout the paper we therefore use the bounded-test class Cb2 (E), which is the formulation needed to insert the one-sided smoothings of the probabilistic resolvent without an artificial spatial cutoff. In the signed problem one allows π and the νi 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 [25, 26], positive-measure BAR characterizations identify stationary probabilities, and in many formulations also the associated boundary occupation measures, once the reflected diffusion and its stationary regime are already well posed [6, 7, 27]. These results do not, by themselves, exclude sign-changing finite measures whose interior and boundary terms cancel in the BAR. Our positive theorem addresses exactly that finite-signed nullspace question in the stable Harrison–Reiman nonsingular-M -matrix class, and it identifies the full boundary tuple as well as the interior coordinate. Much of the stationary SRBM literature concerns explicit formulas, transforms, asymptotics, or numerical computation rather than signed uniqueness. Product-form and skew-symmetry results originate with Harrison and Williams [26]; 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 [8, 9, 11, 18, 19]. The present proof uses none of these explicit analytic representations. Its role is instead structural: it proves that, in the stated M -matrix class, the finite signed BAR has no hidden zero-mass directions. Skorokhod-map Differentiability. Lipschitz, convex-duality and differentiability properties of oblique reflection maps were developed in deterministic form by Dupuis–Ishii, Dupuis–Ramanan, Mandelbaum–Ramanan, and Lipshutz–Ramanan [13–15, 28, 31]. We use the reflected-diffusion version of this theory, namely the pathwise differentiability and sensitivity results of Lipshutz and Ramanan [29, 30], only after verifying their assumptions for the normalized Harrison–Reiman data. The negative result is complementary to the existence and stability literature for completely-S data: Taylor–Williams and Dai–Williams give the relevant SRBM existence frameworks [10, 34], while Lyapunov and recurrence criteria for SRBMs are developed for example in [2, 16, 33]. Section 6 shows that existence and recurrence alone do not replace invertibility of every active principal block. Technical Overview Our positive result answers the signed Dai–Dieker problem for stable Harrison–Reiman data with R ∞ a nonsingular M matrix reflection matrix. The proof is organized around a resolvent invariant identity. Let Rλ h = 0 e−λt Pt h dt be the probabilistic resolvent of the reflected semigroup. Our core contribution is proving the fact that every finite signed BAR tuple satisfies Z (RI) (λRλ h − h) dπ̄ = 0, h ∈ C0 (E), λ > 0. E
This identity R ∞ says exactly that the interior signed measure is invariant under the reflected semigroup. Indeed, using Rλ h = 0 e−λt Pt h dt, (RI) says that the Laplace transform of t 7→ π̄(Pt h) − π̄(h) vanishes for every h ∈ C0 (E). Strong
Signed BAR uniqueness conjecture
3
continuity of the Feller semigroup upgrades this to π̄Pt = π̄ for all t ≥ 0. If π̄ = π̄ + − π̄ − is the Jordan decomposition, positivity of the Markov kernel gives |π̄Pt | ≤ |π̄|Pt ; equality of total masses then makes |π̄| invariant, and hence both Jordan components are invariant positive finite measures. After normalization, every nonzero component is an invariant probability, so uniqueness of the invariant probability gives π̄ = cπ0 . Subtracting c times the stationary BAR leaves a pure boundary identity, and the nonsingular principal reflection blocks identify the boundary measures by an induction over strata. The only nontrivial point in this chain is the derivation of (RI). Formally, if g = Rλ h were an admissible Cb2 test satisfying Di g = 0 on Fi , then (RI) would follow by inserting g into the BAR and using (λ − L)g = h. This formal argument is misleading because at corners the resolvent need not be a classical C 2 function on the closed orthant; Section A gives a stable Harrison–Reiman example where such C 2 regularity is impossible. The proof therefore works in the topology actually seen by finite signed measures: uniform convergence of the interior equation and vanishing of the boundary terms after integration against arbitrary finite signed boundary measures. The approximation used in the proof is intentionally simple. We do not insert g = Rλ h itself into the BAR. Instead we replace it by the one-sided smoothing Z gε (x) = ρ(w)g(x + εw) dw. The mollifier is supported strictly inside the positive orthant, so the value of gε (x) only uses values of g at interior points x + εw. This smoothing supplies the required bounded C 2 regularity for each fixed ε. The only delicate point is to show that these legitimate Cb2 tests have asymptotically zero boundary contribution. The projected boundary derivative of the resolvent gives Di gε (x) −→ 0,
x ∈ Fi ,
with a uniform bound sufficient for dominated convergence against an arbitrary finite signed boundary measure. Thus the functions gε approximate the resolvent in exactly the topology seen by the BAR: the interior equation converges to (λ − L)Rλ h = h, while all boundary terms vanish. The paper also explains why the M -matrix hypothesis is not merely a proof artifact. In the completely-S existence class, a singular proper principal block may cancel all active normal components of a boundary gauge supported on a lower-dimensional stratum. The remaining tangential derivative produces a centered interior source. Under a quantitative recurrence assumption, the zero potential of this source gives a nonzero signed BAR tuple with zero interior mass. Thus signed uniqueness fails in a natural completely-S extension. The Role of AI-assistance The proof given here was not produced by an AI system in a single pass; it is the outcome of an extended, human-directed collaboration (for 3 weeks) in which large language models served as an exploratory and organizational aid, while every mathematical decision and all verification rested with the authors. By shifting the focus from merely verifying the conjecture to characterizing the specific domain where it holds, this study not only reveals the essential divergence between HarrisonReiman Class and Completely-S Class but also demonstrates the vital role of human-AI collaboration in advancing complex mathematical research. Following the program in Dai and Dieker’s open-problem note [5], we first attacked uniqueness in the completely-S class, where the crux is the low regularity of the solution at the boundary. Over many rounds of interaction the model carried out the boundary-layer expansion and tested whether the boundary contribution is sign-definite and whether it can be absorbed by the interior solution. When this cancellation repeatedly failed for d > 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 signedmeasure uniqueness in the Harrison–Reiman class. Our first attempt proceeded through a Kato-type inequality, where the obstruction is the boundary term produced by the integration by parts; to organize the inductive cancellation of this term across the boundary strata, we prompted the model to adopt a homological-algebra–style bookkeeping. This yielded a long (roughly 150-page, see https://drive.google.com/file/d/1QEMTMYR9d0l3ToJtdVHEeYT9TF5Cudui/view?usp=sharing) proof outline that passed an initial screening by an ensemble of ten independent model/agent reviewers. Such consensus is not a proof, and we treated it only as a filter: the argument was subsequently checked by the authors, conclusion by conclusion, with each regularity hypothesis verified for mutual consistency. In the course of this verification the model surfaced the pathwise-differentiability results of Lipshutz and Ramanan [28], which considerably simplified the argument and, after further iteration, produced the proof in its present form. The authors have verified every step and are solely responsible for the correctness of the results. Additionally, we attempted to generate a positive proof via one-shot prompting, leveraging the premise that the conjecture holds true within the Harrison-Reiman class. However, both ChatGPT
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5.5 Pro-extended and Claude Opus 4.8 max failed this task. The chat logs are available at: https://chatgpt.com/share/ 6a44a502-d034-83ea-9608-eecb9ecc898d and https://claude.ai/share/25a16238-360a-4649-935f-b23b4ec500ff(Attempts https://chatgpt.com/share/6a44b084-91dc-83ea-8fc2-49b06770025d to solve the problem, even when prompted with the literature [28, 29], proved unsuccessful.). Surprisingly, contemporary AI approaches even fail to leverage the specific properties of the Harrison–Reiman class, which are essential for the proof of positivity established via the counterexample in the general Completely-S class presented in this paper. We hypothesize that the AI derived meaningful insights from the first 150 pages version of computations, even though these results were not explicitly incorporated into the final proof. This outcome highlights the potential of AI assistance in tackling open mathematical problems, while simultaneously underscoring the indispensable role of human verification and guidance throughout the process. Organization of the Paper We organize the paper as follows: Section 2 states the SRBM and BAR setting, states the main theorem, and reduces the proof to the resolvent identity (RI). Section 3 establishes the two technical properties of g = Rλ h needed later for the approximation: the interior resolvent equation and the projected boundary derivative that will make Di gε vanish on Fi . Section 4 carries out the one-sided smoothing construction, inserts gε ∈ Cb2 (E) directly into the BAR, and proves (RI). Section 5 proves the implication deferred in Section 2: the identity (RI) implies the signed BAR uniqueness conjecture, thus finishing the proof of the main theorem. Section 6 explains why the nonsingular M -matrix condition is structural by giving the completely-S obstruction and an explicit three-dimensional family. Section 7 repackages the positive and negative arguments through a common BAR homotopy lemma and separates the remaining issue into local boundary algebra. 2. Setting, main theorem, and reduction to the resolvent identity This section fixes the data, states the signed-measure theorem, and isolates the central reduction. The conversion of the present standing assumptions into the hypotheses of the reflected-diffusion results is carried out inline, at the point of use, inside the proof of Theorem 3.3: there each source hypothesis is recalled in the present orthant specialization and verified. 2.1. Notation and standing conventions Let J = {1, . . . , d}, E = Rd+ , and E ◦ = (0, ∞)d . For i ∈ J write Fi = {x ∈ E : xi = 0}. For nonempty A ⊂ J , define the relative boundary stratum SA = {x ∈ E : xi = 0 (i ∈ A), xj > 0 (j ∈ / A)}. The sets SA form a disjoint Borel decomposition of ∂E. For a locally compact space B, C0 (B) denotes the continuous real-valued functions vanishing at infinity, and M(B) denotes the finite signed Radon measures on B. For η ∈ M(B), |η| is its total variation measure and ∥η∥TV = |η|(B). We write supp η for the support of a measure and supp f for the support of a function. The symbol 1B denotes the indicator of a set B. We use the closed-domain C 2 convention. Thus C 2 (E) consists of functions f : E → R such that f ∈ C 2 (E ◦ ) and all partial derivatives ∂ α f , |α| ≤ 2, extend continuously from E ◦ to E. The class Cc2 (E) consists of the functions in C 2 (E) with compact support as a subset of E. The class Cb2 (E) consists of the functions in C 2 (E) for which f , ∇f and D2 f are bounded. Since E is the orthant, this closed-domain convention is equivalent to saying that every f ∈ C 2 (E) is the restriction to E of some F ∈ C 2 (U ) on an open neighborhood U ⊃ E. For open subsets of REuclidean space, Cc∞ has ∞ its usual meaning. For the reflected semigroup we write Pt h(x) = E[h(Ztx )] and Rλ h(x) = 0 e−λt Pt h(x) dt, λ > 0, whenever the integral is finite. We call the semigroup Pt Feller if (Pt )t≥0 satisfies Pt C0 (E) ⊂ C0 (E), and is strongly continuous, i.e. ∥Pt h − h∥∞ → 0 as t ↓ 0 for all h ∈ C0 (E). 2.2. SRBM, BAR, and finite signed BAR tuples A semimartingale reflected Brownian motion in E is specified by a drift vector µ ∈ Rd , a symmetric positive definite covariance matrix Σ, and a reflection matrix R = (R1 , . . . , Rd ) whose ith column is the direction of reflection on Fi . Put Q = Σ/2 and Lf = µ · ∇f + Q : D2 f,
Di f = Ri · ∇f.
Throughout the positive part of the paper we work under the following stable nonsingular M -matrix data. The covariance matrix Σ is symmetric positive definite. The reflection matrix R satisfies (2.1)
Rii > 0,
Rij ≤ 0 (i ̸= j),
R−1 ≥ 0.
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Signed BAR uniqueness conjecture
The drift satisfies R−1 µ < 0
(2.2)
componentwise. The phrase “stable” in this paper means exactly (2.2). The linear-algebra consequences of (2.1) are proved in Lemma 3.2; the stochastic consequences used later are stated in Theorem 3.3 and justified in its proof, where every source hypothesis is recalled and checked. Qd A finite signed BAR tuple is a tuple (π̄, ν̄1 , . . . , ν̄d ) ∈ M(E) × i=1 M(Fi ) of finite signed Radon measures satisfying Z Lf dπ̄ +
(2.3) E
d Z X i=1
f ∈ Cb2 (E).
Di f dν̄i = 0,
Fi
The stationary regulator defines finite boundary occupation measures νi0 , and the stationary BAR is Z Lf dπ0 +
(2.4) E
d Z X i=1
Di f dνi0 = 0,
f ∈ Cb2 (E).
Fi
Under (2.1)–(2.2), the normalized reflection matrix is of Harrison–Reiman form, and the associated deterministic Skorokhod problem drains to the origin. Hence [16, Theorem 2.6] and [30, Theorem 3.5] gives provides the existences and the uniqueness of stationary distribution π0 to the SRBM. Obviously, the stationary distribution and the finite stationary boundary measure characterized by the following Proposition 2.1 together provide a solution to the BAR equation (2.4). Proposition 2.1 (Finite stationary boundary measures and stationary BAR). Start the SRBM with Z0 ∼ π0 and write it in the original normalization as Zt = Z0 + µt + Σ1/2 Wt + RYt ,
(2.5)
where each Yi is continuous, nondecreasing, starts from zero, and increases only on Fi . Define, for Borel B ⊂ Fi , νi0 (B) = Eπ0
(2.6)
Z 1 1B (Zs ) dYi (s). 0
Then each νi0 is a finite measure supported on Fi , and (2.4) holds. Proof. Let a = R−T 1. Since R−1 ≥ 0 and no column of the invertible matrix R−1 is zero, a > 0; moreover RT a = 1. For α > 0, set Φα (x) = −
d X
ak e−αxk .
k=1
The function and its first two derivatives are bounded. If x ∈ Fi , then, using Rki ≤ 0 for k ̸= i, xi = 0, and e−αxk ≤ 1, Di Φα (x) = α
d X
Rki ak e−αxk ≥ α
k=1
d X
Rki ak = α.
k=1
Itô’s formula on [0, 1] gives, pathwise, Z 1 Φα (Z1 ) − Φα (Z0 ) =
LΦα (Zs ) ds + M1 + 0
d Z 1 X i=1
Di Φα (Zs ) dYi (s),
0
where M is a square-integrable martingale because ∇Φα 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
d X i=1
Z 1 Eπ0 Yi (1) ≤ −Eπ0
LΦα (Zs ) ds ≤ ∥LΦα ∥∞ . 0
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Thus (2.6) is finite. Its support is contained in Fi because Yi increases only there. Finally, apply Itô’s formula to f ∈ Cb2 (E). The boundedness of f , ∇f and D2 f makes the Brownian and drift terms integrable on [0, 1], and the boundary integrals are integrable by the preceding estimate. Stationarity gives Z 0=
Lf dπ0 + E
d Z X i=1
Di f dνi0 .
Fi
Although Proposition 2.1 provides (π0 , ν10 , . . . , νd0 ) as a solution to the BAR equation, it remains an open question whether the BAR uniquely characterizes the stationary distribution of the diffusion process. 2.3. Signed BAR uniqueness in the Harrison-Reiman Class In the Harrison-Reiman Class, i.e. under the standing assumptions (2.1)–(2.2), we show that the associated BAR uniquely characterizes the stationary distribution of the diffusion process. Theorem 2.2 (Signed BAR uniqueness). Under the standing assumptions (2.1)–(2.2), let π0 and (νi0 )di=1 be the stationary distribution of the process and the corresponding boundary measure constructed in Proposition 2.1. Every finite signed BAR tuple is a scalar multiple of the stationary BAR tuple. More precisely, if (2.3) holds, then there exists c ∈ R such that (2.7)
ν̄i = cνi0 ,
π̄ = cπ0 ,
i = 1, . . . , d.
Consequently the vector space of finite signed BAR tuples is one-dimensional. To prove uniqueness of finite signed BAR tuples, we first show that every BAR tuple satisfies a resolvent identity (RI); we call this identity resolvent insertion. The resolvent insertion identity implies invariance of the interior signed measure under the reflected semigroup, and hence π̄ = cπ0 . After subtracting the interior stationary BAR, the remaining identity is purely on the boundary, and pure boundary injectivity gives ν̄i = cνi0 for all i = 1, . . . , d. Proposition 2.3 (Resolvent identity criterion). Assume that for every finite signed BAR tuple, every h ∈ C0 (E), and every λ > 0, Z (RI) (λRλ h − h) dπ̄ = 0. E
Then the conclusion of Theorem 2.2 holds. The proof of Proposition 2.3 is given in Section 5. Why the resolvent insertion (RI) should hold. The reason for targeting (RI) is transparent from the classical Neumann calculation. Let g = Rλ h. If g were an admissible Cb2 test and if it satisfied Di g = 0 on Fi for i = 1, . . . , d, then inserting g into the BAR would give Z Z XZ 0= Lg dπ̄ + Di g dν̄i = Lg dπ̄. E
i
Fi
E
R
The resolvent equation (λ − L)g = h would therefore imply E (λRλ h − h) dπ̄ = 0. This is only an informal guide. The closed-domain C 2 regularity required for this insertion may fail even in the stable Harrison–Reiman class. Section A gives an explicit stable nonsingular M -matrix example and a smooth compactly supported h for which Rλ h ∈ / C 2 (E). The proof below therefore does not try to show that the resolvent belongs to a classical oblique-Neumann core. 2.4. Making the resolvent insertion rigorous The replacement for the formal insertion is a measure-level Neumann approximation. For smooth compactly supported h we construct tests gε ∈ Cb2 (E) such that, as ε ↓ 0, gε → Rλ h,
(λ − L)gε → h,
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Signed BAR uniqueness conjecture
against every finite signed interior measure, while Z Di gε dν̄i → 0,
i = 1, . . . , d,
Fi
for every finite signed boundary measure. This is exactly what is needed to pass to the limit in the BAR. The convergence is not a pointwise assertion that Rλ h admits a classical oblique derivative Di Rλ h on Fi ; it is an assertion that the boundary pairings seen by the BAR vanish. Proposition 4.4 gives the precise statement, and a density argument then extends (RI) from smooth compactly supported h to all h ∈ C0 (E). The next two sections supply the projected derivative input and the one-sided smoothing construction. Figure 1 summarizes where this approximation sits in the proof: the analytic work proves the target resolvent identity, while the remaining steps are the soft semigroup and boundary-identification arguments. π̄(h) = λπ̄(Rλ h) LEADS TO SIGNED BAR UNIQUENESS
Laplace uniqueness converts resolvents to invariance
Target identity: the hard step For every h ∈ Cc∞ (Rd ) and λ > 0,
Semigroup invariance Thus π̄(Pt h) = π̄(h) for all t ≥ 0. Density of Cc∞ |E in C0 (E), plus contraction of Pt , extends this to all ϕ ∈ C0 (E). Hence π̄Pt = π̄.
Define the defect Ah (t) := π̄(Pt h) − π̄(h). Using R Rλ h = 0∞ e−λt Pt h dt, Fubini turns the target identity into Z ∞ −λt e Ah (t) dt = 0 (λ > 0).
π̄(h) = λ π̄(Rλ h). This is the resolvent form of stationarity.
0
Boundary measures then follow Subtract c times the stationary BAR. −T On each stratum SA , RAA prescribes the active oblique jets (Di f )i∈A . Induction over |A| gives ν̄i = cνi0 .
π̄ = cπ0
Interior measure is then forced For a finite signed invariant measure, positivity gives |µPt | ≤ |µ|Pt . Total mass equality makes the Jordan parts invariant. Uniqueness of π0 gives π̄ = cπ0 .
Signed BAR uniqueness (π̄, ν̄1 , . . . , ν̄d ) = c(π0 , ν10 , . . . , νd0 ). P ROVING π̄(h) = λπ̄(Rλ h): HOW THE MISSING RESOLVENT BOUNDARY REGULARITY IS BYPASSED
Why direct insertion fails The natural test is g = Rλ h, because (λ − L)g = h in the interior. But the BAR accepts bounded C 2 tests and boundary terms Di f . At corners, g need not have a classical ambient gradient, so Di g = 0 is unavailable.
Projected derivative replaces a boundary gradient For feasible inward directions, + ∂w g(x) = Λx (Lx w).
If x ∈ Fi , then Lx Ri = 0, hence the ambient linear extension satisfies ℓx (Ri ) = 0. This is the usable oblique information.
One-sided smoothing turns it into BAR tests Smooth only from inside: Z gε (x) = ρ(w)g(x + εw) dw. The support supp ρ ⋐ (0, ∞)d keeps every sampled direction feasible. Integration by parts and domination give Di gε → 0 on Fi .
Measure–Neumann approximation Use gε ∈ Cb2 (E) directly. Apply the signed BAR and send ε ↓ 0. Boundary integrals vanish for every finite signed ν̄i ; interior terms converge to π̄(h) = λπ̄(Rλ h).
F IG 1. Proof architecture for the uniqueness of the Harrison-Reiman class. The lower half is the analytic insertion mechanism: Proposition 3.1 supplies the projected derivative used by the one-sided smoothing, and Proposition 4.4 turns the smoothed functions into admissible BAR tests. The upper half is the soft reduction: the resulting resolvent identity gives semigroup invariance, then signed uniqueness of the interior measure and finally the boundary measures.
The diagram also shows why the proof first studies the nonsmooth resolvent before carrying out the smoothing. For fixed h and λ, let g = Rλ h. The smoothed BAR tests used later are Z gε (x) = ρ(w)g(x + εw) dw, supp ρ ⊂ (1, 2)d .
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They must approximate g in the interior equation while also satisfying an asymptotic oblique-Neumann condition on each face: Di gε → 0 in pairings with arbitrary finite signed measures on Fi . This is why Section 3 proves a boundary statement for g itself before any smoothing is introduced. Although g need not be C 2 on the closed orthant, its feasible one-sided derivatives exist at boundary points and factor through the active tangent projection; the resulting linear extension ℓx satisfies ℓx (Ri ) = 0 on active faces, acting as an analog to the classical gradient. The one-sided convolution gε in Section 4 is then precisely designed to inherit this first-order oblique flatness in the weaker, measure-level form needed by the BAR. 3. Resolvent regularity and projected boundary derivatives The goal of this section is to prove Proposition 3.1, the input that makes the measure–Neumann approximation in Section 4 possible. Section 4 will construct gε ∈ Cb2 (E) from g = Rλ h and will need three properties: gε → g, (λ − L)gε → h, and Di gε → 0 on Fi after integration against arbitrary finite signed boundary measures. The first two properties come from interior smoothing and the interior resolvent equation. The third property comes from the boundary information proved here: at a boundary point, the feasible directional derivative of g factors through the active tangent projection. Combining this factorization with the identity Lx Ri = 0 gives the usable oblique information ℓx (Ri ) = 0 on Fi , which is exactly what later forces Di gε → 0 in boundary-measure pairings. The proof has two ingredients. The algebraic ingredient is the nonsingularity of every active principal reflection block, which gives the explicit projection Lx . The stochastic ingredient is external: the Lipshutz–Ramanan initial-condition derivative theorem for the normalized Harrison–Reiman reflected diffusion, together with well posedness, strong-continuity property, and the synchronous Lipschitz estimate. The source-to-assumption conversion is carried out in the proof of Theorem 3.3, where each source hypothesis is recalled in the present orthant specialization and verified with a self-contained argument; no unlisted regularity or boundary conclusion is used. For x ∈ E, define I(x) = {i ∈ J : xi = 0}, and put (3.1)
Gx = {w ∈ Rd : wi ≥ 0 for i ∈ I(x)},
Hx = {v ∈ Rd : vi = 0 for i ∈ I(x)}.
Proposition 3.1 (Resolvent regularity and projected derivatives). Let λ > 0 and let h ∈ Cc∞ (Rd ) be regarded as R ∞ boundary −λt x a function on E. Define g(x) = Rλ h(x) := E 0 e h(Zt ) dt , then we have: (i) g is bounded and globally Lipschitz on E. (ii) g is a classical solution of the resolvent equation in E ◦ ; more precisely, g ∈ C ∞ (E ◦ ) and (λ − L)g = h in E ◦ . + (iii) At each x ∈ E, feasible one-sided directional derivatives ∂w g(x) exist for w ∈ Gx . −1 (iv) If A = I(x), then the principal-block projection Lx v = v − RA RAA vA maps Rd onto Hx , and there is a linear functional Λx : Hx → R such that + ∂w g(x) = Λx (Lx w),
w ∈ Gx .
(v) With ℓx (v) = Λx (Lx v), we have ℓx (Ri ) = 0, for i ∈ I(x). 3.1. Matrix normalization and active-set projections Normalize the reflection directions by ∆ = diag(R11 , . . . , Rdd ),
b = R∆−1 , R
bi = Ri /Rii . di = R
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 (3.2)
b = I − PT, R
P ≥ 0,
ρ(P ) < 1.
−1 Every principal submatrix RAA is a nonsingular M -matrix and RAA ≥ 0. In particular, for every nonempty A ⊂ J , the active directions {di : i ∈ A} are linearly independent. For A ⊂ J , define
(3.3)
−1 LA v = v − RA RAA vA ,
with L∅ equal to the identity. If A = I(x), then LA = Lx is the (unique) linear map from Rd to Hx such that Lx v − v ∈ span{Ri : i ∈ A}. Moreover, (3.4)
LA Ri = 0,
i ∈ A,
9
Signed BAR uniqueness conjecture
and −1 CL := max I − RA RAA πA < ∞,
(3.5)
A⊂J
where πA v = vA and the expression for A = ∅ is the identity. b is one and its off-diagonal entries are nonpositive, so P T := I − R b is nonnegative. Also Proof. The diagonal of R b−1 = ∆R−1 ≥ 0. R By Perron–Frobenius, in the standard nonnegative-matrix form summarized for example in [1, Chapter 2], P T has b = 0, contradicting invertibility. If ρ(P ) > 1, then a nonzero vector v ≥ 0 with P T v = ρ(P )v. If ρ(P ) = 1, then Rv −1 b = (1 − ρ(P ))v ≤ 0; multiplying by R b ≥ 0 gives v ≤ 0, again a contradiction. Hence ρ(P ) < 1. Rv For a principal index set A, the principal block (P T )AA is nonnegative and ρ((P T )AA ) ≤ ρ(P T ) < 1. One direct verification of the inequality is that ((P T )AA )n is entrywise bounded by the AA block of (P T )n , after which Gelfand’s formula applies. Therefore (IA − (P T )AA )−1 =
∞ X
((P T )AA )n ≥ 0.
n=0
Since RAA = (IA − (P T )AA )∆A , it follows that −1 T −1 RAA = ∆−1 ≥ 0. A (IA − (P )AA )
P Linear independence of the active normalized columns follows by restricting a relation i∈A ai di = 0 to rows in A. The maximum in (3.5) is finite because the active-set lattice is finite. For the projection claim, a vector of the form v − RA a −1 belongs to Hx exactly when vA − RAA a = 0. The preceding paragraph gives a = RAA vA . If v = Ri with i ∈ A, then vA = RAA ei , which proves (3.4). 3.2. Regularity of SRBM This subsection proves Theorem 3.3, the stochastic regularity statement used in Proposition 3.1. More specifically, for g = Rλ h, we will need the reflected semigroup on C0 (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 ∆ = diag(R11 , . . . , Rdd ),
b = R∆−1 , R
bi = Ri . di = R Rii
Then ⟨di , ei ⟩ = 1. Replacing Ri by the positive multiple di = Ri /Rii only rescales the ith regulator coordinate and leaves b apply to the original the reflected path unchanged. Therefore pathwise statements proved for the normalized matrix R BAR normalization R. Theorem 3.3 records the regularity consequences needed for the proof of Proposition 3.1. Its proof first places the present SRBM into the notation of [29, 30], then verifies the relevant hypotheses under (2.1)–(2.2), and finally applies the corresponding existence, Lipschitz, and derivative results of [29, 30]. Theorem 3.3 (Regularity of SRBM). Under the standing assumptions (2.1)–(2.2), the following hold. (i) For each x ∈ E and each prescribed Brownian motion there is a pathwise unique SRBM Z x , and Z x is strong Markov. (ii) The semigroup (Pt ) maps C0 (E) into itself and is strongly continuous there. (iii) There is a constant KΓ < ∞, depending only on the normalized reflection data, such that synchronous solutions satisfy, for all x, y ∈ E and t ≥ 0, (3.6)
sup |Zsx − Zsy | ≤ KΓ |x − y|
0≤s≤t
almost surely.
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Y. Lu and Y. Zhu
(iv) For every x ∈ E there is an adapted RCLL derivative process Jxt ∈ Lin(Hx , Rd ), t ≥ 0. For each fixed w ∈ Gx , on an event of probability one the derivative Ztx+εw − Ztx ε↓0 ε
∂w Ztx := lim exists for every t ≥ 0. Moreover, for every fixed t > 0, ∂w Ztx = Jxt [Lx w]
(3.7)
almost surely.
(v) For every x ∈ E and every fixed u ∈ Hx , |Jxt [u]| ≤ KΓ |u|
for dt ⊗ P-almost every (t, ω).
To prove Theorem 3.3, we use the following results for reflected diffusions in simple polyhedra [29, 30]. The general framework of [29, 30] is a more flexible version of the same reflected-diffusion equation: it allows a simple polyhedral domain, normalized face directions, and parameter-dependent coefficients. Our orthant SRBM is obtained from that framework by taking constant coefficients and normalized columns di = Ri /Rii ; the only difference from the BAR notation is the harmless positive rescaling of the regulator coordinates. Recall that [29, 30] use the following notation for the general SRBM framework, where we specialized the notation to the spatially homogeneous case. Let parameters α ∈ U , where the parameter family U is open, and let \ G= {x : ⟨x, ni ⟩ ≥ ci }, J = {1, . . . , d}, i∈J
be a minimally represented simple polyhedron with unit inward normals ni , faces Fi = {x ∈ G : ⟨x, ni ⟩ = ci }, and active set IG (x) = {i : x ∈ Fi }. The normalized reflection directions satisfy ⟨di (α), ni ⟩ = 1. In the spatially-homogeneouscoefficient specialization we care about, the family of reflected diffusions parameterized by α is written as X Ztα,x = x + b(α)t + σ(α)Wt + di (α)Yiα,x (t), i∈J
where each Yiα,x is continuous, nondecreasing, starts from zero, and increases only when Z α,x ∈ Fi . Put a(α) = σ(α)σ(α)T , N = (n1 , . . . , nd ), and D(α) = (d1 (α), . . . , dd (α)). For x ∈ G, define CG (x) = {w : ⟨w, ni ⟩ ≥ 0, i ∈ IG (x)} and HG (x) = {v : ⟨v, ni ⟩ = 0, i ∈ IG (x)}. In the orthant specialization, G = E, ni = ei , CG (x) = Gx , and HG (x) = Hx . Then, [29, 30] gives the following proposition. Proposition 3.4 (Reflected diffusions in simple polyhedra). In the setting just described, fix α ∈ U . Assume the following hypotheses. (A1) G is minimally represented and simple, U is open, and α 7→ di (α), b(α), and σ(α) are C 1 with bounded first derivatives and local Hölder regularity. (A2) a(α) is uniformly elliptic: v T a(α)v ≥ θ|v|2 for some θ > 0 and all v ∈ Rd . (A3) N T D(α) is a nonsingular M -matrix. (A4) The reflection matrix is constant in the parameter, or more generally ∂α D(α) is bounded. Then the following conclusions are available under the assumptions indicated. (C1) (Well posedness; uses (A1) and (A3), [30, Theorem 2.8].) For each x ∈ G and each prescribed Brownian motion, there is a pathwise unique reflected diffusion Z α,x , and it is strong Markov. (C2) (Lipschitz extended Skorokhod map; uses (A1) and (A3), [29, Proposition 2.6].) The extended Skorokhod problem associated with (G, di (α)) is well posed, and its extended Skorokhod map Γ̄α is Lipschitz on compact time intervals: sups≤t Γ̄α (f )(s) − Γ̄α (g)(s) ≤ KΓ sups≤t |f (s) − g(s)| for some KΓ < ∞, all continuous inputs f, g, and all t ≥ 0. (C3) (Boundary jitter; uses (A2) in the above setting, [29, Theorem 3.3].) Uniform ellipticity implies the boundary jitter property required for the pathwise derivative theorem. (C4) (Derivative projection; uses (A1) and (A3), [29, Lemma 3.11].) For each x ∈ G there is a unique linear projection d α d Lα x : R → HG (x) such that Lx v − v ∈ span{di (α) : i ∈ IG (x)} for every v ∈ R .
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Signed BAR uniqueness conjecture
(C5) (Pathwise differentiability; uses (A1)–(A4) and (C3)–(C4), [29, Theorem 3.13 and Corollary 3.15].) For this theP orem, note that (A3) implies Condition 2.10 of [29]: if A ⊂ J and i∈A ci di (α) = 0, then (N T D(α))AA cA = 0, and every principal submatrix of a nonsingular M -matrix is nonsingular, so cA = 0. Hence [29, Lemma 3.9] gives the exceptional set of this theorem W α = ∅. Therefore, for each x ∈ G\W α = G there is an adapted RCLL derivative process Jtα,x ∈ Lin(HG (x), Rd ). For every fixed w ∈ CG (x), we have almost surely, ∂w Ztα,x := limε↓0 ε−1 (Ztα,x+εw − Ztα,x ) exists for every t ≥ 0, is continuous at every t > 0 such that Ztα,x ∈ G◦ , and its right-continuous regularization satisfies lims↓t ∂w Zsα,x = Jtα,x [Lα x w] for all t ≥ 0. (C6) (Fixed-time interior statement; uses (A1)–(A4) and (C3)–(C4), [29, Lemma 4.13].) For every fixed t > 0, P(Ztα,x ∈ G◦ ) = 1. Proof of Theorem 3.3. We apply Proposition 3.4 to the constant parameter family G = E,
ni = e i ,
ci = 0,
di (α) = di =
Ri , Rii
b(α) = µ,
σ(α) = Σ1/2 ,
α ∈ U = (−1, 1).
b and N = I. Here constant parameter family means that the domain, reflection directions, drift, and Thus D(α) = R dispersion do not depend on α. Now we verify that the Harrison-Reiman Class satisfies all assumptions (A1)-(A4). T Verification of (A1). The orthant is the minimally represented simple cone E = i {x : ⟨x, ei ⟩ ≥ 0}. Simplicity follows because the coordinate normals are linearly independent on every active set. Minimality follows because, if the ith half-space is removed, then the point −ei satisfies all remaining half-space inequalities but does not belong to E. The parameter set U = (−1, 1) is open. The maps di (α), b(α), and σ(α) are constant, hence C 1 ; all first derivatives are zero, and therefore bounded and locally Hölder. The normalization ⟨di , ei ⟩ = 1 holds by definition. This verifies (A1). Verification of (A2). Here a(α) = Σ. Since Σ is symmetric positive definite, v T Σv ≥ λmin (Σ)|v|2 ,
v ∈ Rd .
Thus (A2) holds with θ = λmin (Σ) > 0. b is a nonsingular b = I − P T , and hence N T D(α) = R. b By Lemma 3.2, R Verification of (A3). Here N = I, D(α) = R M -matrix. This verifies (A3). b is constant in α, so ∂α D(α) = 0. This verifies (A4). Verification of (A4). The reflection matrix D(α) = R All assumptions (A1)–(A4) of Proposition 3.4 have now been verified. Conclusion (C1) gives pathwise existence, uniqueness, and the strong Markov property for the normalized reflected diffusion. Since Ri = Rii di with Rii > 0, replacing the normalized local time by the correspondingly rescaled regulator leaves the reflected path unchanged. Hence the same pathwise existence, uniqueness, and strong Markov conclusions also hold for the original BAR normalization R. This proves assertion (i). For assertion (iii), let the two processes start from x and y and be driven by the same Brownian path. Their free inputs are fx (s) = x + µs + Σ1/2 Ws and fy (s) = y + µs + Σ1/2 Ws , so sups≤t |fx (s) − fy (s)| = |x − y|. Applying conclusion (C2) gives (3.6), proving assertion (iii). Assertion (ii) follows from (3.6) and Brownian continuity. Put Xt = µt + Σ1/2 Wt . Comparing the input x + X with the constant input x gives sup |Zsx − x| ≤ KΓ sup |Xs |
(3.8)
s≤t
almost surely.
s≤t
If h ∈ C0 (E), then h is uniformly continuous. Hence (3.6) gives continuity of x 7→ Pt h(x). If h is supported in the ball of radius r, then |Pt h(x)| ≤ ∥h∥∞ P KΓ sup |Xs | ≥ |x| − r , s≤t
which tends to zero as |x| → ∞; approximation by compactly supported functions gives Pt C0 (E) ⊂ C0 (E). Finally, if ωh is the modulus of continuity of h, then (3.8) gives sup |Pt h(x) − h(x)| ≤ E ωh KΓ sup |Xs | −→ 0 (t ↓ 0) x∈E
s≤t
by bounded convergence. Thus (Pt ) is strongly continuous on C0 (E). This proves assertion (ii).
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Y. Lu and Y. Zhu
We now prove assertions (iv)–(v). By conclusion (C4), the derivative projection at x is the unique linear map onto Hx whose difference from the identity lies in span{di : i ∈ I(x)}. Since span{di : i ∈ A} = span{Ri : i ∈ A} for every A ⊂ J , uniqueness and Lemma 3.2 identify this projection with Lx . Conclusion (C5), applied to the constant parameter family and with parameter direction equal to zero, gives the derivative process Jxt and the directional derivative ∂w Ztx for every fixed w ∈ Gx . It also gives continuity of s 7→ ∂w Zsx at every t > 0 such that Ztx ∈ E ◦ , together with the projected right-continuous regularization. Conclusion (C6) gives P(Ztx ∈ E ◦ ) = 1 for every fixed t > 0. Therefore, for every fixed w ∈ Gx and t > 0, on an event of probability one, ∂w Ztx = lim ∂w Zsx = Jxt [Lx w]. s↓t
This proves assertion (iv). For assertion (v), fix u ∈ Hx . Then u ∈ Gx and Lx u = u. For small ε > 0, x + εu ∈ E. Applying (3.6) with y = x + εu, dividing by ε, and letting ε ↓ 0 gives |∂u Ztx | ≤ KΓ |u| on the event where the directional derivative exists for all t ≥ 0. Combining this bound with (3.7) gives |Jxt [u]| ≤ KΓ |u| for every fixed t > 0, almost surely. Since Jx is RCLL, Fubini gives the same bound for dt ⊗ P-almost every (t, ω). This proves assertion (v) and completes the proof. 3.3. The probabilistic resolvent and its boundary directional derivative The purpose of this subsection is to convert the pathwise derivative package into a boundary identity for the probabilistic resolvent. We prove only that the resolvent is a classical solution of the resolvent equation in the interior, then differentiate the time integral in feasible directions. The resulting boundary derivative is an algebraic linear functional; no classical gradient at a corner is assumed. Fix λ > 0 and h ∈ Cc∞ (Rd ). We regard h as a function on E and define Z ∞ (3.9) g(x) = Rλ h(x) := E e−λt h(Ztx ) dt . 0
Lemma 3.5 (Boundedness and Lipschitz continuity). The function g is bounded and globally Lipschitz on E, with (3.10)
∥g∥∞ ≤
∥h∥∞ , λ
Lip(g) ≤
KΓ Lip(h) . λ
Proof. The first estimate follows immediately from (3.9). For the second, couple Z x and Z y with the same Brownian path. By (3.6), |h(Ztx ) − h(Zty )| ≤ Lip(h)KΓ |x − y| . Integrating against e−λt dt proves the claim. Lemma 3.6 (Interior classical solution of the resolvent equation). The function g is a classical solution of the resolvent equation in E ◦ ; more precisely, g ∈ C ∞ (E ◦ ) and (3.11)
(λ − L)g = h
in E ◦ .
Proof. Fix concentric balls B ′ ⋐ B ⋐ E ◦ and let τB be the first exit time from B. Before τB , the reflected process is the unconstrained diffusion with generator L. The strong Markov property gives, for x ∈ B, Z τB −λt −λτB (3.12) g(x) = Ex e h(Zt ) dt + e g(ZτB ) . 0
Since g ∈ C(B) and ∂B is compact, Stone–Weierstrass applied to the restrictions of polynomials on Rd to ∂B gives polynomials pn with ∥pn − g∥L∞ (∂B) → 0. Setting φn = pn |B , we have φn ∈ C ∞ (B) ⊂ C 2,α (B) and φn → g|∂B uniformly. To solve the interior Dirichlet problems we use the classical Schauder solvability theorem, which we recall in the form used. Theorem 3.7 ([20, Theorem 6.14]). Let Ω ⊂ Rd be a bounded C 2,α domain and let A = aij Dij + bi Di + c be strictly 2 elliptic on Ω, that is, aij (y)ξi ξj ≥ θ0 |ξ| for some θ0 > 0 and all y ∈ Ω, ξ ∈ Rd , with coefficients aij , bi , c ∈ C α (Ω) and c ≤ 0 on Ω. Then for every f ∈ C α (Ω) and every φ ∈ C 2,α (Ω) the Dirichlet problem Au = f in Ω, u = φ on ∂Ω, has a unique solution u ∈ C 2,α (Ω).
13
Signed BAR uniqueness conjecture
We apply this with Ω = B, A = L − λ, so that in coordinates aij = 12 Σij , bi = µi , c = −λ, together with f = −h and φ = ϕn . The four hypotheses hold in the present setting: (a) B is an open Euclidean ball, hence a C ∞ and a fortiori C 2,α domain. 2 (b) For all ξ ∈ Rd , aij ξi ξj = 12 ξ ⊤ Σξ ≥ 21 λmin (Σ) |ξ| , and λmin (Σ) > 0 because Σ is symmetric positive definite; thus 1 A is strictly elliptic with θ0 = 2 λmin (Σ). (c) The coefficients 12 Σij , µi , −λ are constants, hence lie in C α (B) with vanishing Hölder seminorm; and c = −λ < 0 ≤ 0 since λ > 0. (d) The source f = −h ∈ Cc∞ (Rd ) ⊂ C α (B), and each φ = ϕn ∈ C ∞ (B) ⊂ C 2,α (B). Therefore [20, Theorem 6.14] yields a unique un ∈ C 2,α (B) satisfying (λ − L)un = h
in B,
un = ϕn
on ∂B.
Apply Itô’s formula to e−λ(t∧τB ) un (Zt∧τB ). Since un and its first derivatives are bounded on B, the stopped stochastic integral has mean zero. Letting t → ∞ is justified by bounded convergence. Indeed, before τB the process is x + µt + Σ1/2 Wt ; a nonzero one-dimensional projection is a Brownian motion with drift and exits the bounded projection of B almost surely, so τB < ∞ almost surely. This gives the Feynman–Kac representation Z τB −λt −λτB un (x) = Ex e h(Zt ) dt + e ϕn (ZτB ) . 0
Comparing it with (3.12) yields ∥un − g∥L∞ (B) ≤ ∥ϕn − g∥L∞ (∂B) −→ 0. For n, m, the difference w = un − um ∈ C 2,α (B) solves the homogeneous equation (λ − L)w = h − h = 0 in B. To pass to the limit we use the interior Schauder estimate, recalled in the form used. Next, we use interior Schauder estiamte to prove un is Cauchy in C 2,α (B ′ ): Theorem 3.8 (Interior Schauder estimate [20, Theorem 6.2]). Let A = aij Dij + bi Di + c be strictly elliptic on a domain Ω ⊂ Rd with ellipticity constant θ0 > 0 and coefficients bounded in C α (Ω) by a constant Θ. If u ∈ C 2,α (Ω) satisfies Au = f with f ∈ C α (Ω), then for every subdomain Ω′ ⋐ Ω, ∥u∥C 2,α (Ω′ ) ≤ C ∥u∥L∞ (Ω) + ∥f ∥C α (Ω) , C = C d, α, θ0 , Θ, dist(Ω′ , ∂Ω) . The operator A = L − λ satisfies these hypotheses with the ellipticity constant θ0 = 21 λmin (Σ) of (b) and the coefficient bound Θ = max{ 12 ∥Σ∥ , |µ| , λ}, both independent of n, m. Applying it to w = un − um , with Ω = B, Ω′ = B ′ , and f ≡ 0, gives ∥un − um ∥C 2,α (B ′ ) ≤ CB ′ ,B ∥un − um ∥L∞ (B) , CB ′ ,B = C d, α, 12 λmin (Σ), Θ, dist(B ′ , ∂B) , where the constant CB ′ ,B does not depend on n, m. Because un → g uniformly on B, the right-hand side tends to zero as n, m → ∞. Thus (un ) is Cauchy in C 2,α (B ′ ) and converges there to some u ∈ C 2,α (B ′ ). The same sequence converges uniformly to g on B, so the C 2,α (B ′ ) limit must be u = g. Passing to the limit in the equations satisfied by the classical solutions un gives g ∈ C 2,α (B ′ ) and (3.11) on B ′ . Since the coefficients of L are constant and h is smooth, standard interior elliptic regularity, equivalently the usual bootstrapping by interior estimates, gives g ∈ C ∞ (B ′ ). Since B ′ ⋐ E ◦ was arbitrary, the conclusion follows. Proposition 3.9 (Directional factorization of the resolvent). Let x ∈ E. There is a bounded linear functional Λx : Hx → R such that, for every w ∈ Gx , the one sided directional derivative exists and satisfies (3.13)
+ ∂w g(x) := lim ε↓0
g(x + εw) − g(x) = Λx (Lx w). ε
The linear functional ℓx (v) := Λx (Lx v) is bounded by (3.14)
|ℓx (v)| ≤
KΓ CL ∥∇h∥∞ |v| λ
for all v ∈ Rd . If i ∈ I(x), then ℓx (Ri ) = 0. Note that no continuity or measurability of the map x 7→ ℓx is asserted.
14
Y. Lu and Y. Zhu
R∞ Proof. Fix x ∈ E. For u ∈ Hx , define Λx (u) := E 0 e−λt ∇h(Ztx ) · Jxt [u] dt. We first record the measurability and integrability facts needed to define Λx . Since Jx is adapted and RCLL with values in the finite dimensional space Lin(Hx , Rd ), for every fixed u ∈ Hx the process (t, ω) 7→ Jxt [u](ω) is progressively measurable, hence B([0, ∞)) ⊗ F measurable. By Theorem 3.3, for each fixed u ∈ Hx , we have |Jxt [u]| ≤ KΓ |u| holds for dt ⊗ P-almost every (t, ω). Hence R ∞ −λt ∞ E 0 e |∇h(Ztx ) · Jxt [u]| dt ≤ KΓ ∥∇h∥ |u| < ∞, i.e. the integral defining Λx (u) is absolutely convergent with respect λ −λt to e dt ⊗ P, and |Λx (u)| ≤
KΓ ∥∇h∥∞ |u|, λ
u ∈ Hx .
Since Jxt ∈ Lin(Hx , Rd ), linearity of Λx follows from linearity of Jxt and the preceding bound. Thus Λx is a bounded linear functional on Hx . Now fix w ∈ Gx . For all sufficiently small ε > 0, x + εw ∈ E. By the definition of g, Z ∞ h(Ztx+εw ) − h(Ztx ) g(x + εw) − g(x) =E e−λt dt. (3.15) ε ε 0 The synchronous Lipschitz estimate gives the deterministic domination (3.16)
h(Ztx+εw ) − h(Ztx ) ≤ KΓ ∥∇h∥∞ |w|, ε
uniformly for all sufficiently small ε > 0, all t ≥ 0, and all sample paths. On the probability one event in Theorem 3.3 corresponding to this fixed pair (x, w), the pathwise directional derivative ∂w Ztx = limε↓0
Ztx+εw −Ztx exists for every t ≥ 0. Since h ∈ Cc∞ (Rd ), the mean value formula gives ε
h(Ztx+εw ) − h(Ztx ) −→ ∇h(Ztx ) · ∂w Ztx ε for dt ⊗ P-almost every (t, ω). Dominated convergence with respect to e−λt dt ⊗ P therefore yields Z ∞ + (3.17) ∂w g(x) = E e−λt ∇h(Ztx ) · ∂w Ztx dt. 0
For every fixed t > 0, Theorem 3.3 gives ∂w Ztx = Jxt [Lx w] almost surely. For fixed x and w, the map (t, ω) 7→ ∂w Ztx (ω) is B([0, ∞)) ⊗ F -measurable, since it is the pointwise limit of the continuous-in-t difference quotients ε−1 Ztx+εw − Ztx . The process t 7→ Jxt [Lx w] is measurable because Jx is RCLL. Hence the fixed-time almost sure identity can be integrated in t, giving the identity dt ⊗ P-almost everywhere. By Fubini, this identity holds for dt ⊗ P-almost every (t, ω). The value at t = 0 is irrelevant for the time integral. Substituting into the preceding display gives Z ∞ + (3.18) ∂w g(x) = E e−λt ∇h(Ztx ) · Jxt [Lx w] dt = Λx (Lx w), 0
which proves the factorization. Finally, by Lemma 3.2, we have |Lx v| ≤ CL |v|, holds for all v ∈ Rd . Therefore |ℓx (v)| = |Λx (Lx v)| ≤
KΓ ∥∇h∥∞ KΓ CL ∥∇h∥∞ |Lx v| ≤ |v|. λ λ
If i ∈ I(x), then Lemma 3.2 gives Lx Ri = 0, and hence ℓx (Ri ) = Λx (Lx Ri ) = 0. This completes the proof. Remark 3.10 (Regularity at the boundary). At a boundary point x, the proposition gives a bounded linear functional ℓx extending the feasible one-sided derivative. Since g is Lipschitz and w 7→ ℓx (w) is linear, the ray derivatives imply the cone-wise first-order expansion g(x + ϵw) = g(x) + ϵℓx (w) + ow (ϵ)
as ϵ → 0, x + ϵw ∈ E.
This expansion is local at the fixed boundary point. It does not assert that x 7→ ℓx is continuous towards the boundary, nor that the interior gradient ∇g(y) has a limit as y → x from E ◦ . The smoothing argument uses only the algebraic value ℓx (Ri ) = 0 on active reflection directions, after the feasible-direction limit has been averaged against the one-sided mollifier.
Signed BAR uniqueness conjecture
15
Proof of Proposition 3.1. Boundedness and Lipschitz continuity are Lemma 3.5. The statement that g is a classical solution of the resolvent equation in E ◦ is Lemma 3.6. The projection formula and the identity Lx Ri = 0 for i ∈ I(x) are Lemma 3.2. Finally, Proposition 3.9 gives the feasible one-sided derivatives, the factorization through Lx , and the linear extension ℓx satisfying ℓx (Ri ) = 0 on active faces. 4. One-sided smoothing and the measure–Neumann approximation In this section we prove the resolvent insertion theorem, Theorem 4.1. We first prove the identity for h ∈ Cc∞ (Rd ), regarded as a function on E, by a one-sided smoothing argument. At the end of the proof, a density argument extends the identity to all h ∈ C0 (E). Thus, until this final density step throughout this whole section, we fix λ > 0, h ∈ Cc∞ (Rd ), and write g = Rλ h as in (3.9). Theorem 4.1 (Resolvent insertion theorem). For every finite signed BAR tuple (π̄, ν̄1 , . . . , ν̄d ), every h ∈ C0 (E), and every λ > 0, Z (RI) (λRλ h − h) dπ̄ = 0. E
First we prove the insertion for smooth compactly supported h, then extend it to C0 (E) by uniform approximation. A convolution supported strictly inside the orthant produces bounded C 2 functions on a neighborhood of the closed state space. Integration by parts in the convolution variable proves vanishing of every oblique boundary derivative, with a bound uniform in the smoothing scale. Since the BAR is imposed on Cb2 (E), no spatial cutoff is needed in the resolvent insertion. ∞ d R Now we one-sided smoothing the function g. We seek a mollifier ρ ∈ Cc ((1, 2) ) that satisfies ρ ≥ 0 and ρ(w) dw = 1. For ε > 0, define Rd Z Z (4.1) gε (x) = ρ(w)g(x + εw) dw, hε (x) = ρ(w)h(x + εw) dw, x ∈ E. Rd
Rd
Because the support of ρ lies strictly inside the positive orthant, gε is defined and smooth on an open neighborhood of E. Lemma 4.2 (One-sided smoothing). gε ∈ Cb2 (E) for each fixed ε > 0, gε is smooth on an open neighborhood of E, gε → g uniformly, gε is uniformly bounded and Lipschitz, and (4.2)
(λ − L)gε = hε
on E.
Moreover, (4.3)
∥gε − g∥∞ ≤ Cρ ε Lip(g),
(4.4)
∥hε − h∥∞ −→ 0,
(4.5)
∥gε ∥∞ ≤ ∥g∥∞ ,
(4.6)
∥∇gε ∥∞ ≤ Cρ Lip(g),
(4.7)
D2 gε ∞ ≤ Cρ ε−1 Lip(g).
Proof. We first justify the smoothness of gε . Let δρ := dist(supp ρ, ∂Rd+ ) > 0. For each fixed ε > 0, define Uε := {x ∈ Rd : x + εw ∈ E ◦ for every w ∈ supp ρ}. Then Uε is an open neighborhood of E, because supp ρ ⋐ (0, ∞)d . Thus gε is well defined on Uε . Although g is only known to be globally Lipschitz on E, the derivatives of gε may be computed by integration by parts in the convolution variable. For every multiindex α, Z α |α| −|α| α (4.10) ∂x gε (x) = (−1) ε ∂w ρ(w) g(x + εw) dw, x ∈ Uε . Rd
This identity is first obtained in the sense of distributions on Uε . Since the right hand side is continuous in x, it is the classical derivative. Iterating the same argument gives derivatives of all orders; hence gε ∈ C ∞ (Uε ). In particular, gε ∈ C 2 (E) in the closed domain sense. For fixed x ∈ E, the compact set x + ε supp ρ lies in E ◦ . Same as the proof of Lemma 3.6, g is a classical solution of (λ − L)g = h on this compact subset of the interior. Since L has constant coefficients, differentiating under the integral on
16
Y. Lu and Y. Zhu
R R this interior compact set gives (λ − L)gε (x) = Rd ρ(w)(λ − L)g(x + εw) dw = Rd ρ(w)h(x + εw) dw = hε (x), which proves (4.2). Next, the global Lipschitz continuity of g gives Z |gε (x) − g(x)| ≤ ε Lip(g) |w| ρ(w) dw, which is (4.3). Since h ∈ Cc∞ (Rd ), it is uniformly continuous, and therefore ∥hε − h∥∞ → 0 which proves gives (4.4). R The bound (4.5) follows from ρ ≥ 0 and ρ = 1: Z |gε (x)| ≤ ρ(w)|g(x + εw)| dw ≤ ∥g∥∞ . Rd
For the gradient, integration by parts in w and
R
1 ε
Z
(4.8)
∂xj gε (x) = −
Rd
∂wj ρ(w) dw = 0 gives ∂wj ρ(w) g(x + εw) − g(x) dw.
R Hence |∂xj gε (x)| ≤ Lip(g) Rd |∂wj ρ(w)| |w| dw. The Lipschitz bound on the difference proves (4.6). Differentiating once more in the same distributional-convolution formula and subtracting the constant g(x) gives Z 1 2 ρ(w) g(x + εw) − g(x) dw, ∂x2j xk gε (x) = 2 ∂w j wk ε R 2 and therefore |∂x2j xk gε (x)| ≤ ε−1 Lip(g) Rd |∂w ρ(w)| |w| dw. Taking the maximum over j, k proves (4.7). Thus gε , j wk its first derivatives, and its second derivatives are bounded on E, so gε ∈ Cb2 (E). The next proposition is where the projected derivative information for g is used at the boundary. It proves two facts on each face Fi : first, Di gε (x) → 0 for every x ∈ Fi ; second, the uniform bound in (4.9) holds. Together these imply the boundary measure convergence in (4.10) by dominated convergence. The pointwise limit is obtained from Proposition 3.9: if x ∈ Fi and w ∈ supp ρ ⊂ (0, ∞)d , then w ∈ Gx and (g(x + εw) − g(x))/ε → ℓx (w). After integration by parts in the smoothing variable, the limit of Di gε (x) becomes ℓx (Ri ), which is zero because Lx Ri = 0 on active faces. Proposition 4.3 (Vanishing oblique derivative after one sided smoothing). For every i ∈ J and every x ∈ Fi , Di gε (x) −→ 0 as ε ↓ 0. At the same time, the convergence is pointwise in x. There is a constant Cρ,R < ∞, independent of x and ε, such that (4.9)
|Di gε (x)| ≤ Cρ,R Lip(g),
x ∈ Fi ,
0 < ε < 1.
Consequently, for every finite signed measure ηi on Fi and every bounded Borel function a : Fi → R fixed independently of ε, we have Z (4.10) a(x)Di gε (x) dηi (x) −→ 0. Fi
Proof. Since ρ ∈ Cc∞ ((1, 2)d ), integration by parts in the w-variable gives, for x ∈ E, Z 1 ∇gε (x) = − ∇ρ(w) g(x + εw) dw. ε Rd R Also, Rd Ri · ∇ρ(w) dw = 0. Therefore, for x ∈ Fi , Z g(x + εw) − g(x) (4.11) Di gε (x) = − (Ri · ∇ρ)(w) dw. ε Rd Fix x ∈ Fi . Then i ∈ I(x). Since supp ρ ⊂ (1, 2)d , every w ∈ supp ρ belongs to Gx . Proposition 3.9 gives, for each such fixed w, g(x+εw)−g(x) −→ ℓx (w). Moreover, ε (Ri · ∇ρ)(w)
g(x + εw) − g(x) ≤ |(Ri · ∇ρ)(w)| Lip(g)|w|. ε
17
Signed BAR uniqueness conjecture
The right hand side is integrable over Rd , because ρ is smooth and compactly supported. Dominated convergence in the mollifier variable w gives Z (Ri · ∇ρ)(w)ℓx (w) dw. (4.12) lim Di gε (x) = − ε↓0
Rd
Since ℓx is linear and ρ has compact support, another integration by parts gives Z Z (4.13) − (Ri · ∇ρ)(w)ℓx (w) dw = ρ(w)ℓx (Ri ) dw = ℓx (Ri ). Rd
Rd
There is no boundary term because ρ ∈ Cc∞ ((1, 2)d ). Since i ∈ I(x), Proposition 3.9 gives ℓx (Ri ) = 0. This proves the pointwise convergence. The same representation and the Lipschitz bound give Z |Di gε (x)| ≤ Lip(g) |(Ri · ∇ρ)(w)| |w| dw. Rd
R Thus the uniform estimate holds with Cρ,R := max1≤k≤d Rd |(Rk · ∇ρ)(w)| |w| dw < ∞. Finally, let ηi ∈ M (Fi ), and let a : Fi → R be bounded Borel and fixed independently of ε. Since Di gε is continuous on Fi , the product aDi gε is Borel. The pointwise convergence just proved and the bound |a(x)Di gε (x)| ≤ ∥a∥∞ Cρ,R Lip(g) R allow dominated convergence with respect to |ηi |. Hence Fi a(x)Di gε (x) dηi (x) −→ 0. This completes the proof. Proposition 4.4 (Measure–Neumann resolvent approximation). Let m be a finite signed measure on E and let ηi be finite signed measures on Fi . Then, as ε ↓ 0, Z Z (4.14) (λ − L)gε dm −→ h dm, E
E
Z
Z gε dm −→
(4.15)
g dm,
E
E
Z Di gε dηi −→ 0,
(4.16)
i = 1, . . . , d.
Fi
Proof. The first assertion follows from (λ − L)gε = hε and the uniform convergence hε → h. The second follows from the uniform convergence gε → g. Since m is finite signed, uniform convergence is sufficient in both cases. For the boundary terms, take the bounded Borel multiplier a ≡ 1 in (4.10). This gives (4.16) for each finite signed boundary measure ηi . Proof of Theorem 4.1. First assume h ∈ Cc∞ (Rd ), regarded as a function on E, and let g = Rλ h. By Lemma 4.2, gε ∈ Cb2 (E), so it is an admissible BAR test. Applying the BAR to gε and rearranging gives Z
Z (λ − L)gε dπ̄ = λ
(4.17) E
gε dπ̄ + E
d Z X i=1
Di gε dν̄i .
Fi
By Proposition 4.4, letting ε ↓ 0 yields Z
Z h dπ̄ = λ
E
Rλ h dπ̄. E
R Equivalently, E (λRλ h − h) dπ̄ = 0 holds for every smooth compactly supported h. Now let h ∈ C0 (E). The restrictions to E of functions in Cc∞ (Rd ) are uniformly dense in C0 (E): extend a function from the closed set E to C0 (Rd ), cut it off, and mollify on Rd . Choose hn ∈ Cc∞ (Rd ) with ∥hn − h∥∞ → 0 on E. Since (Pt ) is a contraction on bounded functions, ∥Rλ (hn − h)∥∞ ≤ λ−1 ∥hn − h∥∞ . The finiteness of π̄ therefore permits passage to the limit in the smooth identity, proving (RI) for h ∈ C0 (E).
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Y. Lu and Y. Zhu
5. Proof of Proposition 2.3: From the resolvent identity to signed BAR uniqueness We now complete the proof of resolvent identity criterion (Proposition 2.3) and then complete the proof of the main Theorem (Theorem 2.2). We first use the uniqueness of Laplace transforms to show that the signed measure that satisfies (RI) is invariant for the semigroup (Pt ). Lemma 5.1 (Uniqueness of Laplace transforms[12, Chapter II]). RLet a : [0, ∞) → R be locally integrable and of at most ∞ exponential growth. Suppose that its Laplace transform b a(λ) = 0 e−λt a(t) dt vanishes for every λ in some interval (λ∗ , ∞). Then a(t) = 0 for Lebesgue almost every t ≥ 0. Proposition 5.2 (Resolvent identity implies semigroup invariance). Assume that a finite signed measure π̄ satisfies (RI) for every h ∈ C0 (E) and every λ > 0. Then π̄ is invariant for (Pt ): Z Z (5.1) Pt φ dπ̄ = φ dπ̄, t ≥ 0, φ ∈ C0 (E). E
E
Proof. From (RI), for every h ∈ C0 (E) and every λ > 0, Z Z (5.2) Rλ h dπ̄ = λ−1 h dπ̄. E
Fix h ∈ C0 (E) and define Fh (t) =
R E
E
Pt h dπ̄ (t > 0). We first record the elementary regularity of Fh . By the Feller t↓0
statement (ii) in Theorem 3.3, Pt C0 (E) ⊂ C0 (E), and ∥Pt h − h∥∞ −−→ 0. The semigroup property and the contraction property imply norm continuity of t 7→ Pt h on all of [0, ∞). Indeed, ∥Ps h − Pt h∥∞ = ∥Pmin{s,t} (P|s−t| h − h)∥∞ ≤ ∥P|s−t| h − h∥∞ −→ 0 as s → t. Since π̄ is finite signed, it follows that Fh is continuous: |Fh (s) − Fh (t)| ≤ |π̄|(E) ∥Ps h − Pt h∥∞ . Moreover, we have |Fh (t)| ≤ ∥h∥∞ |π̄|(E), for |Pt h| ≤R∥h∥ . We now pass from the resolvent identity to a Laplace R∞ ∞ transform identity. Since |e−λt Pt h(x)| ≤ e−λt ∥h∥∞ and E 0 e−λt ∥h∥∞ dt d|π̄|(x) = λ−1 ∥h∥∞ |π̄|(E) < ∞, Fubini’s theorem gives Z Z Z ∞ Rλ h dπ̄ = e−λt Pt h(x) dt dπ̄(x) E
E
0
Z ∞ =
e 0
−λt
Z
Z ∞
Pt h dπ̄ dt =
E
e−λt Fh (t) dt.
0
R R∞ At the same time, since λ−1 E h dπ̄ = 0 e−λt Fh (0) dt, therefore (5.2) is equivalent to Z ∞ (5.3)
e−λt Fh (t) − Fh (0) dt = 0,
λ > 0.
0
Put ah (t) = Fh (t) − Fh (0). Then ah is continuous and bounded. In particular, ah is locally integrable and of at most exponential growth. Indeed, |ah (t)| ≤ |Fh (t)| + |Fh (0)| ≤ 2∥h∥∞ |π̄|(E),
t ≥ 0.
Equation (5.3) says precisely that the Laplace transform of ah vanishes for every λ > 0. By Lemma 5.1, ah (t) = 0 for Lebesgue almost every t ≥ 0. Since aRh is continuous, R this almost everywhere equality upgrades to equality for every t ≥ 0. Hence Fh (t) = Fh (0) for t ≥ 0, i.e. E Pt h dπ̄ = ER h dπ̄, holds for all h ∈ C0 (E). For a finite signed measure α, we write (αPt )(B) = E Pt (x, B) α(dx) for B ∈ B(E). Then π̄Pt is a finite signed Radon measure, and for every φ ∈ C0 (E), Z Z Z φ d(π̄Pt ) = Pt φ dπ̄ = φ dπ̄. E
E
E
Since C0 (E) separates finite Radon measures on the locally compact space E, this implies π̄Pt = π̄.
Signed BAR uniqueness conjecture
19
Proposition 5.2 indicates that the signed-BAR interior solution π̄ is invariant as a signed measure for the SRBM semigroup, i.e. π̄Pt = π̄. Then we follows the Dai-Dieker [5], using Jordan decomposition to show that π̄ uniquely characterizes the stationary probability distribution π0 of the diffusion process in the sense that π̄ = π̄(E)π0 . Proposition 5.3 (Identification of the interior measure). If π̄ is a finite signed invariant measure for the SRBM semigroup, then π̄ = cπ0 where c = π̄(E). Proof. For a Markov kernel P and a finite signed measure π̄, positivity gives the measure inequality |π̄P | ≤ |π̄|P. R R Indeed, for every Borel set B, | P (x, B) dπ̄(x)| ≤ P (x, B) d|π̄|(x), and the same domination holds for finite measurable partitions, hence for total variation. If π̄Pt = π̄, then (5.3) gives |π̄| ≤ |π̄|Pt . Both positive measures have total mass |π̄|(E), because Pt (x, E) = 1. Thus the domination is actually equality: if finite positive measures α ≤ β have α(E) = β(E), then β − α is a positive measure with total mass zero, hence vanishes. Applying this with α = |π̄| and β = |π̄|Pt gives |π̄|Pt = |π̄|. Consequently the Jordan components π̄ + = 12 (|π̄| + π̄) and π̄ − = 12 (|π̄| − π̄) are invariant positive finite measures. Each nonzero component, after normalization by its total mass, is an invariant probability and therefore equals π0 , which is unique. Thus π̄ = (π̄ + (E) − π̄ − (E))π0 = π̄(E)π0 .
(5.3)
The preceding Propositions 5.2 and 5.3 identifies the uniqueness of the interior measure, but an exact same assessment for boundary measure is not yet established. Therefore, by subtracting the appropriate scalar multiple of the stationary BAR vector, the remaining signed tuple has zero interior measure. Thus the only possible obstruction to the signed uniqueness of BAR solution is a purely boundary one: a collection of finite signed measures (ηi )di=1 , with ηi supported on Fi , whose Pd R boundary pairing vanishes, i.e. i=1 Fi Di f dηi = 0 against every test function f ∈ Cb2 (E). If such boundary measures exist, adding them to an existing solution doesn’t change the validity of the solution. The next proposition shows that no such nontrivial boundary annihilator exists, ruling out any other signed-BAR solutions. The proof is local on the boundary stratification and uses the nonsingular M -matrix assumption only through the invertibility of the principal reflection blocks RAA . On a stratum SA , the active oblique derivatives are determined by T the active normal jet: (Di f |SA )i∈A = RAA a, where a = (∂xj f |SA )j∈A . Since RAA is invertible, we can prescribe these −T oblique derivatives independently. In particular, choosing a = RAA ek ψ gives Di f |SA = δik ψ for i ∈ A, and for any test function ψ ∈ Cc∞ (SA ). This isolates the k-th boundary measure on SA . An induction over the codimension |A| removes all lower-stratum contributions and forces ηk |SA = 0. Since both A and k ∈ A are arbitrary, all boundary measures vanish. Proposition 5.4 (Pure boundary injectivity). Let ηi ∈ M(Fi ) be finite signed measures. If d Z X
(5.4)
i=1
Di f dηi = 0,
f ∈ Cb2 (E),
Fi
then ηi = 0, for all i = 1 . . . , d. Proof. For A ⊂ J and i ∈ A, let ηiA = ηi |SA . Then ηi = We prove by induction on n = |A| that (5.5)
ηiA = 0
A A∋i ηi as a finite sum of mutually singular signed measures.
P
for every A ⊂ J with |A| = n and every i ∈ A.
Assume the claim has been proved for all strata of cardinality less than n, and fix A with |A| = n. Let k ∈ A and let c c ψ ∈ Cc∞ (SA ). Write points as x = (xA , y), where y = xAc ∈ (0, ∞)A . We identify ψ with its extension by zero to RA ; ∞ Ac this extension is smooth because supp ψ is compact in the open orthant. Choose a tangential cutoff ϑ ∈ Cc ((0, ∞) ) that equals one on a neighborhood of supp ψ. When A = J , interpret the tangential space as a point and put ϑ = 1. Choose a normal cutoff ζ ∈ Cc∞ (RA ) that equals one near the origin. Define the A-vector (5.6)
−T a(y) = RAA ek ψ(y)
and the test function (5.7)
f (xA , y) = ζ(xA )ϑ(y)
X j∈A
xj aj (y).
20
Y. Lu and Y. Zhu
After shrinking the support of ϑ if necessary, f is supported away from every face Fj with j ∈ / A. At a point of SA , the tangential derivatives of f vanish and (5.8)
∂xj f (0, y) = aj (y),
j ∈ A.
Therefore, for i ∈ A, (5.9)
Di f |SA =
X
T Rji aj = (RAA a)i = δik ψ.
j∈A
The support condition implies that the only boundary strata meeting supp f are SC with ∅ ̸= C ⊂ A. The contributions from |C| < n vanish by the induction hypothesis. Hence (5.4) and (5.9) give Z XZ 0= Di f dηiA = ψ dηkA . i∈A
SA
SA
Since ψ is arbitrary, ηkA = 0. Since k ∈ A was arbitrary, the induction step is complete. The base case n = 1 is the same argument with no lower strata. Thus all ηiA vanish and hence all ηi vanish. Now, the uniqueness of Signed BAR solution in the Harrison-Reiman Class is the natural conclusion of Proposition 5.2Proposition 5.4. Proof of Proposition 2.3. Assume (RI) for the finite signed BAR tuple (π̄, ν̄1 , . . . , ν̄d ). By Proposition 5.2, we know the signed measure π̄ is invariant under the semigroup, i.e. π̄Pt = π̄ holds for t ≥ 0. Then by Proposition 5.3, we have the uniquness of the interior measure π̄ = cπ0 for c = π̄(E). Define ηi = ν̄i − cνi0 . Subtract c times the stationary BAR (2.4) from the signed BAR (2.3). The interior measures Pd R cancel, and we obtain i=1 Fi Di f dηi = 0, for all test function f ∈ Cb2 (E). Proposition 5.4 gives ηi = 0 for every i, hence ν̄i = cνi0 . This proves Proposition 2.3. Finally, we finish the proof of the signed-BAR problem. Proof of Theorem 2.2. Theorem 4.1 proves (RI) for every finite signed BAR tuple. Proposition 2.3 converts (RI) into full signed BAR uniqueness. Therefore Theorem 2.2 follows. 6. Failure in the completely S class In this section, we demonstrate a family of counterexamples in the completely S class due to the singular proper active block of the relection matrix. The negative construction is summarized in Fig. 2. It has two parts: a boundary gauge identity on the singular stratum, followed by a zero-potential correction that extends the resulting centered source into the interior and adds the matching boundary occupation potential. Recall that a square matrix A is an S matrix if there exists a vector u > 0 such that Au > 0, and is completely-S if every principal submatrix is an S matrix. completely-S reflection matrices are the natural existence class for orthant SRBMs, but they need not have invertible principal blocks. 6.1. A singular-block boundary gauge In this section, we utilize the singular proper active block of the reflection matrix to construct a nonzero null direction on that block and place a signed boundary gauge on the corresponding lower-dimensional boundary stratum. The singularity makes the active normal reflection components cancel, so the gauge leaves only a tangential derivative along the stratum. After integration by parts, this tangential derivative becomes a finite nonzero centered signed source supported on the same stratum. This source will be cancelled by the zero-potential correction in the next subsection. Let J = {1, . . . , d}. For a nonempty proper set A ⊊ J , put T = J \ A and write RBC for the submatrix with rows in B and columns in C. Define the open stratum SA = {x ∈ E : xi = 0 for i ∈ A, xj > 0 for j ∈ T }. We identify SA with (0, ∞)T through the embedding ιA : (0, ∞)T → E that inserts zeros in the coordinates indexed by A. Proposition 6.1 (Boundary gauge from a singular principal block). Assume that R is nonsingular and that RAA is singular for some nonempty proper set A ⊊ J . Choose 0 ̸= v ∈ ker RAA ,
w = RT A v ∈ RT .
21
Signed BAR uniqueness conjecture B OUNDARY GAUGE ON A SINGULAR STRATUM
Normal components cancel Combining active faces gives
Boundary gauge on SA For y ∈ (0, ∞)T and ιA (y) ∈ SA , set
Singular active block Choose ∅ ̸= A ⊊ J and 0 ̸= v ∈ ker RAA . With T = J \ A, put w := RT A v ̸= 0. The nonzero vector w is tangent to the stratum SA .
dζi (ιA (y)) = vi φ(y) dy,
X
i ∈ A,
vi Ri = (RAA v, RT A v) = (0, w).
i∈A
Thus the gauge sees only the tangential derivative w · ∇T f on SA .
and set ζi = 0 for i ∈ / A.
Tangential integration by parts Because φ ∈ Cc∞ ((0, ∞)T ), Z XZ Di f dζi = f dχ, i
Fi
E
dχ(ιA (y)) = −(w · ∇T φ(y)) dy. The source is supported on SA , nonzero, and centered: χ(E) = 0. E XTENSION INTO THE INTERIOR BY A ZERO - POTENTIAL
+
R
f dχ
Interior zero-potential Spread the centered source by the reflected semigroup: Z ∞ π̄ = χPt dt.
Boundary occupation correction Use the one-unit regulator kernels Ki and set X θi = (χPn )Ki . n≥0
0
The one-step regulator bound makes each θi finite on Fi .
Exponential ergodicity and χ(E) = 0 make this finite and give π̄(E) = 0.
Poisson identity cancels the source Itô’s formula over integer intervals gives Z XZ Lf dπ̄ + Di f dθi = − f dχ.
Z E
i
Fi
E
Adding the boundary-gauge identity from the upper half cancels
R
f dχ.
Signed BAR tuple with zero interior mass Z ν̄i := θi + ζi ,
XZ
Lf dπ̄ + E
Di f dν̄i = 0,
π̄(E) = 0.
Fi
i
Since π̄ ̸= 0 but π̄(E) = 0 whereas π0 (E) = 1, the interior coordinate cannot be a scalar multiple of the stationary probability. Varying φ on disjoint supports gives the infinite-dimensional failure.
F IG 2. Counterexample construction in the completely-S class. The upper group is the boundary algebra of Proposition 6.1: a gauge supported on a lower-dimensional stratum has its active normal components killed by RAA v = 0, leaving a tangential derivative and hence a centered source χ. The R lower group is the zero-potential correction of Proposition 6.2: the semigroup potential π̄ = 0∞ χPt dt and the boundary occupation potentials θi cancel the source and produce a nonzero zero-mass signed BAR tuple.
Then w ̸= 0. Let ϕ ∈ Cc∞ ((0, ∞)T ) satisfy w · ∇T ϕ ̸≡ 0. For i ∈ A, define a finite signed measure ζi on Fi by Z Z (6.1) g(x) dζi (x) = vi g(ιA (y))ϕ(y) dy, (0,∞)T
Fi
and set ζi = 0 for i ∈ / A. Define χ ∈ M(E) by Z Z (6.2) g(x) dχ(x) = −
g(ιA (y)) w · ∇T ϕ(y) dy.
(0,∞)T
E
Then χ is finite, nonzero, supported on SA , and satisfies χ(E) = 0. Moreover, (6.3)
d Z X i=1
Fi
Z Di f dζi =
f dχ, E
f ∈ Cb2 (E).
22
Y. Lu and Y. Zhu
Proof. If w = 0, extend v to ve ∈ Rd by setting its coordinates in T equal to zero. Then RAA v Re v= = 0, RT A v contradicting the nonsingularity of R. Hence w ̸= 0, and a compactly supported smooth ϕ with nonzero directional derivative along w exists. For f ∈ Cb2 (E), combine the face labels before integrating: d Z X i=1
!
Z Di f dζi =
ϕ(y) (0,∞)T
Fi
X
vi R i
· ∇f (ιA (y)) dy
i∈A
Z ϕ(y) [(RAA v) · ∇A f (ιA (y)) + (RT A v) · ∇T f (ιA (y))] dy
= (0,∞)T
Z ϕ(y)w · ∇T f (ιA (y)) dy
= (0,∞)T
Z
w · ∇T ϕ(y) f (ιA (y)) dy.
=− (0,∞)T
There is no boundary term because ϕ is compactly supported in the open stratum. This proves (6.3). Taking a test function equal to one on a neighborhood of supp ϕ gives χ(E) = 0, and the choice of ϕ gives χ ̸= 0. The singular block cancels the components normal to the active faces. The remaining vector w is tangent to SA , and tangential integration by parts turns the boundary gauge into the centered source χ. 6.2. Interior correction by a zero-potential In this section, we take that centered source and spread it through the reflected Brownian semigroup by a zero potential, while also adding the matching boundary occupation potentials generated by the regulator. Under exponential ergodicity and a one-step regulator bound, these potentials are finite. Ito’s formula then shows that the zero potential contributes exactly the negative of the source created in the previous section. Adding the original boundary gauge cancels the defect and produces a genuine finite signed BAR tuple. Its interior part has total mass zero but is not the zero measure, so it cannot be a scalar multiple of the stationary distribution, whose mass is one. Let (Pt )t≥0 be the transition semigroup of theR SRBM, and let Y = (Y1 , . . . , Yd ) be its regulator. For a finite signed measure α, we define the semigroup (αPt )(B) = E Pt (x, B) α(dx). For each face define the boundary occupation kernel Z 1 Ki (x, B) = Ex
(6.4)
1B (Z(s)) dYi (s), 0
which is supported on Fi . Proposition 6.2 (Interior correction by a zero-potential). Assume that the SRBM is a strong Markov process whose transition semigroup is Feller on C0 (E), and that it has stationary probability π0 . Suppose that there are a locally bounded function V : E → [1, ∞) and constants M, κ > 0 such that (6.5)
∥Pt (x, ·) − π0 ∥TV ≤ M V (x)e−κt ,
x ∈ E, t ≥ 0.
Suppose also that ci := sup Ex Yi (1) < ∞,
(6.6)
i = 1, . . . , d.
x∈E
Let χ ∈ M(E) satisfy χ(E) = 0 and (6.7)
R E
V d|χ| < ∞, and let ζi ∈ M(Fi ) satisfy
d Z X i=1
Fi
Z Di f dζi =
f dχ, E
f ∈ Cb2 (E).
23
Signed BAR uniqueness conjecture
Define Z ∞ π=
(6.8)
χPt dt, 0
θi =
(6.9)
∞ X
(χPn )Ki ,
n=0
ν i = θi + ζ i .
(6.10)
Then all measures in (6.8)–(6.10) are well defined in total variation and finite. They form a signed BAR tuple for all f ∈ Cb2 (E), hence also for all f ∈ Cc2 (E), and π(E) = 0. If χ ̸= 0, then π ̸= 0. Proof. Because χ(E) = 0, Z χPt =
Pt (x, ·) − π0 χ(dx).
E
Therefore ∥χPt ∥TV ≤ M e−κt
(6.11)
Z V d|χ|. E
The integral in (6.8) converges in total variation, and π(E) = 0 because χPt (E) = χ(E) = 0. For a finite signed measure α and the positive kernel Ki , Z (6.12) ∥αKi ∥TV ≤ Ki (x, E) |α|(dx) ≤ ci ∥α∥TV . E
Combining (6.11) at integer times with (6.12) proves absolute convergence of (6.9). Each θi is supported on Fi . Fix f ∈ Cb2 (E). Itô’s formula up to an integer time N , followed by integration against the Jordan decomposition of χ, gives Z N (6.13)
χPN (f ) − χ(f ) =
(χPt )(Lf ) dt + 0
d Z X i=1
Z N Di f (Z(t)) dYi (t) χ(dx).
Ex
E
0
Splitting the boundary integral into unit intervals and using the strong Markov property yields Z N
Z (6.14)
Di f (Z(t)) dYi (t) χ(dx) =
Ex E
0
N −1 X
(χPn )Ki (Di f ).
n=0
The estimates above justify passage to the limit. Since χPN (f ) → 0, equations (6.13) and (6.14) give Z Lf dπ + E
d Z X i=1
Z Di f dθi = −
Fi
f dχ. E
Adding (6.7) proves the BAR for (π, ν 1 , . . . , ν d ). It remains to prove that the zero potential is injective on this centered class. Let A be the generator of the Feller semigroup on C0 (E). For h ∈ D(A), semigroup differentiation gives Z T π(Ah) = lim
T →∞ 0
χPt (Ah) dt = lim χ(PT h − h) = −χ(h), T →∞
where the last limit follows from (6.11). If π = 0, then χ(h) = 0 for every h ∈ D(A). To conclude that χ = 0 we use the following density theorem. Theorem 6.3 ([17, Chapter 1, Section 2]). The infinitesimal generator A of a strongly continuous contraction semigroup on a Banach space has domain D(A) dense in that space. In particular, for a Feller semigroup on C0 (E), the domain D(A) is dense in C0 (E).
24
Y. Lu and Y. Zhu
Its hypothesis is exactly the standing assumption of the present proposition: (Pt ) is Feller on C0 (E). Hence D(A) is dense in C0 (E). Since M(E) = C0 (E)∗ , the identity χ(h) = 0 on D(A) implies χ = 0. Therefore χ ̸= 0 implies π ̸= 0. Theorem 6.4 (Failure in the completely-S class). Consider an SRBM in E = Rd+ with positive definite covariance matrix, nonsingular completely-S reflection matrix R, and stationary probability π0 . Assume that the process is strong Markov, is Feller on C0 (E), and satisfies the quantitative recurrence conditions (6.5) and (6.6). If RAA is singular for some nonempty proper set A ⊊ J , then signed BAR uniqueness fails. More precisely, there exists a finite signed BAR tuple (π, ν 1 , . . . , ν d ) such that π(E) = 0,
(6.15)
π ̸= 0.
Consequently π is not a scalar multiple of π0 . R Proof. Apply Proposition 6.1. Its source χ is compactly supported, and the local boundedness of V gives E V d|χ| < ∞. Proposition 6.2 then produces the required BAR tuple. If π = cπ0 , total masses give c = 0, contradicting π ̸= 0. Corollary 6.5 (Infinite-dimensional failure). Under the assumptions of Theorem 6.4, the set of interior BAR coordinates of total mass zero contains an infinite-dimensional linear subspace. In particular, the full vector space of finite signed BAR tuples is infinite-dimensional. Proof. Choose functions ϕm ∈ Cc∞ ((0, ∞)T ) with pairwise disjoint supports and w · ∇T ϕm ̸≡ 0. Let χm and π m be PN the corresponding sources and zero potentials. If m=1 am π m = 0, linearity and the injectivity identity in the proof of PN Proposition 6.2 imply m=1 am χm = 0. The sources are nonzero and have pairwise disjoint supports, so every am is zero. The theorem concerns a singular proper principal block. Singularity of an arbitrary rectangular or nonprincipal submatrix does not yield the cancellation RAA v = 0 needed in Proposition 6.1. Conversely, every principal block of a nonsingular M -matrix is nonsingular by Lemma 3.2, so this obstruction is absent from the class covered by Theorem 2.2. 6.3. A checkable three-dimensional family The general obstruction is useful only if the recurrence assumptions can be verified without solving the stationary distribution. The next criterion is a direct way to do this for a broad positive-reflection subclass. A Z matrix means a matrix with nonpositive off-diagonal entries. Corollary 6.6 (A checkable completely-S subclass). Assume in addition that Rii = 1 and Rij ≥ 0 for all i, j. Suppose there is a symmetric positive definite matrix H such that HR is a Z matrix and Hµ < 0 componentwise. If R is nonsingular, completely S, and has a singular proper principal block, then signed BAR uniqueness fails and the conclusion of Corollary 6.5 holds. Proof. We verify the standing hypotheses of Theorem 6.4: existence, the strong Markov property, the C0 -Feller property, the recurrence certificate (6.5), and the regulator bound (6.6). First, for the existence, the strong Markov property, the C0 -Feller property, as well as (6.6), we have Proposition 6.7 ([34]). For a symmetric positive definite covariance Σ, a drift µ, and a reflection matrix R with unit diagonal, the orthant SRBM with data (Σ, µ, R) exists and is unique in law if and only if R is completely-S; when it exists it is a Feller continuous strong Markov process, and x 7→ Pt h(x) is continuous for every h ∈ Cb (E). Since Σ is positive definite, R has unit diagonal, and R is completely-S, the orthant SRBM exists, is unique in law, is strong Markov, and x 7→ Pt h(x) is continuous for every h ∈ Cb (E) (Feller continuity). Now we check the C0 Feller property, i.e. Pt C0 (E) ⊂ C0 (E) and that ∥Pt h − h∥∞ → 0 for h ∈ C0 (E). Write Z x (t) = x + µt + B(t) + RY x (t), where B(t) = Σ1/2 W (t) and each Yix is nondecreasing. Since Rij ≥ 0 and Yjx ≥ 0, the one-dimensional Skorokhod formula gives, for 0 ≤ s ≤ t, we have Yix (s) ≤ |µi |t + sup0≤u≤t |Bi (u)|. (This also verifies (6.6) since the right side has finite expectation independent of the Pinitial state x.) Hence, with Mt := |µ|∞ t + sup0≤u≤t |B(u)|∞ and CR := 1 + maxi j Rij , we have the uniform displacement bound sup |Z x (s) − x|∞ ≤ CR Mt . 0≤s≤t
25
Signed BAR uniqueness conjecture
Let h ∈ Cc (E) and suppose supp h ⊂ {|y|∞ ≤ a}. Then |Pt h(x)| ≤ ∥h∥∞ P{CR Mt ≥ |x|∞ − a} −→ 0
as |x|∞ → ∞.
Thus Pt h ∈ C0 (E) for h ∈ Cc (E). By contraction and approximation of C0 (E) by compactly supported continuous functions, the same holds for every h ∈ C0 (E). Finally, every h ∈ C0 (E) is uniformly continuous. If ωh is its modulus of continuity, then sup |Pt h(x) − h(x)| ≤ E ωh (CR Mt ). x∈E
Since Mt → 0 almost surely as t ↓ 0 and 0 ≤ ωh ≤ 2∥h∥∞ , dominated convergence gives ∥Pt h − h∥∞ → 0. Therefore (Pt ) is a strongly continuous positive contraction semigroup on C0 (E). Then we verify (6.5). Proposition 6.8 ([33, Corollary 3.2]). Let (Σ, µ, R) be orthant SRBM data with Σ symmetric positive definite and R completely-S. If there is a symmetric positive definite matrix H with HR a Z matrix and Hµ < 0 componentwise, then the SRBM is positive recurrent with a unique stationary probability π0 and is V -uniformly exponentially ergodic: there exist a locally bounded V : E → [1, ∞) and constants M, κ > 0 such that ∥Pt (x, ·) − π0 ∥TV ≤ M V (x)e−κt ,
x ∈ E, t ≥ 0.
The matrix H in the statement of the corollary is exactly such a certificate: it is symmetric positive definite, HR is a Z matrix, and Hµ < 0 componentwise, while Σ is positive definite and R is completely-S. Hence Sarantsev’s criterion supplies the stationary probability π0 and the recurrence certificate (6.5). Thus Theorem 6.4 applies. Set 11a R(a, b, c, d) = 1 1 b , cd1
(6.16)
−11 µ = −7 , −11
Σ = I3 ,
and let P be the parameter region (6.17)
0<b≤
11 , 20
56b + 25 3b + 3 ≤a≤ , 60 5
0<c≤
4 , 25
2 35 ≤d≤ . 3 46
The subset obtained by making all inequalities strict is nonempty, so P contains a genuine four-dimensional region. Theorem 6.9 (Four-parameter family). For every (a, b, c, d) ∈ P, the SRBM data in (6.16) define a nonsingular completely S, exponentially ergodic SRBM with a unique stationary probability. Its reflection matrix has the singular proper principal block 11 R{1,2},{1,2} = . 11 For every such parameter choice, signed BAR interior uniqueness fails, and the space of zero-mass interior BAR coordinates is infinite-dimensional. Proof. Every entry of R(a, b, c, d) is positive. Hence every principal submatrix is an S matrix, with the all-ones vector as a witness, and R is completely S. A direct calculation gives det R(a, b, c, d) = (b − a)(c − d). The parameter bounds imply a > b and d > c, so the determinant is positive. The block indexed by A = {1, 2} is singular, with 1 v= ∈ ker RAA , w = R{3},A v = c − d ̸= 0. −1 Consider the symmetric matrix 1 3 3 2 − 10 − 10 3 7 1 H = − 10 25 8 . 3 1 23 − 10 8 100
(6.18)
26
Y. Lu and Y. Zhu
1 79 Its leading principal minors are 12 , 20 , 80000 , so H is positive definite. Multiplication gives
(6.19)
1
3c 5 − 10
1 3d 5 − 10
c 1 HR = 8 − 50
1 d 8 − 50
a 3b 3 2 − 10 − 10
7b 1 . − 3a 10 + 25 + 8
23c 7 23d 7 3a b 23 100 − 40 100 − 40 − 10 + 8 + 100
The six off-diagonal entries are nonpositive by (6.17). Also 1 − 10 7 (6.20) Hµ = − 200 < 0. 21 − 200 Corollary 6.6 completes the proof. For this entire family the algebraic source is explicit. Let S = {(0, 0, y) : y > 0} and choose a nonconstant ϕ ∈ Cc∞ ((0, ∞)). Define Z Z ∞ (6.21) g dζ1 = g(0, 0, y)ϕ(y) dy, F1
0
Z ∞
Z g dζ2 = −
(6.22) F2
g(0, 0, y)ϕ(y) dy,
ζ3 = 0.
0
Since R1 − R2 = (0, 0, c − d)T , (6.23)
3 Z X i=1
Z ∞ Di f dζi = (d − c)
Fi
ϕ′ (y)f (0, 0, y) dy.
0
Thus (6.24)
χ(dx) = (d − c)ϕ′ (x3 ) dx3 δ0 (dx1 )δ0 (dx2 )
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 1 1 35 (6.25) R = 1 1 16 , Σ = I3 , 2 2 1 15 3
−11 µ = −7 , −11
there is a finite signed BAR tuple (π, ν 1 , ν 2 , ν 3 ) such that π(E) = 0 and π ̸= 0. Proof. The parameter choice belongs to P. Exact arithmetic gives
(6.26)
det R =
52 > 0, 225
365 − 104 203 −1 R µ = − 104 < 0. − 120 13
The matrix H in (6.18) satisfies 4 1 0 − 20 25 1 19 1 − 120 HR = − 300 300 , 433 13 17 − 3000 − 600 240
(6.27)
and (6.20) holds. Thus all assumptions in Corollary 6.6 are verified.
27
Signed BAR uniqueness conjecture
For complete explicitness, take the standard bump 1 exp − , (y − 1)(2 − y) (6.28) ϕ(y) = 0,
1 < y < 2, otherwise.
Define ζ1 , ζ2 , ζ3 by (6.21)–(6.22). Here
0 R1 − R2 = 0 , 8 − 15 so (6.29)
χ(dx) =
8 ′ ϕ (x3 ) dx3 δ0 (dx1 )δ0 (dx2 ), 15
χ(E) = 0,
χ ̸= 0.
Let Pt be the reflected semigroup and let Ki be the kernels in (6.4). Set Z ∞ ∞ X χPt dt, (6.30) π= ν i = ζi + (χPn )Ki . 0
n=0
By Proposition 6.2, all measures in (6.30) are finite and satisfy Z Lf dπ + E
3 Z X i=1
Di f dν i = 0,
f ∈ Cb2 (E).
Fi
Moreover π(E) = 0 and π ̸= 0. Therefore π cannot be a scalar multiple of the stationary probability π0 . 7. Unified understanding: RI defects and boundary algebra This section isolates resolvent insertion as the common mechanism behind both the signed BAR uniqueness theorem and the completely-S obstruction. We view a signed BAR tuple through its interior coordinate and measure the failure of this coordinate to satisfy resolvent insertion by its RI defect. After quotienting out the stationary BAR direction and the pure boundary kernel, the remaining part of the BAR kernel is exactly the RI defect quotient. In the Harrison–Reiman nonsingular M -matrix class this quotient vanishes, while in the completely-S singular-block regime the boundary source and its zero-potential lift produce a nonzero class in this quotient. We denote
ME = M(E),
M∂ =
d Y
M(Fi ),
T = Cb2 (E).
i=1
R Pd R For (π, ν) ∈ ME × M∂ , define the BAR functional A(π, ν)(f ) = E Lf dπ + i=1 Fi Di f dνi , for f ∈ T . Thus finite Pd R signed BAR tuples are precisely ker A. Define the pure boundary operator ∂R ζ(f ) = i=1 Fi Di f dζi . Since one can R identify a finite signed measure χ ∈ M(E) with the functional f 7→ E f dχ on T , we define the relation ∂R ζ = χ by the R Pd R condition that i=1 Fi Di f dζi = E f dχ holds for all f ∈ T . 7.1. The RI-defect quotient Let ZBAR := ker A and ΠE (π, ν) = π, and define the space of interior coordinates of signed BAR tuples by (7.1)
IBAR := ΠE ZBAR = {π ∈ M(E) : there exists ν ∈ M∂ with (π, ν) ∈ ker A}.
For a finite signed measure m ∈ M(E), define its resolvent-insertion defect by Z (7.2) r(m)(λ, h) := (λRλ h − h) dm, λ > 0, h ∈ C0 (E). E
Thus m satisfies the resolvent identity precisely when r(m) = 0. Set IRI := IBAR ∩ ker r, and define the RI-defect quotient QRI := IBAR /IRI . Equivalently, QRI ≃ r(IBAR ). This quotient records exactly the part of the signed BAR kernel not killed by resolvent insertion.
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Y. Lu and Y. Zhu
Lemma 7.1 (BAR quotient by RI defects). Assume that the reflected semigroup is C0 -Feller and strongly continuous, that it has a unique invariant probability π0 , and that s0 = (π0 , ν 0 ) is the stationary BAR tuple. Then there is a canonical exact sequence 0 −→ ker ∂R −→ ker A/Rs0 −→ QRI −→ 0.
(7.3) Equivalently,
ker A ≃ QRI . Rs0 ⊕ ({0} × ker ∂R )
(7.4)
Proof. Let HBAR := ker A/Rs0 . Define Γ : HBAR → QRI , and Γ([(π, ν)]) = [π]. Replacing (π, ν) by (π, ν) + c(π0 , ν 0 ) changes the interior coordinate by cπ0 . Since π0 is invariant, Z Z ∞ Z Z Rλ h dπ0 = e−λt Pt h dπ0 dt = λ−1 h dπ0 , E
0
E
E
so r(π0 ) = 0 and π0 ∈ IRI . Hence [π] is unchanged in QRI . The map Γ is surjective by the definition of QRI . We compute its kernel. Suppose Γ([(π, ν)]) = 0. Then π ∈ IRI , so Z (λRλ h − h) dπ = 0, h ∈ C0 (E), λ > 0. E
By Proposition 5.2, πPt = π for all t ≥ 0. By Proposition 5.3, we have π = cπ0 , where c = π(E). Since both (π, ν) and c(π0 , ν 0 ) are BAR tuples, (0, ν − cν 0 ) = (π, ν) − c(π0 , ν 0 ) ∈ ker A. Equivalently, ∂R (ν − cν 0 ) = 0. Thus [(π, ν)] = [(0, η)] for some η ∈ ker ∂R . Conversely, if η ∈ ker ∂R , then (0, η) ∈ ker A and its interior coordinate has zero RI defect. Hence ker Γ = {[(0, η)] : η ∈ ker ∂R }. The map η 7→ [(0, η)] is injective because if [(0, η)] = 0 in ker A/Rs0 , then (0, η) = c(π0 , ν 0 ) for some c ∈ R; the interior coordinate gives cπ0 = 0, hence c = 0 and η = 0. This proves the exact sequence (7.3). The quotient isomorphism (7.4) is the corresponding first-isomorphism statement. The sum in the denominator is direct by the same interior-coordinate argument. Remark 7.2 (Strength of the C0 Feller Assumption). The only assumption we make in this abstraction is the C0 Feller property used in the RI defect quotient which is only a soft semigroup input instead of a boundary regularity assumption. It means that Pt C0 (E) ⊂ C0 (E),
∥Pt h − h∥∞ → 0
as t ↓ 0, h ∈ C0 (E).
It is much weaker than strong Feller smoothing, the existence of transition densities, differentiability of Pt h, or closed domain C 2 regularity of the probabilistic resolvent. In particular, it does not assert that Rλ h has a classical oblique Neumann trace on the boundary. In the present argument this input is used only through standard semigroup consequences. In Proposition 5.2, it R makes t 7−→ E Pt h dπ̄ bounded and continuous for h ∈ C0 (E). The Laplace transform identity then upgrades an almost everywhere conclusion to equality for every t ≥ 0, using uniqueness of the Laplace transform [12, Chapter II]. In Proposition 6.2, strong continuity is used through the density of the generator domain in C0 (E), a standard fact for strongly continuous Feller semigroups [17, Chapter 1, Section 2]. Thus the exact sequence in the unified section could be formulated with these consequences directly: RI null interior coordinates must be signed invariant measures, and the zero potential must be injective on the centered source class. For the positive Harrison–Reiman nonsingular M matrix class, the C0 Feller property is not an extra boundary smoothness input. It is proved in Theorem 3.3. The proof uses the synchronous Lipschitz estimate for the Skorokhod map, imported from [29, Proposition 2.6], together with Brownian path continuity. The same mechanism applies more generally to any orthant SRBM for which one has a pathwise unique continuous construction and a finite horizon Lipschitz estimate of the form sup0≤s≤T |Zsx − Zsy | ≤ CT |x − y| under a synchronous coupling. Then x 7→ Pt h(x) is continuous, Pt h vanishes at infinity, and ∥Pt h − h∥∞ → 0 follow by the same argument as in the proof of Theorem 3.3. For the completely S counterexample regime, the matrix condition R completely S should not be read as a substitute for the analytic inputs in the RI reduction. It is an existence geometry condition. In the checkable subclass of Corollary 6.6, Taylor–Williams provide the orthant SRBM existence and Feller framework for completely S reflection data [34], while Sarantsev’s Lyapunov criterion supplies the stationary probability and the V uniform total variation exponential ergodicity
29
Signed BAR uniqueness conjecture
used in (6.5) [33, Corollary 3.2]. The one step regulator bound is then checked directly in Corollary 6.6. These assumptions are sufficient for the zero potential construction, but they do not imply the global RI reduction for all signed BAR tuples. Indeed, in the singular block case the boundary gauge and zero potential construction produce a nonzero RI defect class. Thus the completely S hypothesis alone is not a uniqueness regularity assumption. The minimal zero potential input is also source specific. For a given centered source χ, it is enough to have finite signed measures U χ and Θi χ satisfying A(U χ, Θχ) = −χ, and (U χ)(E) = 0 together with injectivity U χ = 0 ⇒ χ = 0 on the source class under consideration. The quantitative recurrence and in Proposition 6.2 are a convenient R ∞ regulator assumptions P sufficient package: they imply total variation convergence of 0 χPt dt and n≥0 (χPn )Ki , justify the Poisson identity, and give the injectivity needed to show that a nonzero centered source yields a nonzero zero mass interior BAR coordinate. The positive theorem in the Harrison–Reiman Class. In the Harrison–Reiman nonsingular M -matrix setting, Theorem 4.1 says that every signed BAR tuple satisfies the resolvent identity. Equivalently, r(IBAR ) = 0 and QRI = 0. Then Lemma 7.1 reduces the quotient ker A/Rs0 to the pure boundary kernel. The latter is killed by Proposition 5.4, i.e. ker ∂R = {0}. Consequently ker A = Rs0 , which is the signed uniqueness conclusion of Theorem 2.2. −1 This formulation separates the two uses of active-block invertibility. First, RAA defines the active projection −1 LA v = v − RA RAA vA ,
LA Ri = 0,
i ∈ A.
−T This is the algebraic input behind the measure–Neumann resolvent insertion. Second, RAA prescribes active oblique jets on −T T SA . If the active normal gradient is a, then (Di f |SA )i∈A = RAA a, so invertibility of RAA permits the choice a = RAA ψ. The induction over strata in Proposition 5.4 uses exactly this prescription.
The completely-S obstruction.. In this section, we advance our understanding of the zero potential construction as a device for cancelling the source term in the BAR but as a way of placing explicit nonzero elements into QRI . Starting from the singular boundary gauge, one has a boundary source R ∞identity ∂R ζ = χ. Thus the boundary gauge alone has BAR defect +χ. The zero potential Qχ = (Uχ , Θχ ), where Uχ = 0 χPt dt, is constructed so that its BAR contribution is exactly the opposite defect: A(Qχ ) = −χ. Therefore we have Qχ + (0, ζ) = (Uχ , Θχ + ζ) ∈ ker A, so the zero potential turns the boundary gauge into a genuine signed BAR tuple. However, this cancellation does not make the source χ disappear. R Instead, we shows that χ reappears as the resolvent insertion defect of the interior measure Uχ : r(Uχ )(λ, h) = − E Rλ h dχ. Hence, if χ ̸= 0, then Uχ cannot satisfy the resolvent insertion identity; otherwise the strong continuity of the semigroup would imply that χ vanishes on all functions in C0 (E), forcing χ = 0. Consequently, the zero potential is the mechanism that converts the singular boundary source into a concrete nonzero class. 0 ̸= [Uχ ] ∈ QRI . This is why the completely S counterexample is best understood as an RI defect: the boundary algebra creates the centered source χ, and the zero potential lifts that source into a genuine BAR tuple whose interior coordinate carries a nonzero resolvent insertion defect. Proposition 7.3 (Boundary sources give RI-defect classes). Assume the semigroup is strongly continuous on C0 (E) and that the zero-potential construction of Proposition 6.2 is available for a centered source χ. Write Z ∞ Qχ = (U χ, Θχ),
Uχ =
χPt dt,
Θi χ =
0
∞ X
(χPn )Ki .
n=0
If ζ ∈ M∂ satisfies ∂R ζ = χ, then Qχ + (0, ζ) = (U χ, Θχ + ζ) ∈ ker A.
(7.5)
Its image under the map in Lemma 7.1 is the class [U χ] ∈ QRI , and Z (7.6) r(U χ)(λ, h) = − Rλ h dχ, λ > 0,
h ∈ C0 (E).
E
In particular, if χ ̸= 0, then [U χ] ̸= 0 in QRI . R R Proof. By Proposition 6.2 and fact that ∂R ζ = χ, thus A(Qχ)(f ) = − E f dχ and A(0, ζ)(f ) = E f dχ holds for all f ∈ T . Adding the two identities gives (7.5). Hence U χ ∈ IBAR R and its class in the RI-defect quotient is [U χ]. It remains to identify its defect. For h ∈ C0 (E) put F (t) = E Pt h dχ. The total-variation convergence in Proposition 6.2 justifies the following Fubini calculation: Z ∞Z Z ∞ Z ∞ r(U χ)(λ, h) = Pt (λRλ h − h) dχ dt = λ e−λs F (t + s) ds − F (t) dt 0
E
0
0
30
Y. Lu and Y. Zhu
Z ∞
−λu
F (u)(1 − e
=
Z ∞ ) du −
0
Z ∞ F (u) du = −
0
e
−λu
Z F (u) du = −
0
Rλ h dχ. E
If [U χ] = 0 in QRI , then r(U χ) = 0, hence χ(Rλ h) = 0 for all λ > 0 and h ∈ C0 (E). Since the semigroup is strongly continuous on C0 (E), Z ∞ λRλ h = e−s Ps/λ h ds −→ h in C0 (E) 0
as λ → ∞. Therefore χ(h) = 0 for every h ∈ C0 (E), and the finite Radon measure χ is zero. Thus χ ̸= 0 implies [U χ] ̸= 0. Suppose that R is nonsingular and that RAA is singular for some nonempty proper subset A ⊊ J . Put T = J \ A and choose 0 ̸= v ∈ ker RAA , where w = RT A v. Since R is nonsingular, w ̸= 0. On SA , the local boundary symbol is X (7.7) vi Di f = (RAA v) · ∇A f + (RT A v) · ∇T f. i∈A
Because RAA v = 0, only the tangential derivative w · ∇T f remains. Proposition 6.1 turns this symbol calculation into a boundary source: for a compactly supported smooth density ϕ on the open stratum with w · ∇T ϕ ̸≡ 0, it constructs boundary measures ζi and a finite nonzero centered measure χ supported on SA such that ∂R ζ = χ. Under the recurrence and regulator hypotheses of Proposition 6.2, Proposition 7.3 gives (U χ, Θχ + ζ) ∈ ker A and places its interior coordinate into the RI-defect quotient as 0 ̸= [U χ] ∈ QRI . 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). Therefore a global resolvent insertion theorem cannot hold in this singular-block regime once the zero-potential lift is available. Acknowledgment The authors would like to thank Jose Blanchet for bringing this open problem to our attention and encouraging us to pursue an AI-based solution. We are also grateful to Jose Blanchet, Yufan Chen and Wenhao Yang for their valuable feedback on this manuscript. The authors are also grateful to Bin Dong, Xiao Ma, Jiajin Li and Jianfeng Lu for their insightful discussions regarding the application of AI in mathematical proving and the formulation of our AI usage disclosure. Appendix A: A stable example with g = Rλ h ∈ / C 2 (E) This appendix gives a concrete example in which the probabilistic resolvent g = Rλ h is not in C 2 (E). The point is that interior smoothness and one-sided oblique flatness on open faces do not guarantee closed-domain C 2 regularity at a corner. We choose a smooth nonnegative source h ∈ Cc∞ (E ◦ ) such that h ̸≡ 0 but h(0) = 0. We first justify that g(0) > 0. Since h ≥ 0, h ̸≡ 0, and supp h ⊂ E ◦ , choose z∗ ∈ E ◦ , r > 0, and ch > 0 such that B(z∗ , r) ⊂ E ◦ and h ≥ ch on B(z∗ , r). Let Γ be the Harrison–Reiman Skorokhod map. On [0, 2], Γ is Lipschitz with constant KΓ . Define γ(t) = tz∗ ,
0 ≤ t ≤ 1,
γ(t) = z∗ ,
1 ≤ t ≤ 2.
Since γ stays in E, Γ(γ) = γ. For the SRBM started from zero, the free input is X(t) = µt + W (t). Put b(t) = γ(t) − µt. Since Brownian motion has full support in C0 ([0, 2], R3 ), the event A = sup0≤t≤2 |W (t) − b(t)| < r/KΓ has positive probability. On A, the free input X is within r/KΓ of γ, and therefore the reflected path Z 0 = Γ(X) is within r of γ. R2 Since γ(t) = z∗ for 1 ≤ t ≤ 2, we have Z 0 (t) ∈ B(z∗ , r) throughout [1, 2] on A. Hence g(0) ≥ P(A)ch 1 e−λt dt > 0. On the other hand, if g were C 2 up to the corner and satisfied the exact face conditions, those conditions would force ∇g(0) = 0 and D2 g(0) = 0. The resolvent equation at the corner would then give 0 = h(0) = (λ − L)g(0) = λg(0), contradicting g(0) > 0. Consider d = 3, Σ = I3 , and 1 0 − 12 824 1 R−1 = 4 8 2 ≥ 0. R = − 21 1 0 , 7 1 248 0 −2 1 Thus R is a nonsingular M -matrix. Let µ = −R1; then R−1 µ = −1 < 0, so the data are stable in the sense of (2.2).
Signed BAR uniqueness conjecture
31
Choose h ∈ Cc∞ (E ◦ ) with h ≥ 0 and h ̸≡ 0, and set g = Rλ h for some λ > 0. Since the Brownian input has full support on compact time intervals and the Harrison–Reiman Skorokhod map is continuous, the SRBM started from the origin has positive probability of entering a ball on which h > 0 and then remaining there for a nonzero time interval. Hence R∞ g(0) = E0 0 e−λt h(Zt ) dt > 0. Also h(0) = 0, because h is supported in the interior. Assume, for contradiction, that g ∈ C 2 (E) in the closed-domain sense. On each open face Fi◦ , the one-sided derivative identity of Proposition 3.9 applies in the feasible direction Ri . Since Lx Ri = 0 for x ∈ Fi◦ , it gives Di g = 0 on Fi◦ . By continuity of the first derivatives, these identities extend to the origin as RT ∇g(0) = 0. Since R is invertible, ∇g(0) = 0. Let H = D2 g(0). For j ̸= i, differentiating Di g = Ri · ∇g = 0 in the tangential direction ej along Fi◦ and then letting the tangential point tend to the origin gives RiT Hej = 0. Equivalently, offdiag(RT H) = 0. Thus RT H = diag(d1 , d2 , d3 ) for some real numbers d1 , d2 , d3 , and therefore 8d1 4d2 2d3 1 H = R−T diag(d1 , d2 , d3 ) = 2d1 8d2 4d3 . 7 4d1 2d2 8d3 Since H is symmetric, comparison of the (1, 2), (2, 3), and (1, 3) entries gives d1 = 2d2 , d2 = 2d3 , and d3 = 2d1 . Hence d1 = d2 = d3 = 0, so H = 0. The interior resolvent equation gives (λ − L)g = h on E ◦ . If g ∈ C 2 (E), the left-hand side extends continuously to the origin. Since h(0) = 0, ∇g(0) = 0, D2 g(0) = 0, and Σ = I3 , this gives 0 = h(0) = (λ − L)g(0) = λg(0), contradicting g(0) > 0. Consequently, for this stable Harrison–Reiman SRBM and this smooth compactly supported interior source, Rλ h ∈ / C 2 (E). References [1] B ERMAN , A. and P LEMMONS , R. J. (1994). Nonnegative Matrices in the Mathematical Sciences. Classics in Applied Mathematics 9. Society for Industrial and Applied Mathematics, Philadelphia. https://doi.org/10.1137/1.9781611971262 [2] B RAMSON , M., DAI , J. 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