{"id":"a12d9e96-b817-4875-bc50-ba7ebdefbc72","arxiv_id":"2411.17216","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For strong-Feller Markov processes killed on exiting a domain, the occupation measure conditioned on survival obeys a large deviation principle with unique rate-function zero at the quasi-ergodic distribution.","lead":"This mathematics paper proves a precise large deviation principle for where a Markov process spends its time, conditioned to stay inside a region. The rate function vanishes exactly at the quasi-ergodic distribution, giving exponential rates of convergence for occupation measures in metastable dynamics.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Step 4 of the proof of Theorem 2 does not establish (2.13): the displayed chain drops a logarithm and uses the wrong normalization, so the Jensen lower bound only gives a sublinear rate, and the claimed inequality Λ_D(V) ≥ π_D(V) fails at V=0.","rationale":"The reader's weakest_assumption pointed at the imported Perron-Frobenius/essential-spectral-radius machinery in Theorem 1; that is a legitimate verification burden, but it is not where the argument visibly breaks. The decisive defect is in Step 4 of the proof of Theorem 2, which the reader flagged as a normalization error but which is in fact a missing-logarithm/sublinear-rate error: the Jensen lower bound gives A_T/B_T ∼ Tπ_D(V), so (1/T) log(A_T/B_T) → 0, and the claimed inequality Λ_D(V) ≥ π_D(V) is also false at V=0. Since (2.13) is the theorem's advertised uniqueness-of-the-zero statement, the paper as written does not establish Theorem 2(B). I do not read this as a fatal flaw: Step 2 already proves analyticity/Gateaux differentiability of Λ_D, and the first-order perturbation formula for the eigenvalue yields exactly π_D(V) with no extra hypotheses, so a corrected Step 4 is available. The examples and the spectral-gap theorem are not the main risk. Consequently the conditional verdict stands; the needed revision is a replacement of Step 4, not a change of the overall assessment.","tokens_in":24902,"tokens_out":11781,"duration_ms":110362,"concrete_test":"Re-derive Step 4 via perturbation theory: for V ∈ bB(D), set Λ(ε) = Λ_D(εV), with eigenpair (μ_ε, φ_ε) from Theorem 1 normalized by μ_ε(φ_ε) = 1. Differentiate the eigen-equation P_1^{D,εV} φ_ε = e^{Λ(ε)} φ_ε at ε = 0 and verify that Λ'(0) = μ_D(Vφ_D) = π_D(V). Then combine with convexity of Λ_D, established from (4.4), to check that Λ_D(V) − Λ_D(0) ≥ π_D(V) for all V, whence I_D(π_D) = 0 and no other measure achieves zero. If this perturbation computation fails or requires extra regularity of V beyond measurability, the central zero-set claim (2.13) is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 2(B) is not completed as written. In Step 4 the goal is to show π_D lies in the subdifferential of Λ_D at 0, i.e. Λ_D(V) − Λ_D(0) ≥ π_D(V) for all bounded measurable V. The paper instead states the equivalent target as Λ_D(V) ≥ π_D(V), which at V=0 would give Λ_D(0) ≥ 0, contradicting Theorem 1(1), where Λ_D(0) < sup_D 0 = 0. The Jensen lower bound also contains a missing logarithm: from log E ≥ log(A_T/B_T) with their own asymptotic A_T/B_T ∼ Tπ_D(V)ν(φ_D)/ν(φ_D), the limit of (1/T) log(A_T/B_T) is 0, not π_D(V); the displayed chain evaluates (1/T) A_T/B_T instead of (1/T) log(A_T/B_T). Moreover, the expansion applies the spectral rate Λ_D(V) to the unperturbed killed semigroup P^D_t, whose true rate is Λ_D(0). These are not cosmetic normalization typos: the displayed computation cannot yield a positive linear lower bound. The intended claim is nevertheless likely repairable through Step 2: by first-order perturbation of the simple eigenpair, d/dε Λ_D(εV)|_{ε=0} = μ_D(Vφ_D) = π_D(V), and convexity of Λ_D (from (4.4) as a limit of convex conditional log-Laplace transforms) then gives the tangent lower bound and hence (2.13). But that argument is not what is written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper proves a large deviation principle (LDP) for the occupation measures L_T of a strong Feller Markov process conditioned to survive in a domain D up to time T. The rate function I_D is expressed as the Legendre transform of the difference Λ_D(V)-Λ_D(0), where Λ_D(V) is the log spectral radius of the killed Feynman-Kac semigroup P^{D,V}_t on a weighted space. Theorem 2 asserts that I_D is good, has a unique zero equal to the quasi-ergodic distribution π_D=φ_D μ_D, and that the LDP holds on P(D) with the τ-topology. Theorem 1 establishes existence, uniqueness, and exponential convergence of the principal eigenpair of P^{D,V}_t for bounded measurable V. Corollary 2 identifies I_D with the Dirichlet form minus the ground-state eigenvalue in the reversible case. Examples include elliptic SDEs driven by rotationally invariant α-stable processes, the kinetic Langevin process, and the overdamped Langevin process.","tokens_in":25105,"tokens_out":16686,"duration_ms":132407,"significance":"If completed, the result would be a substantial contribution to quasi-stationary/ergodic theory and the large-deviation analysis of killed Markov processes: it gives a τ-topology LDP with an explicitly identified unique minimizer, provides a spectral-gap result for Feynman-Kac semigroups that is of independent interest, and covers non-reversible and jump-type models. The paper's strengths are its coherent Gärtner-Ellis architecture, the explicit use of the spectral data to characterize the zero set, and the detailed verification of assumptions for the stable-driven SDEs. The examples are nontrivial and the paper is generally well organized. However, two load-bearing points in the proof need repair: Step 4 of the proof of Theorem 2 contains a normalization/log error, and the import of the Perron-Frobenius theorem from [35] for arbitrary measurable V is not fully justified in the text.","major_comments":[{"comment":"The proof of (2.13) is not valid as written. The subdifferential condition should be Λ_D(V) ≥ Λ_D(0)+π_D(V); the displayed (4.7), Λ_D(V)≥π_D(V), fails at V=0 since Λ_D(0)<0 by Theorem 1(1). In the following chain, the Jensen lower bound is formed correctly, but the subsequent asymptotics normalize the unperturbed semigroup P^D_T by e^{-Λ_D(V)T} although its true rate is Λ_D(0), and the final limit computes (1/T)·(ratio) instead of (1/T)·log(ratio). Thus the displayed computation cannot yield the claimed positive linear lower bound. The intended argument is available from Step 2: first-order perturbation of the simple eigenpair gives d/dε Λ_D(εV)|_{ε=0}=μ_D(Vφ_D)=π_D(V), and convexity of Λ_D from (4.4) yields the tangent lower bound; this replacement should be written out.","section":"Section 4, Step 4 (proof of Theorem 2(B))"},{"comment":"The proof of Theorem 1 invokes [35, Theorem 4.1] to obtain the unique eigenpair and exponential convergence for P^{D,V}_1, but the hypotheses of that theorem are not stated or verified for arbitrary bounded measurable V. The manuscript only checks strong Fellerity of P^{D,V}_t (Step 2) and the weighted norm bound in Section 2.2.1; it does not show that the Lyapunov and irreducibility conditions required by [35, Theorem 4.1] hold for the twisted kernel. Since Theorem 1 is the basis for the log-Laplace limit (4.4), the authors should either state the precise theorem and verify its assumptions for all V∈bB(D), or prove the spectral gap directly.","section":"Section 3, Step 3"}],"minor_comments":[{"comment":"The rate function in Theorem 3 is given as sup_{V∈C_b(D)}, while Theorem 2, (2.10), uses sup_{V∈bB(D)}. Under the τ-topology the Legendre transform should be over bB(D); please reconcile this mismatch, for instance by stating Theorem 3 with bB(D), which is consistent with condition (1) and with the proof in Step 1.","section":"Section 2.2.3 and Theorem 3"},{"comment":"The phrase 'expotential *-tightness' should read 'exponential *-tightness'.","section":"Section 4, Step 3"},{"comment":"The existence of an initial distribution ν=hπ with h∈L2(S,π), h1_{D^c}=0 π-a.e., and ν(W)<∞ is asserted without proof; it holds for instance for h=c1_K with c>0 and K a compact subset of D of positive π-measure, so a brief justification would be helpful.","section":"Section 5, Step 2"},{"comment":"The inserted remark about Patrick Cattiaux is not appropriate for a research paper and should be removed.","section":"Section 2.2.5, footnote 3"},{"comment":"The notation is inconsistent: the solution is sometimes written X_t(x) and sometimes X_t; please unify. Also, the proof uses T both for the fixed time horizon and as a variable inside the Gronwall argument, which can confuse the reader.","section":"Section 6.2.3, Proposition 3"}],"recommendation":"major_revision","confidential_remarks":"The paper is heavily self-referential: Theorem 1, the spectral engine, is imported from the authors' prior papers [35, 61]. This is not circular, but the revision should make the imported hypotheses checkable by stating the theorems. The core LDP result is plausible and fits the journal's scope; with the Step 4 repair and the Theorem 1 verification, the paper would be acceptable. The footnote about Patrick Cattiaux should also be removed before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real advance and worth engaging, but the proof of the main theorem's central claim is not completed as written. The LDP for conditioned occupation measures of strong-Feller processes, with rate function vanishing only at the quasi-ergodic distribution, is genuinely new. Theorem 1 extending the spectral gap to bounded measurable potentials is also substantial and appears new. The examples—alpha-stable-driven SDEs, kinetic Langevin, overdamped Langevin—are rich and nontrivial.\n\nThe proof architecture is a standard Gartner-Ellis with a spectral engine, and Steps 1–3 (log-Laplace limit, Gateaux differentiability, exponential tightness) are coherent. The problem is Step 4. The displayed target (4.7), Λ_D(V) ≥ π_D(V), is not equivalent to π_D ∈ ∂Λ(0) because Λ(V) = Λ_D(V) − Λ_D(0). At V=0, (4.7) gives Λ_D(0) ≥ 0, contradicting Theorem 1(1). The correct inequality is Λ_D(V) − Λ_D(0) ≥ π_D(V). The Jensen computation also misses a logarithm: they bound log E ≥ log(A_T/B_T) but then evaluate (1/T)A_T/B_T instead of (1/T)log(A_T/B_T). With their own expansions A_T/B_T ∼ Tπ_D(V), the limit is 0. They also normalize the unperturbed killed semigroup by e^{-Λ_D(V)T} rather than e^{-Λ_D(0)T}. So the displayed chain cannot prove the positive linear lower bound. That is not a cosmetic typo: the uniqueness-of-zero claim (2.13) is unproved as written.\n\nThat said, the intended claim is likely repairable. Gateaux differentiability plus convexity of Λ_D (from (4.4)) means the subdifferential at 0 is a singleton, and first-order perturbation of the simple eigenpair gives d/dε Λ_D(εV)|_{ε=0} = π_D(V). That route is suggested by their Step 2 but not written. The reliance on [35] and [61] for Perron–Frobenius and essential spectral radius is not circular—those are independent published theorems—and the V≡0 case of Theorem 1 is essentially their prior result, so citing it is legitimate. The delegation of support-theorem verification in the stable example to a sketch is thin but acceptable.\n\nThis paper is for specialists in quasi-stationary theory, metastability, and large deviations. It deserves a serious referee: send it out, but require a corrected Step 4 and a check that the perturbation argument yields exactly (2.13). My verdict would be conditional acceptance.","headline":"Genuinely new LDP for quasi-ergodic distributions with a strong spectral engine, but Step 4 of Theorem 2's proof has a normalization error that leaves the uniqueness claim unproved as written; repairable and worth refereeing.","tokens_in":25785,"tokens_out":3393,"would_cite":true,"duration_ms":27962,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F10","60J25","60J60","60J75"],"pacs":[],"model":"deepseek-v4-flash","headline":"Conditioned occupation measures of a strong Feller Markov process satisfy a large deviation principle whose rate function vanishes exactly at the quasi-ergodic distribution.","keywords":["large deviation principle","quasi-ergodic distribution","quasi-stationary distribution","killed Markov process","strong Feller semigroup","Feynman-Kac semigroup","Dirichlet form","Langevin dynamics"],"falsifier":"Take a strong Feller Markov process satisfying (C1)-(C5) and compute the limit in (4.4) directly for a specific bounded measurable $V$ on a reversible diffusion in an interval; if the limit differs from $\\Lambda_D(V)-\\Lambda_D(0)$, the spectral gap step fails. Alternatively, exhibit $V\\in bB(D)$ for which the killed Feynman-Kac semigroup on $b\\mathcal{W}_B(D)$ has essential spectral radius strictly positive, contradicting Theorem 1 and invalidating the rate function's form.","tokens_in":123,"feed_emoji":"🎲","tokens_out":6984,"duration_ms":60721,"temperature":0.7,"pith_summary":"The paper aims to establish a large deviation principle (LDP) for the empirical occupation measure of a strong Feller Markov process, conditioned on the event that the process has not left an open subset $D$ before the observation time. The rate function is shown to be good and to vanish at exactly one measure, the quasi-ergodic distribution $\\pi_D$ of the process in $D$, which is the product of the quasi-stationary distribution and the principal eigenfunction of the killed semigroup. This turns a qualitative conditional ergodic theorem into exponential-rate control: conditioned on survival, the empirical measure converges to $\\pi_D$ exponentially fast in probability, in a topology stronger than weak convergence. In the reversible case the paper identifies the same rate function with the Dirichlet form shifted by the principal Dirichlet eigenvalue, and it verifies the hypotheses for several concrete dynamics, including elliptic SDEs driven by rotationally invariant $\\alpha$-stable processes and kinetic and overdamped Langevin processes.","feed_headline":"Trapped Markov paths obey a large-deviations law with one attractor","feed_subtitle":"The unique zero of the rate function is the quasi-ergodic distribution, so survival-conditioned averages concentrate exponentially fast.","key_machinery":"The load-bearing object is the killed Feynman-Kac semigroup $P_t^{D,V} f(x)=E_x[f(X_t)e^{\\int_0^t V(X_s)ds}1_{t<\\sigma_D}]$ acting on the weighted space $b\\mathcal{W}_B(D)$ of functions bounded by a constant multiple of the Lyapunov function $W$. Its log spectral radius $\\Lambda_D(V)$ is the principal eigenvalue-type quantity that the log-Laplace transform of the conditional empirical measure is shown to equal: $\\Lambda(V)=\\Lambda_D(V)-\\Lambda_D(0)$. The key structural fact is a one-dimensional spectral gap: a unique left eigenmeasure $\\mu_{D,V}$, a unique positive continuous right eigenfunction $\\phi_{D,V}$, and exponential convergence after rescaling, which follows from zero essential spectral radius on the weighted space. This gap simultaneously gives Gateaux differentiability of the log-Laplace functional and the uniqueness of the zero of the rate function.","core_discovery":"Let $P_t^{D,V}$ be the killed Feynman-Kac semigroup with bounded measurable potential $V$, and let $\\Lambda_D(V)$ be its log spectral radius on the weighted space $b\\mathcal{W}_B(D)$. Theorem 2 asserts that for every initial law $\\nu$ with finite $W$-mass, the conditional laws $P_\\nu[L_T \\in \\cdot \\mid T < \\sigma_D]$ satisfy an LDP on $\\mathcal{P}(D)$ with the $\\tau$-topology, speed $T$, and rate $I_D(\\beta)=\\sup_{V\\in bB(D)}\\{\\beta(V)-(\\Lambda_D(V)-\\Lambda_D(0))\\}$. The rate function is good, and $I_D(\\beta)=0$ if and only if $\\beta=\\pi_D:=\\phi_D\\mu_D$, where $\\mu_D$ is the quasi-stationary distribution and $\\phi_D$ the positive eigenfunction of the killed semigroup normalized by $\\mu_D(\\phi_D)=1$. The proof runs through a spectral gap theorem (Theorem 1) for $P_t^{D,V}$: one eigenmeasure, one eigenfunction, and exponential convergence at rate $e^{-\\delta t}$ after rescaling by $e^{\\Lambda_D(V)t}$. This spectral gap is what makes the log-Laplace transform computable and Gateaux differentiable, so the generalized Gartner-Ellis theorem applies.","pith_inferences":["Extension not in the paper: the general-case inequality $I_D\\ge J+\\Lambda_D(0)$ suggests that for non-reversible processes the empirical-measure fluctuations may be strictly stronger than those of the unkilled path entropy; one could test equality or strictness on a non-reversible hypoelliptic example.","Extension not in the paper: Remark 2's approximation by $V_n=-n1_{D^c}$ points to a possible alternative proof; verifying the limit exchange numerically for a disconnected domain $D$ (where the paper notes the convex-rate LDP can fail without connectedness) would delimit the true scope.","Extension not in the paper: because the rate is a supremum over all bounded measurable potentials, the LDP carries information about every bounded measurable set, so one could in principle read off sharp asymptotics for functionals such as last-exit locations or boundary occupation under the same hypotheses."],"forward_implications":["Conditioned on survival, $L_T$ converges to $\\pi_D$ exponentially fast in probability in the $\\tau$-topology, with explicit exponential bounds for any measurable neighborhood of $\\pi_D$ (Corollary 1).","In the reversible case the rate function is the Dirichlet form: $I_D(\\beta)+\\lambda_D=\\mathcal{E}(\\sqrt{h},\\sqrt{h})$ when $\\beta=h\\pi$, and $+\\infty$ otherwise, so fluctuations are governed by the principal Dirichlet eigenvalue $\\lambda_D$.","The whole set of conclusions—spectral gap, LDP, and exponential convergence—holds for elliptic SDEs driven by rotationally invariant $\\alpha$-stable noise satisfying (6.3), for kinetic Langevin processes, and for overdamped Langevin diffusions.","Because the rate function has a unique zero, the quasi-ergodic distribution is singled out as the only possible limit of conditional empirical measures, strengthening the usual conditional ergodic theorem from convergence in law to exponential concentration."],"supporting_citations":[{"why":"Supplies the Perron-Frobenius and essential-spectral-radius facts for the killed semigroup, including $\\beta_w(Q_0)=0$, on which Step 1 of Theorem 1 relies.","marker":"[35, Theorem 3.5]"},{"why":"Provides the generalized Perron-Frobenius theorem that yields the unique eigenmeasure-eigenfunction pair and exponential convergence used here for arbitrary bounded $V$.","marker":"[35, Theorem 4.1]"},{"why":"Gives the formula for the essential spectral radius of positive kernels in terms of $\\beta_w$, used to conclude that the essential spectral radius of the killed Feynman-Kac kernel is zero.","marker":"[61, Theorem 3.5]"},{"why":"Provides the variational characterization of the path-space entropy functional $J(\\beta)$ and the large-deviation framework used in the proof of Theorem 2.","marker":"[27]"},{"why":"Provides the generalized Gartner-Ellis theorem with exponential $*$-tightness on the $\\tau$-topology, the engine of the LDP proof.","marker":"[59]"},{"why":"Gives the unnormalized empirical-measure LDP upper bound with rate $J(\\beta)$, used to establish exponential $*$-tightness for the conditional measures.","marker":"[60]"}],"fun_headline_variants":["Quasi-ergodic distribution is the unique zero of a large-deviations rate","Large deviations for trapped Markov paths yield a single attractor","Rate function zero pinpoints quasi-ergodic distribution for trapped Markov","Trapped Markov processes: LDP has unique minimizer at quasi-ergodic law","Survival-conditioned occupation measures obey large deviations with one limit"],"cache_read_input_tokens":27648,"weakest_assumption_plain":"The proof imports a Perron-Frobenius and essential-spectral-radius theorem from earlier work that guarantees a one-dimensional spectral gap for every bounded measurable potential; if that theorem actually needs the potential or the coefficients to be more regular than measurability, the log-Laplace limit (4.4) and hence the large deviation principle would not follow as written.","fun_headline_variants_meta":{"raw":{"variants":["Quasi-ergodic distribution is the unique zero of a large-deviations rate","Large deviations for trapped Markov paths yield a single attractor","Rate function zero pinpoints quasi-ergodic distribution for trapped Markov","Trapped Markov processes: LDP has unique minimizer at quasi-ergodic law","Survival-conditioned occupation measures obey large deviations with one limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.0007,"raw_usage":{"total_tokens":3163,"prompt_tokens":951,"completion_tokens":2212,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":2116}},"tokens_in":567,"tokens_out":2212,"duration_ms":12685,"temperature":1.0,"reasoning_tokens":2116,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T12:26:31.063866+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a strong Feller Markov process satisfying (C1)-(C5) and compute the limit in (4.4) directly for a specific bounded measurable $V$ on a reversible diffusion in an interval; if the limit differs from $\\Lambda_D(V)-\\Lambda_D(0)$, the spectral gap step fails. Alternatively, exhibit $V\\in bB(D)$ for which the killed Feynman-Kac semigroup on $b\\mathcal{W}_B(D)$ has essential spectral radius strictly positive, contradicting Theorem 1 and invalidating the rate function's form.","supporting_citations":[{"cited_title":"Deuschel and D.W","cited_arxiv_id":null,"evidence_quote":"Provides the variational characterization of the path-space entropy functional $J(\\beta)$ and the large-deviation framework used in the proof of Theorem 2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the generalized Gartner-Ellis theorem with exponential $*$-tightness on the $\\tau$-topology, the engine of the LDP proof."}],"review_version":1}