{"id":"157f3866-85d7-48b4-b996-0567309a2650","arxiv_id":"1908.00465","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":2.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The expected value of e^{-λτ}φ(B_τ) over Brownian exit times solves Δh = λh with boundary value φ on negatively curved manifolds.","lead":"This paper fills in the missing proof of a known result: on a curved space, averaging a boundary condition over the random exit time of Brownian motion produces a smooth eigenfunction of the Laplacian. The proof matters because it supplies a citable reference for a technique researchers use to build eigenfunctions with prescribed boundary values from random paths.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof is coherent, and the only load-bearing external input is a standard cited heat-kernel bound.","rationale":"I compared the main theorem against each step of Sections 2-6. The construction of Brownian motion from the heat semigroup is standard, and the Feller property, stochastic completeness, and continuity of sample paths are justified correctly. The generator computation in Proposition 4.1 has a typo in Eq. (8), but the intended Green identity is valid with compactly supported test functions. The Dirichlet heat kernel estimates and Lemmas 5.2 and 6.2 are standard consequences of (1), (2), and volume comparison. The key Lemma 6.3 correctly writes e^{tΔ}h in terms of the shifted process and splits off the events {τ < t}; both error terms are o(t) uniformly on compacts because Brownian motion starting at distance δ from ∂D cannot exit D within time t with probability decaying exponentially in δ²/t, and the L^p shift estimate uses Proposition 6.1 for a slightly larger real part. Proposition 6.4 then identifies Δh = λh distributionally, and elliptic regularity gives smoothness. The boundary-limit proof is the standard three-region estimate: near the boundary point the continuity of φ controls the first region, uniform small-time exit estimates control the second, and Lemma 6.5 controls the tail region. I found no circularity, missing case, or unjustified interchange. The only genuine dependency is the long-time heat kernel upper bound (2), which the reader also identified as the weakest assumption; if that bound failed, the tail control in Proposition 6.1 and Lemma 6.5 would fail. However, this is a cited theorem rather than a flaw in the paper's reasoning, and it is compatible with the hypotheses. The small-time part of Lemma 6.5 does require the short-time bound (1) in addition to (2); the text mentions only (2) in that paragraph, but the omitted split is straightforward and does not change the verdict. The ACCEPT verdict stands.","tokens_in":15374,"tokens_out":33914,"duration_ms":327794,"concrete_test":"Re-derive the exit-time tail bound of Proposition 6.1 using estimate (2) for s > 1 and the short-time estimate (1) for 0 < s ≤ 1, and check that the resulting constant Cλ is finite uniformly for x ∈ D whenever Re λ > λ1; this verifies the one external input on which the main proof depends.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is Theorem 1.1: for Re λ > λ1, h(x) = E_x(e^{-λτ}φ(Bτ)) is a C∞ eigenfunction on D with eigenvalue λ and boundary value φ. I traced the full argument and found no internal inconsistency or unsupported step that threatens this claim. The most load-bearing assumption is the long-time heat kernel upper bound (2) from Grigoryan, which gives the exponential tail P_x(τ ≥ s) ≤ C e^{λ1 s} used in Proposition 6.1 and Lemma 6.5. This is an external cited estimate rather than a gap: in the pinched-negative-curvature setting it is standard and consistent with McKean's bound λ1 < 0. The small-time regime in Lemma 6.5 is handled by combining the short-time bound (1) for s < 1 with (2) for s > 1, although the text mentions only (2) in that paragraph; the split is easily supplied. The only textual slip I noticed is Eq. (8), where the integration-by-parts identity is printed as a tautology; the intended standard Green identity is valid for compactly supported φ and does not affect the argument. The Feller construction, the Dirichlet heat kernel estimates, the o(t) error terms in Lemma 6.3, the distributional identification Δh = λh, and the three-region boundary-limit estimate are mutually consistent. I therefore do not raise a substantive objection.","agreement_with_reader":"agree"},"referee_report":null,"author_rebuttal":null,"desk_editor":null,"rs_alignment":null,"lean_confirmation":null,"pith_extraction":null,"created_at":"2026-08-14T15:56:07.863198+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":null,"supporting_citations":[],"review_version":1}