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REVIEW 2 major objections 5 minor 67 references

Large deviations of the empirical measures of a strong-Feller Markov process inside a subset and quasi-ergodic distribution

T0 review · 2 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Conditioned occupation measures of a strong Feller Markov process satisfy a large deviation principle whose rate function vanishes exactly at the quasi-ergodic distribution.

desk verdict 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. read the letter →

arxiv 2411.17216 v1 pith:EVL4YJIJ submitted 2024-11-26 math.PR

Arnaud Guillin (LMBP) , Boris Nectoux (LMBP) , Liming Wu (LMBP) This is my paper · ORCID
classification math.PR MSC 60F1060J2560J6060J75
keywords largedeviationprinciplequasi-ergodicdistributionquasi-stationarykilledMarkovprocessstrongFellersemigroupFeynman-KacDirichletformLangevindynamics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

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.

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 (2)
  1. [Section 4, Step 4 (proof of Theorem 2(B))] 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.
  2. [Section 3, Step 3] 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.
minor comments (5)
  1. [Section 2.2.3 and Theorem 3] 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.
  2. [Section 4, Step 3] The phrase 'expotential *-tightness' should read 'exponential *-tightness'.
  3. [Section 5, Step 2] 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.
  4. [Section 2.2.5, footnote 3] The inserted remark about Patrick Cattiaux is not appropriate for a research paper and should be removed.
  5. [Section 6.2.3, Proposition 3] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the rate function is computed from spectral data via Gartner-Ellis, not fitted; the unique zero is tied to the q.e.d.

full rationale

The derivation chain runs: Theorem 1 establishes a spectral gap for the killed Feynman-Kac semigroup P^{D,V}_t on bW_B(D); equality (4.4) then computes the conditional log-Laplace limit as Lambda(V) = Lambda_D(V) - Lambda_D(0) from that spectral data; the generalized Gartner-Ellis theorem (Theorem 3) turns this into the LDP with rate function (2.10); and the zero set of the Legendre transform is the subdifferential of Lambda_D at V = 0. The central claim (2.13) therefore reduces to proving Lambda_D'(0)(V) = mu_D(V phi_D) = pi_D(V) for all bounded measurable V. That is a genuine spectral statement, not an input: pi_D = phi_D mu_D is defined in Section 1.1 from the eigenpair of the unperturbed killed Dirichlet semigroup P^D_t, independently of the rate function, and the derivative identity is the standard first-order perturbation formula for a simple eigenvalue. Nothing is fitted to data and no prediction is renamed from an input; the rate function is computed, not postulated. The paper leans on the authors' own earlier theorems [35, Theorems 3.5 and 4.1] (Perron-Frobenius and essential spectral radius, JEMS 2022), [61, Theorem 3.5] (beta_w / beta_tau machinery, PTRF 2004), and [60] (Donsker-Varadhan upper bound, SPA 2001), but these are published, parameter-free statements whose assumptions do not contain the LDP conclusion, so per the reviewing rules they are independent evidence and do not raise the circularity score. Flag as a proof gap, not circularity: Section 4, Step 4 of the proof of Theorem 2 displays a chain ending with Lambda_D(V) >= lim sup (1/T) log E_nu[e^{int V}|T<sigma_D] >= lim (1/T) [e^{-Lambda_D(V)T} int int P^D_t V P^D_{T-t}1 dt dnu]/[e^{-Lambda_D(V)T} int P^D_T 1 dnu] = pi_D(V). As written this cannot establish (4.7): the target Lambda_D(V) >= pi_D(V) fails at V = 0, where it would give Lambda_D(0) >= 0, contradicting Theorem 1(1) (Lambda_D(V) < sup_D V, so Lambda_D(0) < 0); the correct subdifferential target is Lambda_D(V) - Lambda_D(0) >= pi_D(V). The Jensen estimate gives A_T/B_T ~ T pi_D(V) nu(phi_D)/nu(phi_D), whose (1/T)-log-rate is 0, not pi_D(V), since the displayed chain drops the logarithm and evaluates (1/T)(A_T/B_T) instead of (1/T) log(A_T/B_T); and the spectral expansion applies the rate Lambda_D(V) to the unperturbed semigroup P^D_t whose true rate is Lambda_D(0).

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The central claim depends on the standing hypotheses (C1)-(C5) and on several imported theorems from the published literature, chiefly the authors' previous spectral work [35] and Wu's essential-spectral-radius results [61]. These are theorems with proofs, not fitted parameters, so the circularity burden is low. No free parameters are calibrated to data; the exponents in the Lyapunov constructions of Section 6 are constraint parameters, not fitted values. No new entities are postulated; the quasi-ergodic distribution is defined from objects established in Theorem 1. The main fragility is the extent to which the imported Perron-Frobenius theorem applies to bounded measurable potentials V, which the paper only partially re-derives.

assumptions (8)
  • domain assumption (C1)-(C5) hold for the process and the domain D
    Standing hypotheses of Theorems 1 and 2: strong Feller, continuity of x to P_x(X[0,T] in .), Lyapunov condition -LW^p >= r_n W^p - b_n 1_{K_n}, killed-semigroup continuity, and irreducibility plus non-absorption. If any fails, the spectral gap and the LDP are not proved.
  • domain assumption V is bounded and measurable (HV)
    Required for the killed Feynman-Kac semigroup and the supremum over bB(D) in the rate function.
  • standard math [35, Theorem 3.5] and [35, Theorem 4.1] supply essential spectral radius 0 and a one-dimensional Perron-Frobenius theorem for killed Feynman-Kac operators on bW_B(D)
    Cited rather than reproved; Step 1 of Theorem 1 reduces to them. Extension to bounded measurable V is sketched, so this is the most fragile import.
  • standard math [61, Theorem 3.5] characterises the essential spectral radius of positive kernels via beta_w, beta_tau
    Used in Step 1 of Theorem 1 to get ress(P^{D,V}_t|bW_B)=0 from beta_w(Q)=0.
  • standard math [60] provides the Donsker-Varadhan LDP for the non-killed process on (P(S),tau)
    Used in Step 3 of Theorem 2 for exponential tau-tightness of the conditioned measures.
  • standard math [45, Theorem 2.1] support theorem for Levy-driven SDEs
    Used in Proposition 4 to prove positivity of killed transition probabilities for the stable-driven SDE.
  • standard math [27] Donsker-Varadhan formula J(beta)=E(sqrt(h),sqrt(h)) in the reversible case
    Used in the proof of Corollary 2 to identify the rate function with the Dirichlet form.
  • standard math [40] Kato perturbation theory for isolated simple eigenvalues
    Used in Step 2 of Theorem 2 to prove Gateaux differentiability of Lambda_D(V).

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Pith. "Pith review of Large deviations of the empirical measures of a strong-Feller Markov process inside a subset and quasi-ergodic distribution." pith.science (2026). https://pith.science/paper/EVL4YJIJ

@misc{pith2026241117216,
  author       = {Pith},
  title        = {Pith review of: Large deviations of the empirical measures of a strong-Feller Markov process inside a subset and quasi-ergodic distribution},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EVL4YJIJ}},
  note         = {Machine review of arXiv:2411.17216}
}
abstract

In this work, we establish, for a strong Feller process, the large deviation principle for the occupation measure conditioned not to exit a given subregion. The rate function vanishes only at a unique measure, which is the so-called quasi-ergodic distribution of the process in this subregion. In addition, we show that the rate function is the Dirichlet form in the particular case when the process is reversible. We apply our results to several stochastic processes such as the solutions of elliptic stochastic differential equations driven by a rotationally invariant $\alpha$-stable process, the kinetic Langevin process, and the overdamped Langevin process driven by a Brownian motion.

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Pith tools

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