{"id":"c629ef7c-d8d0-4543-8750-e37c630372a0","arxiv_id":"1908.03481","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"A large deviation principle is claimed for the slow variable in fully coupled slow-fast jump diffusions, with three different rate functions depending on the time-scale separation exponent.","lead":"This paper derives large deviation rate functions for the slow variable in a two-scale stochastic system driven by both Brownian motion and Lévy jumps, with different rate functions depending on the speed of the fast variable. The result would matter for rare-event estimates in finance and geophysical models, but the proof leaves key conditions unverified.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Operator inequality (4.31) is only proved in the critical case; for the genuinely new supercritical and subcritical regimes it is asserted, so the identification of H0 and hence the LDP in Theorem 2 is not established.","rationale":"The reader's verdict of REJECT is appropriate. The paper's claimed novelty over [26] is precisely the supercritical and subcritical regimes, and in both regimes the decisive operator inequality (4.31) is asserted rather than proved. A referee could try to fill the gap, but that would be original work, not verification of the submitted proof. The uniform-continuity gap in Theorem 1, flagged by the reader, compounds the problem; I regard (4.31) as the single most load-bearing gap because without it the candidate H0 is not shown to be the effective Hamiltonian, so the comparison principle would only compare solutions of an equation whose Hamiltonian may be wrong. I do not claim the result is false; a correct proof of (4.31) along the lines of the critical case may exist. But the submitted manuscript does not provide it, so the central claim is not established. I therefore keep the reader's rejection rather than recommending minor revisions.","tokens_in":25730,"tokens_out":10309,"duration_ms":117105,"concrete_test":"Verify (4.31) for the subcritical case in an explicit periodic model: take rho=0, b2=0, sigma2=1, k2=0, and choose the Levy measure nu1 so that V_{x,p}(y)=sin(2pi y). Compute the left side inf_{zeta in C_c^2, 0<theta<1} sup_{y in [0,1]} {V_{x,p}(y)+(1-theta)(zeta'(y))^2+theta(xi'(y))^2} with the fixed Lyapunov function xi from (C5), and compare with the right side H0=max_y V=1. If the infimum is strictly smaller, (4.31) fails and the subcritical rate function is wrong. For an analytic check, solve the one-dimensional cell problem H0=V+(W')^2 with periodic W and determine whether the minimal level is max_y V; this settles whether the asserted inequality is compatible with the Hamiltonian derived in (3.15).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central theorem depends on Lemma 4, whose proof uses the comparison principle and the operator inequality (4.31) to show that the semilimits of U^epsilon solve the limit equation with H0. Section 4.4 proves (4.31) for alpha=2 and then states: 'For the supercritical case (alpha > 2) and subcritical case (alpha < 2), the operator inequality (4.31) can be proven with similar ideas.' This is a placeholder, not a proof, and these are exactly the cases not covered by the cited prior work [26]. The formal derivation of H0 = max_y V in (3.15) also hides a cell-problem step: one must prove that a periodic W exists exactly at that level and that no smaller level supplies a subsolution; the one-line completion 'If H0 >= V, we have ... then we obtain' does not supply that proof. In addition, Theorem 1 assumes H0 is uniformly continuous on compact sets, with verification deferred to Remark 2. Without (4.31), the comparison argument only shows convergence to a solution of an equation whose Hamiltonian is not proved to be the effective Hamiltonian, so Theorem 2 does not follow as written.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper aims to establish a large deviation principle for the slow component X^ε_t of a fully coupled slow-fast stochastic system driven by both Brownian and Lévy noises, in the scaling δ = ε^α with α > 1. The proof follows the Bryc–Varadhan/viscosity-solution route: the log-moment functionals U^ε are characterized as viscosity solutions of a nonlinear partial integro-differential equation (2.12), their relaxed semilimits are shown to converge to the unique solution of a limit Hamilton–Jacobi equation (3.16), and the rate function is given by the Legendre-type transform (5.3). Three regimes are treated: supercritical α > 2, critical α = 2, and subcritical α < 2, with explicit limit Hamiltonians (3.4), (3.11), and (3.15).","tokens_in":25936,"tokens_out":10418,"duration_ms":99542,"significance":"If fully established, the result would be a natural extension of Kumar–Popovic [26] to all α > 1 in the fully coupled case, and the explicit Hamiltonians would make the LDP potentially applicable to multiscale models with non-Gaussian noise. The paper contains a substantial rigorous argument in the critical case, including the proof of the operator inequality (4.31) via Donsker–Varadhan variational formulas and occupation-measure estimates, and the exponential-tightness argument in Lemma 5. However, the most novel parts—the supercritical and subcritical regimes—rest on unproved assertions, and the comparison principle is conditional on an unverified regularity hypothesis.","major_comments":[{"comment":"The operator inequality (4.31) is proved only for the critical case α = 2. For the supercritical (α > 2) and subcritical (α < 2) cases, the text states that (4.31) 'can be proven with similar ideas', but no proof is given. Since (4.31) is the step that identifies the effective Hamiltonian H0 in the comparison argument of Lemma 4, Theorem 2 is not established for the two regimes that constitute the paper's main new contribution. A proof cannot be replaced by an assertion, especially because the Hamiltonians in (3.4) and (3.15) are not principal eigenvalues and do not obviously follow from the critical-case reasoning.","section":"§4.4, inequality (4.31)"},{"comment":"Theorem 1 proves the comparison principle for (3.16) only under the explicit assumption that H0 is uniformly continuous on compact sets. No verification of this assumption is supplied for any of the three Hamiltonians (3.4), (3.11), (3.15); Remark 2 merely defers the verification, and Remark 3 covers only a special x-independent case with ρ = 0. Because Lemma 4 and hence Theorem 2 rely on this comparison principle, the main theorem is conditional on an unverified regularity property.","section":"Theorem 1 and Remark 2"},{"comment":"The derivation of H0(x,p) = max_y V_{x,p}(y) in the subcritical regime is not a proof. The line 'If H0 ≥ V, we have ... then we obtain' asserts the equality of the square-root expression without justification; one must prove existence of a periodic solution W to the cell problem at exactly that level and exclude smaller levels. As written, (3.15) is an unproved ansatz that feeds directly into the rate function of Theorem 2.","section":"§3, subcritical case, Eq. (3.15)"},{"comment":"Lemma 4 states 'Suppose the comparison principle holds for the Cauchy problem (3.16)', but its proof proceeds by invoking Theorem 1 to obtain the comparison principle. This makes the lemma logically inconsistent: either the hypothesis is redundant and should be removed with the uniform-continuity condition verified, or the proof does not use the stated hypothesis. In either case, the convergence result is not established under the assumptions (C1)-(C5) alone, contrary to the statement of Theorem 2.","section":"Lemma 4"}],"minor_comments":[{"comment":"The word 'sloutions' should be 'solutions'.","section":"§1, Introduction"},{"comment":"The word 'supermatingale' should be 'supermartingale'.","section":"Lemma 5"},{"comment":"The phrase 'the expression as on the right-hand side of (4.30)' is awkward; it should be 'the expression on the right-hand side of (4.30)'.","section":"Remark 2"},{"comment":"The proof of Lemma 1 is omitted with a reference to 'similar arguments' in [17,19,26]; for a self-contained journal paper, a precise citation to the exact theorem or a short proof should be included.","section":"Lemma 1"},{"comment":"The claim that the proof for system (5.6) is 'similar to that of system (1.1)' needs more detail, since the drift modification Kν1, Kν2 changes the Hamiltonian in the formal expansions.","section":"Example 1"}],"recommendation":"major_revision","confidential_remarks":"The two missing proofs concern exactly the regimes that go beyond the existing literature: the critical case α = 2 was already covered in [26]. If the authors can supply a rigorous proof of (4.31) for α > 2 and α < 2, and verify the uniform continuity of H0, the paper may become publishable; otherwise the main theorem is not supported as written. I would ask the editor to weigh whether 'similar ideas' can deliver the subcritical Hamiltonian max_y V, which is structurally different from the principal-eigenvalue formula."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know up front. First, the paper's genuinely new results—the supercritical and subcritical large deviations—are not actually proved. The operator inequality (4.31) that identifies the effective Hamiltonian is proved only in the critical case alpha=2, which is exactly the case already handled by Kumar-Popovic. For the new regimes the paper says it \"can be proven with similar ideas.\" That is a placeholder, not a proof. Second, the formal picture—average over the invariant measure for alpha>2, principal eigenvalue for alpha=2, max over y of V_{x,p} for alpha<2—is natural and probably right. So the paper is worth reading for the heuristic, but Theorem 2 should not be accepted as established.\n\nWhat the paper does well: the three-regime formal asymptotic derivation is clean, and the distinction between the regimes is interesting. The critical-case operator inequality is a real argument using Donsker-Varadhan variational formulas, with the occupation-measure lower bound spelled out. The exponential tightness lemma is also in place. That is substantive work.\n\nThe soft spots are exactly the load-bearing ones. (i) Theorem 1 proves comparison only under uniform continuity of H0 on compact sets; Remark 2 defers verification, but no verification appears for any of the three Hamiltonians. Lemma 4 then assumes comparison, and its proof immediately invokes Theorem 1, so the gap is internal. (ii) The subcritical derivation of H0 = max_y V hides a cell problem: one has to show that a periodic corrector exists exactly at that level and that no lower level gives a subsolution. The one-line completion in (3.15) does not supply that. (iii) Lemma 1, the viscosity property of U^epsilon, is stated without proof; this is probably standard and fixable, so I treat it as minor. The abstract's claim that the comparison principle is \"verified\" is too strong—it is verified conditionally.\n\nNone of this means the paper is nonsense. The framework is the well-trodden Bryc/Varadhan route, and the critical case shows the authors can do the hard estimate when they need to. But as written, the central theorem relies on exactly the cases they did not prove.\n\nWho it is for: people working on averaging and large deviations for jump processes will want to know the three-regime classification. It deserves a serious referee, and I would engage with it. My recommendation to the editor: send it to review, but tell the referee the priorities are the missing verification of (4.31) for alpha>2 and alpha<2, and the uniform continuity needed by Theorem 1. A major revision could make this a solid paper.","headline":"The three-regime picture is plausible and the critical-case proof is real, but the new regimes ride on an asserted operator inequality, so the main theorem is not proven as written.","tokens_in":26497,"tokens_out":2813,"would_cite":false,"duration_ms":30034,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60F10","37H10"],"pacs":[],"model":"deepseek-v4-flash","headline":"Slow variables in Lévy-driven slow-fast diffusions obey a large deviation principle with explicit rate function.","keywords":["large deviations","slow-fast systems","Lévy noise","viscosity solutions","comparison principle","non-Gaussian noise","multiscale stochastic dynamics","rare events"],"falsifier":"For a concrete two-scale Ornstein-Uhlenbeck-type model with Lévy jumps, compute $H^0$ in each regime and compare Monte Carlo estimates of $-\\varepsilon\\log P(X^\\varepsilon_t\\in A)$ with the predicted $\\inf_{x\\in A}I(x,x_0,t)$ over decreasing $\\varepsilon$; a mismatch would falsify the large deviation claim. A cheaper check is whether the three $H^0$ forms are uniformly continuous on compact sets, the condition the comparison-principle proof needs.","tokens_in":25443,"feed_emoji":"🎲","tokens_out":9837,"duration_ms":98722,"temperature":0.7,"pith_summary":"Large deviations describe how likely the slow variable is to wander far from its typical path, on the exponential scale. This paper proves that in a coupled slow-fast diffusion with both Brownian and Lévy noise, these probabilities decay at rate $1/\\varepsilon$ with a cost $I(x,x_0,t)$ determined by a limiting Hamilton-Jacobi equation. The limiting Hamiltonian takes one of three explicit forms depending on the time-scale exponent $\\alpha$: an average over fast states, a principal eigenvalue, or a maximum over fast states. If the claim holds, modelers can compute rare-event asymptotics in multiscale systems with non-Gaussian noise without solving the full two-scale dynamics.","feed_headline":"Lévy-noise slow-fast systems get explicit large deviation rates","feed_subtitle":"Three time-scale regimes yield three explicit Hamiltonians, so rare-event costs are computable from one limit equation.","key_machinery":"The load-bearing identity is the logarithmic transform $U^\\varepsilon(t,x,y)=\\varepsilon\\ln\\mathbb{E}[e^{h(X^\\varepsilon_t)/\\varepsilon}\\mid X^\\varepsilon_0=x,Y^\\varepsilon_0=y]$, which converts rare-event expectations into viscosity solutions of a nonlinear partial integro-differential equation of Hamilton-Jacobi type. The limit is controlled by the Hamiltonian $H^0$; its correct form is forced by an operator inequality $\\inf_\\lambda \\hat{H}^\\lambda \\le H^0 \\le \\sup_\\lambda \\check{H}^\\lambda$, where $\\hat{H}^\\lambda$ and $\\check{H}^\\lambda$ are regularized operators built from test functions, a Lyapunov function, and an indexing set. The comparison principle for the limit equation then identifies the upper and lower semilimits, yielding uniform convergence of $U^\\varepsilon$ to $U^0$. This uniform convergence, plus exponential tightness of the slow process, is what produces the large deviation principle.","core_discovery":"The central claim is that the slow variable $X^\\varepsilon_t$ satisfies a large deviation principle (Theorem 2). Under conditions (C1)-(C5), the probability that $X^\\varepsilon_t$ lies near $x$ decays as $\\exp(-I(x,x_0,t)/\\varepsilon)$, where $I(x,x_0,t)=\\sup_{h\\in C_b(\\mathbb{R})}\\{h(x)-U^0(t,x_0)\\}$ and $U^0$ is the unique viscosity solution of the limiting Cauchy problem $\\partial_t U=H^0(x,\\partial_x U)$ with $U(0,x)=h(x)$. The paper identifies $H^0$ separately for $\\alpha>2$, $\\alpha=2$, and $\\alpha<2$: respectively, the integral of the local Hamiltonian density $V_{x,p}(y)$ against the invariant measure of the fast motion; the principal eigenvalue of the generator $L_{x,p}+V_{x,p}$ expressed through a variational formula over probability measures; and $\\max_y V_{x,p}(y)$. The derivation passes through the convergence of the logarithmic functionals $U^\\varepsilon$ to $U^0$ on compact sets, obtained by viscosity sub- and supersolution methods and the comparison principle.","pith_inferences":["The paper leaves uniform continuity of the three $H^0$ forms as an open verification; a direct check for each regime would turn the conditional convergence theorem into an unconditional one.","The subcritical form $H^0=\\max_y V_{x,p}(y)$ suggests that, in that regime, the rare-event cost is controlled by the single fast state most favourable to the fluctuation, so model reduction could be possible by maximizing over $y$ alone.","The same semilimit-plus-comparison route should extend to multidimensional slow variables, with the Hamiltonian forms re-derived coordinate-wise, but the Lyapunov conditions would need to be adapted."],"forward_implications":["If the theorem is correct, rare-event probabilities for the slow variable are asymptotically computable from the limiting equation, bypassing the two-scale dynamics.","In the supercritical regime ($\\alpha>2$) the rate is an average over the fast invariant measure; in the subcritical regime ($\\alpha<2$) it is a maximum over fast states, so the two regimes can have quite different rare-event costs.","When the coefficients do not depend on $x$, the rate function simplifies to $t Q^0((x_0-x)/t)$, an explicit Legendre-transform form.","The three-regime distinction gives a practical rule: the time-scale ratio $\\delta=\\varepsilon^\\alpha$ changes not just the averaging but the structure of the effective Hamiltonian."],"supporting_citations":[{"why":"It supplies the multi-scale jump-diffusion large deviation method that this paper adapts to three regimes and Lévy noises.","marker":"[26]"},{"why":"It provides the comparison-principle and operator-inequality framework used to prove uniform convergence of the logarithmic functionals.","marker":"[17]"},{"why":"It gives the logarithmic-transform Cauchy problem for fast mean-reverting volatility and the Legendre-transform lemma used for x-independent coefficients.","marker":"[19]"},{"why":"It establishes the three-regime viscosity approach for fast volatility that is extended here to Lévy-driven slow-fast systems.","marker":"[8]"},{"why":"It states the inverse large-deviation theorem that turns convergence of exponential functionals and exponential tightness into an LDP.","marker":"[14]"},{"why":"It supplies the uniform LDP for occupation measures of the fast process used in the critical-case lower bound.","marker":"[16]"},{"why":"It identifies the principal eigenvalue of a perturbed generator through a variational formula, which becomes the critical-case Hamiltonian.","marker":"[15]"},{"why":"It guarantees strong existence, uniqueness, and the Markov property under the Lipschitz and growth conditions.","marker":"[1]"}],"fun_headline_variants":["Explicit large deviation rates for slow-fast Lévy systems","Three regimes, three Hamiltonians for slow-fast Lévy dynamics","Lévy slow-fast systems: rare-event costs from one limit equation","Large deviations for slow-fast diffusions with non-Gaussian noise","Slow-fast Lévy: explicit rate function via viscosity solutions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the comparison principle for the limit Cauchy problem (3.16), and the paper proves that principle only under a uniform-continuity condition on $H^0$ that is not verified for the three explicit Hamiltonians.","fun_headline_variants_meta":{"raw":{"variants":["Explicit large deviation rates for slow-fast Lévy systems","Three regimes, three Hamiltonians for slow-fast Lévy dynamics","Lévy slow-fast systems: rare-event costs from one limit equation","Large deviations for slow-fast diffusions with non-Gaussian noise","Slow-fast Lévy: explicit rate function via viscosity solutions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000245,"raw_usage":{"total_tokens":1509,"prompt_tokens":893,"completion_tokens":616,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":509,"completion_tokens_details":{"reasoning_tokens":530}},"tokens_in":509,"tokens_out":616,"duration_ms":6188,"temperature":1.0,"reasoning_tokens":530,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:12:02.326294+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a concrete two-scale Ornstein-Uhlenbeck-type model with Lévy jumps, compute $H^0$ in each regime and compare Monte Carlo estimates of $-\\varepsilon\\log P(X^\\varepsilon_t\\in A)$ with the predicted $\\inf_{x\\in A}I(x,x_0,t)$ over decreasing $\\varepsilon$; a mismatch would falsify the large deviation claim. A cheaper check is whether the three $H^0$ forms are uniformly continuous on compact sets, the condition the comparison-principle proof needs.","supporting_citations":[{"cited_title":"Kumar, L","cited_arxiv_id":null,"evidence_quote":"It supplies the multi-scale jump-diffusion large deviation method that this paper adapts to three regimes and Lévy noises."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the comparison-principle and operator-inequality framework used to prove uniform convergence of the logarithmic functionals."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It gives the logarithmic-transform Cauchy problem for fast mean-reverting volatility and the Legendre-transform lemma used for x-independent coefficients."},{"cited_title":"Bardi, A","cited_arxiv_id":null,"evidence_quote":"It establishes the three-regime viscosity approach for fast volatility that is extended here to Lévy-driven slow-fast systems."},{"cited_title":"Dembo, O","cited_arxiv_id":null,"evidence_quote":"It states the inverse large-deviation theorem that turns convergence of exponential functionals and exponential tightness into an LDP."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It supplies the uniform LDP for occupation measures of the fast process used in the critical-case lower bound."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It identifies the principal eigenvalue of a perturbed generator through a variational formula, which becomes the critical-case Hamiltonian."},{"cited_title":"Applebaum, L´ evy Processes and Stochastic Calculus","cited_arxiv_id":null,"evidence_quote":"It guarantees strong existence, uniqueness, and the Markov property under the Lipschitz and growth conditions."}],"review_version":1}