{"id":"54b03327-b7c0-495b-9f77-6b0e49717379","arxiv_id":"2502.01200","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For reflected constrained dynamics, the deterministic Mortensen estimator is the small-noise limit of the stochastic filter, and its value function solves a Hamilton-Jacobi equation with two different boundary conditions.","lead":"This paper studies how to estimate the position of a moving object that is forced to stay inside a bounded region, using noisy real-time measurements. It proves that the deterministic minimum-energy estimator and the small-noise limit of the probabilistic filter agree, and it gives convergence rates for a practical penalized approximation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 2.8 is a Laplace principle for the deterministic PDE (24) driven by a C1 observation path, while actual stochastic-filter observation paths are a.s. nowhere differentiable; the paper gives no argument transferring the result to the filter.","rationale":"The deterministic PDE results (Theorems 2.2, 2.5) and the Laplace principle for smooth observations may be correct as stated; the paper's quantitative penalisation convergence and viscosity analysis are substantial. But the central advertised claim—equivalence between the constrained observer and the small-noise stochastic filter—requires applying Theorem 2.8 to the filter's observation process. The paper freezes a C1 realisation (Section 2.3.3) and never returns to the actual Brownian observation path. In the classical James-Baras/Fleming framework the same issue arises, but there the transfer is either stated as a conditional LDP for smooth paths or handled by a joint LDP over the observation noise; this paper does neither. The h=0 example shows the two objects are genuinely different: for a rough path, the PDE cost ∫|dot y|^2 is not defined and any approximation can produce arbitrary limits, while the filter limit is y-independent. Hence this is the most load-bearing concern. The reader's weakest_assumption (boundary regularity of q~^ε from [Fri08]/[Huc90]) is related but distinct; even if that regularity is granted, the observation-path mismatch remains. Verdict stays CONDITIONAL because the deterministic theorems may still stand, but the stochastic-filtering claim needs an explicit additional statement and proof.","tokens_in":20570,"tokens_out":26629,"duration_ms":292530,"concrete_test":"Take the solvable case G=[0,1], b=0, h=0, q^ε_0≡1, Φ=0. The robust Zakai density is then the reflected-Brownian transition density and its log-Laplace limit is independent of y; compute it explicitly. For a fixed non-differentiable observation path y (e.g. a Brownian path), compare this limit with the limit obtained from Theorem 2.8 applied to a sequence y_n∈C^1 converging to y in C[0,T] with ∫|dot y_n|^2→∞. If the limits differ, Theorem 2.8 cannot be the small-noise limit of the stochastic filter for actual observation paths.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 2.8 (Section 4) works with the deterministic no-flux PDE (24), whose potential is (1/2ε)|dot y(t)-h(t,x)|^2, and Section 2.3.3 says a C1 realisation y is frozen before deriving (24). However, the observation process in the stochastic filtering model is dY^ε_t = h(t,X^ε_t)dt + √ε dB'_t (Eq. 18), whose paths are almost surely nowhere differentiable; no C1 realisations exist. Thus Theorem 2.8, as stated, concerns a PDE driven by a synthetic smooth path, not the conditional law of the stochastic filter. To make the advertised small-noise connection, one would need to prove that the robust/pathwise Zakai solution, which is well defined for every continuous observation path, satisfies the same Laplace principle for the actual (non-differentiable) paths, or to formulate a joint large-deviation statement over observation paths; neither appears in the paper. This is not a minor regularity issue: for h=0 the filter is observation-independent, while the PDE limit for smooth y depends on ∫|dot y|^2, so the two limits cannot coincide for rough y without an additional limiting or regularisation prescription.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper considers deterministic minimum-energy state estimation for the reflected differential inclusion (1) in a bounded convex domain G. It defines the cost-to-come V in (10), proves a quantitative convergence rate κ^{-1/4} for a Moreau–Yosida penalised estimator (Theorem 2.2), and characterises V as a viscosity solution of the HJB equation (14) with different Neumann-type boundary conditions for sub- and super-solutions, with uniqueness when b·n≤0 (Theorem 2.5). It also states a Laplace principle (Theorem 2.8) claiming that the small-noise reflected filtering density concentrates on minimisers of V, via a duality argument for the no-flux PDE (24).","tokens_in":20769,"tokens_out":10877,"duration_ms":114307,"significance":"The deterministic results are a substantial extension of earlier work [Cha+23; JB88a; Fle97] to reflected dynamics: the explicit convergence rate, the asymmetric boundary conditions, and the uniqueness under an inward-pointing condition are valuable, and the viscosity proofs are detailed. However, the advertised link to stochastic filtering is not proven: Theorem 2.8 concerns the deterministic PDE (24) with a frozen C1 observation path, not the conditional law of the filter (17)-(18). Since actual observation paths are a.s. nowhere differentiable, and for h=0 the filter is independent of Y while the PDE limit contains ∫|ẏ|^2, the stochastic claim is currently unsupported. The deterministic part remains a solid contribution, but the title and abstract overstate the achieved connection.","major_comments":[{"comment":"The theorem is a statement about the deterministic no-flux problem (24) driven by a fixed C1 path y, and the proof in Section 4 uses only the PDE (24) and the duality identity (34). The observation process in (18) does not have C1 paths, so this theorem does not describe the conditional law of the stochastic filter. The sentence after Theorem 2.8 ('As ε→0, this tells that the non-normalised density q̃^ε concentrates...') is not justified, and the claim in the section introduction that π^ε_t concentrates on minimisers of V is not proved. Moreover, for h=0 the filter is independent of Y, whereas the limit of the PDE expression depends on (1/2)∫|ẏ|^2; hence the two limits cannot coincide without an additional regularisation or joint large-deviation argument. The paper must either prove a Laplace principle for the robust/pathwise Zakai equation valid for arbitrary continuous observation paths, or explicitly restrict the announced stochastic filtering claim.","section":"Section 2.3.3, Theorem 2.8"},{"comment":"The assertion that for every ᾱ∈R^n there exists an admissible pair (x_ω(0),ω) ∈ A^G_{t,x} with continuous ω(t)=ᾱ and x_ω(s)∈G for s<t, and for x∈∂G under [b(t,x)+ᾱ]·n(x)≥0, is used to derive the sub-solution property by taking ᾱ=∇φ(t,x), but no proof is given. This is a nontrivial controllability statement for a sweeping dynamics with unilateral constraint; it should be stated and proved as a lemma. Without it, the sub-solution part of Theorem 2.5 is incomplete.","section":"Section 3.2, proof of Theorem 2.5(i)"},{"comment":"The proof cites [PR14, Proposition 4.16-I/II] for moment bounds on reflected SDEs, but the passage from these process estimates to the claimed uniform bound sup |V^ε_Φ - V^0_Φ| ≤ C ε^{1/4} is not fully justified. The infimum in (35) is over adapted control processes, while V^0_Φ is a deterministic optimal control value; comparing the two requires an argument (e.g., a measurable selection or a dynamic programming estimate) that (35) is stable under ε→0. The one-sentence 'we then plug this into the minimisation' with a Cauchy–Schwarz bound is insufficient as written.","section":"Section 4, Lemma 4.1"}],"minor_comments":[{"comment":"The bibliography lists the identical reference as [JB88a] and [JB88b]; these should be consolidated into one entry.","section":"References"},{"comment":"In the display of the observation process, 'h(t,X^κ,ε_t)' should read 'h(t,X^ε_t)'.","section":"Section 2.3.3"},{"comment":"The citation 'Lemma 3.2-(i)' is incorrect: Lemma 3.2 is a single statement, and the boundedness of V^κ used there follows from Lemma 3.1.","section":"Proof of Theorem 2.2"},{"comment":"The codomain of Φ is written as 'Φ : G → 0'; this should be 'Φ : G → R'.","section":"Theorem 2.8 and its proof"}],"recommendation":"major_revision","confidential_remarks":"The deterministic estimation part is solid and publishable, but the stochastic filtering result as advertised is not established. If the authors can reframe the paper as a deterministic observer/viscosity paper with a formal connection, a revision might work; otherwise the stochastic claim needs a genuinely new argument. I would urge the editor to require the stochastic claim to be either proved or explicitly retracted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: Theorem 2.5 is a genuine new result and the proof looks sound. Theorem 2.8 is not, as stated, a result about the stochastic filter—it is a Laplace principle for a deterministic PDE with a frozen C1 observation path, and the paper does not bridge the gap to Brownian observations.\n\nWhat is actually new and valuable: the two different Neumann boundary conditions for subsolutions and supersolutions of the constrained cost-to-come, and the comparison/uniqueness under b·n ≤ 0. That is a real advance over the earlier one-dimensional half-line setting. The quantitative penalisation convergence in Theorem 2.2 is also carefully done, and the Section 3 proofs are reasonably detailed.\n\nSoft spots. First, the technical assertion in the subsolution proof that one can find admissible trajectories with prescribed terminal control, including at the boundary, is stated without proof or reference. It may be true, but it needs a lemma. Second, the numerical illustration promised in the introduction is not in v1; that is minor. Third, and more important, Theorem 2.8 is oversold. The derivation fixes a C1 realisation y of the observation process. Brownian observation paths are almost surely nowhere differentiable, so the q̃ε in the theorem is not the conditional density for any actual observation path. The paper gives no argument, through robust filtering, joint large deviations, or path approximation, that the Laplace principle transfers to the real filtering problem. The h=0 case makes the gap visible: the true filter is independent of Y and concentrates according to the signal's own LDP, while the PDE rate function still contains ∫|dot y|². So the theorem is a correct statement about a controlled PDE, but not the stochastic filtering connection advertised in the abstract. The imported regularity from [Fri08] and duality from [Huc90] are probably fine for the PDE reading; they are not the issue.\n\nOverall, the HJB part deserves a serious referee and is likely publishable. The filtering part needs either a real transfer argument or an honest reframing. I would send it to review and ask the authors to address the path-regularity gap; as it stands, I would not repeat the filtering claim in print.","headline":"The HJB boundary-condition analysis is solid and new, but the advertised small-noise filtering limit only holds for frozen smooth observation paths, not for the actual Brownian observation process.","tokens_in":21341,"tokens_out":4197,"would_cite":true,"duration_ms":48453,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49L25","93E11","60F10","49J40"],"pacs":[],"model":"deepseek-v4-flash","headline":"A reflected, non-smooth constrained observer is characterized by an HJB equation with mismatched boundary conditions, and a small-noise stochastic filter concentrates on its minimizers.","keywords":["constrained estimation","sweeping process","viscosity solution","Neumann boundary condition","stochastic filtering","large deviations","Laplace principle","Mortensen observer"],"falsifier":"A concrete check: take G=(0,1), drift b(x)=1, observation h=0, observation rate ˙y≡0, and initial log-density ψ(x)=(x-1/2)^2/2; solve the one-dimensional no-flux parabolic problem (24) exactly by spectral expansion and compare the left side of Theorem 2.8 with inf_{x∈[0,1]}(Φ(x)+V(t,x)) for a nonconstant continuous Φ such as Φ(x)=x. If -ε log ∫ $e^{{-Φ/ε}}$ q~ε(t,x)dx fails to approach inf(Φ+V), or if the limit instead equals the infimum of a function satisfying only the 1/2 super-solution boundary condition, the Laplace principle as stated is false.","tokens_in":20354,"feed_emoji":"🎯","tokens_out":8103,"duration_ms":83017,"temperature":0.7,"pith_summary":"This paper studies how to estimate the hidden state of a dynamical system that is forced to stay inside a bounded convex domain by a normal-cone reflection, when noisy observations arrive in real time. The central result is that the deterministic cost-to-come function for this non-smooth constrained dynamics is a viscosity solution of a Hamilton-Jacobi-Bellman equation, but its sub-solutions and super-solutions obey different Neumann boundary conditions; uniqueness holds only under an inward-pointing drift condition. The paper then proves a Laplace principle connecting this deterministic value function to stochastic filtering: as the process and observation noise vanish, the reflected filter density concentrates on the minimizers of the cost-to-come, which is precisely the Mortensen observer. Because the reflecting dynamics are well posed only forward in time, the usual time-reversibility argument used in smooth filtering is unavailable; the proof works around this with a duality identity. If the result is right, deterministic constrained observers and small-noise stochastic filters are asymptotically equivalent for a broad class of non-smooth systems.","feed_headline":"Small-noise filters match constrained observers","feed_subtitle":"Reflected stochastic filtering concentrates on minimizers of a deterministic cost-to-come, proved via a Laplace principle.","key_machinery":"The central mechanism is the logarithmic transformation that turns a linear parabolic equation into a nonlinear HJB equation: with q~ε the solution of the no-flux Zakai-type problem (24), define V^ε = -ε log q~ε. Under the classical existence result [Fri08], V^ε solves the HJB equation with boundary condition b·n + (1/2)∂V^ε/∂n = 0. A duality identity ∫ Φ^ε(0,x) q~ε(t,x)dx = ∫ Φ^ε(t,x) q~ε(0,x)dx from [Huc90] links the filter density to the solution of the backward dual parabolic problem, and a verification argument gives a control representation of V^ε with reflected controlled dynamics. Comparing V^ε with $V^{0}$ yields Lemma 4.1 and then the Laplace principle. Viscosity solutions here are understood in the Barles-Lions Neumann sense, where the boundary condition is relaxed by allowing the equation to hold at the boundary.","core_discovery":"On the paper's own terms, the central discovery is twofold. First, the constrained cost-to-come V defined by the infimum over square-integrable disturbances that drive the reflected trajectory to x at time t satisfies the HJB equation (14) in G, with the sub-solution boundary condition b·n + ∂V/∂n = 0 and the super-solution boundary condition b·n + (1/2)∂V/∂n = 0 on ∂G (Theorem 2.5). Since these boundary conditions differ, a comparison principle is not proved in general; however, when b·n ≤ 0 on ∂G the paper obtains uniqueness and recovers the state-constrained viscosity solution of [CL90]. Second, for the reflected diffusion with small noise √ε, the normalized filter density obeys the Laplace principle -ε log ∫ $e^{{-Φ/ε}}$ q~ε(t,x)dx → inf_x(Φ(x)+V(t,x)) for continuous Φ (Theorem 2.8), showing that the stochastic filter concentrates on the same minimizers as the deterministic observer. A quantitative convergence result, |V^κ - V| ≤ $Cκ^{{-1/4}}$, justifies replacing the non-smooth inclusion by a penalized smooth dynamics.","pith_inferences":["An extension the authors leave implicit: the same Laplace principle should hold for the normalized conditional density π^ε, because the normalizing constant is independent of x and drops out of the infimum; this would make the concentration statement directly usable for particle-filter or ensemble Kalman initializations.","A sharper rate for the penalization convergence is plausible: replacing the crude distance estimate of Lemma 3.1 by a refined reflection-coupling inequality may improve κ^{-1/4} to κ^{-1/2}, testable in one dimension where the penalized and reflected trajectories can be computed explicitly.","The duality technique, which bypasses the missing comparison principle, could be transferred to sweeping processes with time-dependent constraint sets G(t) or to backward reachability problems, where non-reversibility similarly prevents a direct HJB comparison.","The factor-of-two boundary asymmetry predicts that for outward-pointing drift the small-noise filter may be better approximated by the super-solution boundary condition than by the sub-solution one; this could be checked numerically on a half-space example and, if confirmed, would guide filter design in constrained problems."],"forward_implications":["Under an inward-pointing drift, the constrained HJB equation (14)-(16) has a unique viscosity solution, so the deterministic observer is well-defined and can be computed by standard numerical schemes for viscosity solutions.","The reflected filter's conditional density satisfies a large deviation principle with rate function x ↦ Φ(x)+V(t,x), so in the small-noise limit the optimal estimator is the minimizer of the cost-to-come, extending the Kalman-Bucy concentration to non-smooth constrained dynamics.","The O(κ^{-1/4}) uniform convergence of the penalized value function gives a quantitative justification for approximating the reflecting inclusion by the smooth penalized dynamics in numerical implementations.","The boundary-condition mismatch (the factor 1/2 in the normal derivative) is not an artifact: it reflects the fact that reflected diffusions spend zero Lebesgue time on the boundary while constrained deterministic trajectories can slide along it; any numerical filter must respect this asymmetry near ∂G.","If b·n ≤ 0 fails, the deterministic cost-to-come may not be the unique solution of the HJB problem, so the equivalence established here relies on the drift direction at the boundary; practical observers in outward-drift domains should be treated as a separate regime."],"supporting_citations":[{"why":"Supplies the nonlinear-filtering large-deviation method: log-density converges to cost-to-come via a PDE-control approach.","marker":"[JB88a]"},{"why":"Provides the deterministic filtering framework and the log-transform estimate that the paper adapts to reflected dynamics.","marker":"[Fle97]"},{"why":"Gives the duality identity (34) for the reflected Zakai equation, the bridge between the filtered density and the dual parabolic problem.","marker":"[Huc90]"},{"why":"Supplies the classical existence and uniqueness result for the no-flux parabolic problem (24) used for q~ε and Φ^ε.","marker":"[Fri08]"},{"why":"Establishes the reflected filtering density and Zakai equation with boundary condition, the stochastic object being analyzed.","marker":"[Par78b]"},{"why":"Provides the comparison lemma for reflected curves used in the quantitative penalization convergence proof.","marker":"[Tan79]"},{"why":"Gives well-posedness of the perturbed sweeping process, used to define admissible trajectories and the Bellman principle.","marker":"[ET05]"},{"why":"Defines the viscosity notion with Neumann boundary conditions and the comparison principle used in Theorem 2.5.","marker":"[BL91]"}],"fun_headline_variants":["Small noise filters pick constrained observer paths","Filter large deviations hit constrained estimates","Constrained observer emerges from noisy filter","Reflected filters concentrate on observer minima"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the reflected filter density solves the classical no-flux parabolic problem (24) with enough boundary regularity, $C^{1}$ in time and $C^{2}$ in space away from t=0, and that the duality identity from [Huc90] applies; if the density develops singularities at the boundary, the log-transform and the verification argument connecting the stochastic filter to V collapse.","fun_headline_variants_meta":{"raw":{"variants":["Small noise filters pick constrained observer paths","Filter large deviations hit constrained estimates","Constrained observer emerges from noisy filter","Reflected filters concentrate on observer minima"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000643,"raw_usage":{"total_tokens":2960,"prompt_tokens":951,"completion_tokens":2009,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":1958}},"tokens_in":567,"tokens_out":2009,"duration_ms":15620,"temperature":1.0,"reasoning_tokens":1958,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T16:14:41.211872+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check: take G=(0,1), drift b(x)=1, observation h=0, observation rate ˙y≡0, and initial log-density ψ(x)=(x-1/2)^2/2; solve the one-dimensional no-flux parabolic problem (24) exactly by spectral expansion and compare the left side of Theorem 2.8 with inf_{x∈[0,1]}(Φ(x)+V(t,x)) for a nonconstant continuous Φ such as Φ(x)=x. If -ε log ∫ $e^{{-Φ/ε}}$ q~ε(t,x)dx fails to approach inf(Φ+V), or if the limit instead equals the infimum of a function satisfying only the 1/2 super-solution boundary condition, the Laplace principle as stated is false.","supporting_citations":[],"review_version":1}