{"id":"d53aa97c-a55a-4a7e-ac1a-8b8c37d89289","arxiv_id":"2411.09614","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"For the Parabolic Anderson model on Cartan-Hadamard manifolds, the paper establishes a curvature-dependent Dalang condition, exponential moment upper bounds with a spectral-gap term, and asymptotically matching lower bounds.","lead":"The paper proves well-posedness and exponential-in-time moment bounds for the Parabolic Anderson model on non-positively curved manifolds, with explicit curvature-dependent rates. It also claims asymptotically matching lower bounds in the strong-noise limit, showing curvature controls the phase transition and intermittency.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The large-beta asymptotics of Theta_alpha in Theorem 13/abstract invert Lemma 11 incorrectly for alpha<n/4; the correct exponent is 1/(2alpha-n/2+1), not 1-2alpha+n/2, so the central moment exponent claim is false as stated.","rationale":"The reader's verdict identified a contradiction in the displayed Theta exponent and in the p-dependence; my stress-test agrees and localizes the more load-bearing fault. The heat-kernel upper bound (Theorem 3) and the renewal estimate (31)-(34) appear internally consistent, and the well-posedness criterion alpha > (n-2)/4 follows from convergence of I1. Independent support: the derivation of the noise family and the second-moment recursion are coherent, and the paper itself contains the ingredients (Lemma 11(b), invertibility of F_i, Corollary 15) that expose the errors. However, the central quantitative claim--the large-beta exponent of the moment Lyapunov upper bound and its claimed match with the lower bound--depends on the inversion Theta_alpha = F_i^{-1}(1/(C beta^2)). That inversion is miscomputed in the alpha < n/4 regime, and Theorem 14 also writes the p-dependence incorrectly. These are not merely cosmetic: they alter the functional form of the central bound. A correction is likely possible, but as written the abstract, Theorem 13, and Theorem 14 do not support the stated sharp asymptotics. Hence the REJECT verdict stands pending revision.","tokens_in":19127,"tokens_out":13834,"duration_ms":128335,"concrete_test":"Take n=3, alpha=1/2, K1=1. Using the exact expression in the proof of Lemma 11, F1(rho) = I1(rho) + I4(rho), with I1(rho) = rho^{-1/2} gamma(1/2, rho/2) and I4(rho) = integral_{1/2}^infty (1+s)^{-3/2} e^{-rho s} ds. Solve F1(Theta) = 1/(C beta^2) numerically for beta = 10, 100, 1000 and fit Theta proportional to beta^s. Lemma 11 forces s = 1/(2alpha - n/2 + 1) = 2; Theorem 13 and the abstract predict s = 1 - 2alpha + n/2 = 3/2. If the numerical exponent is 2, the paper's asymptotic formula is wrong. A purely symbolic check: inverting rho^{-delta} = epsilon gives epsilon^{-1/delta}, which is (C beta^2)^{1/(2alpha-n/2+1)}, not (C beta^2)^{1-2alpha+n/2} unless delta = 1; this single inversion check settles the load-bearing error.","verdict_should_be":"REJECT","load_bearing_attack":"The most load-bearing step is not the heat-kernel upper bound itself but the inversion of the functions F_i into Theta_alpha in Theorem 13. Lemma 11(b) states that for alpha in ((n-2)/4, n/4), F_1(rho) ~ rho^{-(2alpha-n/2+1)} as rho -> infinity. Since Theta_alpha(beta) is defined as F_i^{-1}(1/(C beta^2)), the correct large-beta behavior is Theta_alpha(beta) ~ (C beta^2)^{1/(2alpha-n/2+1)}. The exponent 1/(2alpha-n/2+1) lies in (1, infinity) and equals 1 only at the boundary alpha = n/4. Theorem 13 and the abstract instead assert (C beta^2)^{1-2alpha+n/2}, i.e. (C beta^2)^{2-(2alpha-n/2+1)}, which is a different exponent except when alpha = n/4. Concretely, for n=3 and alpha=1/2, the inverse is C^2 beta^4, while the paper claims C^{3/2} beta^3. This changes the beta-scaling of the claimed p-th moment Lyapunov exponent, so the advertised sharpness and the 'matching' lower bound are not supported as written. A separate p-dependence error appears in Theorem 14: hypercontractivity gives the convergence condition C beta^2 (p-1) F_i(rho) < 1, so the argument should be beta sqrt(p-1), not beta p/(p-1), as Corollary 15 itself confirms. The Theta_alpha inversion error is the more fundamental obstruction to the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs fractional Gaussian noises on Cartan-Hadamard manifolds via integrals of the heat kernel, proves well-posedness of the parabolic Anderson model in the Itô-Walsh sense under the Dalang-type condition α > (n-2)/4, and derives exponential-in-time upper and lower bounds for p-th moments. The upper bound uses a renewal-type chaos estimate and hypercontractivity; the lower bound uses a Feynman-Kac formula and curvature-dependent heat kernel estimates. The advertised conclusion is that negative curvature shifts the moment Lyapunov exponent by a spectral-gap term and that the large-β behavior is sharp.","tokens_in":19505,"tokens_out":14313,"duration_ms":129079,"significance":"If the stated asymptotics were correct, this would be a valuable first treatment of the parabolic Anderson model on noncompact negatively curved manifolds, connecting intermittency with the bottom of the spectrum and making precise the effect of curvature on the required noise regularity. The paper's strengths are that the arguments are transparent, are built on rigorous heat-kernel estimates (Cheeger-Yau and Davies-Mandouvalos), contain no fitted parameters, and are mostly checkable step by step. However, the central large-β exponents and the p-dependence in the upper bound contain algebraic errors, and the lower-bound matching argument needs to be redone; the framework appears salvageable, but the paper is not acceptable in its current form.","major_comments":[{"comment":"The large-β exponent for Θα is obtained by an incorrect inversion. Lemma 11(b) gives F1(ρ) ∼ ρ^{-(2α-n/2+1)} as ρ→∞, and Θα(β) is defined as F1^{-1}(1/(Cβ^2)); therefore the correct asymptotic is Θα(β) ∼ (Cβ^2)^{1/(2α-n/2+1)}. The exponent 1-2α+n/2 printed in Theorem 13(i) and in the abstract is 2-(2α-n/2+1), which agrees with the correct value only at α=n/4. For example, when n=3 and α=1/2 the paper's formula gives Θα(β)∼(Cβ^2)^{3/2}, whereas the inversion gives (Cβ^2)^2. This changes the claimed β-scaling of the p-th moment Lyapunov exponent, so the advertised asymptotic sharpness is not established as written.","section":"Lemma 11(b); Theorem 13(2)(i); Abstract"},{"comment":"The argument of Θα in the p-th moment bound is wrong. Hypercontractivity applied to the k-th chaos produces a factor (p-1)^{k/2}β^k, so the convergence condition is C(p-1)β^2 F_i(ρ) < 1, i.e. ρ > Θα(√(p-1)β). The statement's Θα(p/(p-1)β) does not follow from the displayed inequality in the proof, and it is inconsistent with Corollary 15, which uses √(p-1)β < β_c. The correct upper bound should involve Θα(√(p-1)β).","section":"Theorem 14"},{"comment":"The large-β lower bound does not have the stated normalization or derivation. With r=cβ^{(4α-n-2)/2} and G_{2α}(r) ≥ C r^{4α-n}, the two terms in Q(r)=β^2(p-1)G_{2α}(r)-c/r^2 have exponents 2+(4α-n-2)(4α-n)/2 and -(4α-n-2), neither of which is the claimed (4α-n-2)/2, and the printed normalization 2/(4α-n-2) is not compatible with a finite limit in the displayed inequality. The balancing choice that matches the corrected upper bound is r ∼ β^{-2/(4α-n+2)}, which gives the scaling β^{4/(4α-n+2)}; the proof of Theorem 22(A) should be redone with this choice.","section":"Theorem 22(A)"}],"minor_comments":[{"comment":"There is a threshold inconsistency: property 1 states Θα ≡ 0 for β < 1/√(C F_i(0)), while the definition in the proof uses β < 1/(C F_i(0)). Since the convergence condition is Cβ^2 F_i(0) < 1, the square-root version is the correct one.","section":"Theorem 13, property 1 and proof"},{"comment":"The exponents in the β-normalization and in the p-normalization should be checked against the corrected large-β scaling; as written they do not match the asymptotic β^{4/(4α-n+2)} obtained from the balancing choice r ∼ β^{-2/(4α-n+2)}.","section":"Theorem 22(A) and Remark 23(A)"}],"recommendation":"major_revision","confidential_remarks":"I would not recommend an outright reject: the inversion error in Theorem 13 and the p-dependence error in Theorem 14 are local and correctable, and the lower-bound section appears reparable with a different choice of r. The paper's reliance on the authors' own prior results is not itself problematic because the relevant lemmas are published and independently derived; no circularity is apparent. The main obstacle is that the central advertised matching asymptotics are not supported by the calculations as written."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know about this paper: it's the right problem and the right framework, but the central asymptotic claims as written are wrong. The paper proves well-posedness for the parabolic Anderson model on Cartan-Hadamard manifolds with a family of colored noises and gives upper and lower moment bounds with curvature-dependent Lyapunov exponents. That's genuinely new: previous work handled Euclidean space, Heisenberg groups, tori, and bounded domains, not general Cartan-Hadamard manifolds with negative curvature. The core machinery—chaos expansion, renewal inequality, hypercontractivity, Feynman-Kac—is imported from [BCH+24] and [BOTW23], but the geometric setting and the curvature corrections (the spectral gap term) are novel.\n\nThe well-posedness part looks solid. Lemma 8's heat kernel estimates are carefully done, and Dalang's condition α>(n-2)/4 emerges correctly from the convergence of the renewal series.\n\nThe soft spots are in the advertised asymptotics. Theorem 13 and the abstract state that for α∈((n-2)/4, n/4), Θα(β) ~ (Cβ^2)^{1-2α+n/2}. But Lemma 11(b) says F1(ρ) ~ ρ^{-(2α-n/2+1)}. Since Θα is defined as F1^{-1}(1/(Cβ^2)), the inverse gives Θα(β) ~ (Cβ^2)^{1/(2α-n/2+1)}. These exponents differ unless α=n/4. For n=3, α=1/2, the paper's formula gives β^3; the correct inversion gives β^4. This is load-bearing: it changes the beta-scaling of the claimed sharp moment exponent.\n\nThere's also a p-dependence slip in Theorem 14. Hypercontractivity introduces a factor (p-1)^{k/2}, so the convergence condition is Cβ^2(p-1)F_i(ρ)<1, meaning the argument should be β√(p-1), not β p/(p-1). Corollary 15 confirms the √(p-1) form.\n\nThese look like correctable typos, not conceptual failures. The lower-bound section (Theorem 22) appears to aim at the same large-β scaling, though I'd want the authors to re-verify the stated normalizers there too. The paper would be a solid contribution once these are fixed.\n\nMy recommendation: send it to a serious referee, but insist the author correct the Θα inversion and the p-dependence before acceptance. The core result is significant enough to warrant referee time.","headline":"Right problem, right framework, but the paper's central large-β asymptotics are wrong as written; the errors look fixable and the core result is worth refereeing.","tokens_in":20022,"tokens_out":12652,"would_cite":false,"duration_ms":89864,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H15","58J65","35R60","60H07","58J35"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that on a Cartan-Hadamard manifold with sectional curvature bounded above by a negative constant, the p-th moment Lyapunov exponent of the stochastic heat equation is bounded by a curvature-corrected expression, and…","keywords":["stochastic heat equation","parabolic Anderson model","Cartan-Hadamard manifold","Lyapunov exponent","intermittency","heat kernel estimates","colored noise","fractional Laplacian"],"falsifier":"Invert the function $F_1(\\rho)$ from Lemma 11 and compare the resulting $\\Theta_\\alpha(\\beta) = F_1^{-1}(1/(C\\beta^2))$ with the asymptotics stated in Theorem 13; the exponents differ (the inverse gives $(C\\beta^2)^{1/(2\\alpha - n/2 +1)}$, while Theorem 13 states $(C\\beta^2)^{1-2\\alpha+n/2}$), so a direct check of this identity settles whether the large-$\\beta$ sharpness claim is correctly stated.","tokens_in":18932,"feed_emoji":"📉","tokens_out":9624,"duration_ms":74521,"temperature":0.7,"pith_summary":"This paper studies the parabolic Anderson model on a Cartan-Hadamard manifold: a complete, simply connected Riemannian manifold of non-positive sectional curvature, driven by a Gaussian noise that is white in time and colored in space. It constructs the fractional noises $W_\\alpha$ parameterized by a regularity exponent $\\alpha$ and proves that a mild solution exists and is unique exactly when $\\alpha > (n-2)/4$, the manifold analogue of Dalang's condition. The main result is an exponential-in-time upper bound for the $p$-th moments: under $\\sec M \\le -K_1 < 0$, the $p$-th moment Lyapunov exponent is at most $\\frac{p}{2}\\big(\\Theta_\\alpha(\\frac{p}{p-1}\\beta) - \\frac{(n-1)^2}{\\max(2,r)}K_1\\big)$, where $\\Theta_\\alpha$ vanishes for small $\\beta$ and grows like a power of $\\beta^2$ for large $\\beta$. Under the additional lower curvature bound $\\sec M \\ge -K_2 > -\\infty$, matching lower bounds are obtained in the large-$\\beta$ limit, so the upper bound is sharp there. Together these bounds imply intermittency: for large enough $p$, the ratio of the $p$-th to the $q$-th moment norms diverges exponentially in time.","feed_headline":"Negative curvature caps stochastic heat moment growth","feed_subtitle":"On Cartan-Hadamard manifolds, p-th moment Lyapunov exponents pick up a spectral-gap correction, with matching lower bounds at high noise.","key_machinery":"The load-bearing object is the Davies–Mandouvalos-type heat kernel bound of Theorem 3, $P_t(x,y) \\le C h(K_1 t, \\sqrt{K_1}d(x,y))$ with $h(t,z) \\asymp t^{-n/2}(1+t+z)^{\\frac{n-3}{2}}(1+z)e^{-z^2/(4t)-(n-1)^2t/4-(n-1)z/2}$, obtained by combining the Cheeger–Yau comparison theorem with the explicit hyperbolic-space kernel. This bound injects the exponential factor $e^{-(n-1)^2 K_1 t/4}$ into every chaos term, which makes the renewal kernel $\\Psi(t) = e^{2bt}\\sup_x\\|(-\\Delta)^{-\\alpha}P_t(x,\\cdot)\\|^2_{L^2}$ integrable and produces the curvature correction in the Lyapunov exponent. The argument runs through a renewal inequality $\\mathcal{N}_{k+1}(t) \\le \\int_0^t \\Psi(t-s)\\mathcal{N}_k(s)\\,ds$, whose Laplace transform yields the functions $F_i$ whose inverse defines $\\Theta_\\alpha$, followed by hypercontractivity in each Wiener chaos to pass from second to $p$-th moments. For the lower bound, the Feynman–Kac formula for moments rewrites $\\mathbb{E}[|u(t,x)|^p]$ as an expectation over $p$ independent Brownian motions with a self-intersection exponential, and the correlation lower bound of Lemma 18 supplies the small-distance singularities of $G_{2\\alpha}$ that drive the large-$\\beta$ growth.","core_discovery":"The paper's central claim is that negative sectional curvature measurably slows the growth of moments of the stochastic heat equation, and that this slowdown is exactly the spectral-gap quantity $(n-1)^2 K_1/4$ entering through the heat kernel. Concretely, Theorem 14 asserts that for $p\\ge 2$ and $u_0 \\in L^\\infty(M)\\cap L^r(M)$, $\\limsup_{t\\to\\infty} \\frac{1}{t}\\ln \\mathbb{E}[u(t,x)^p] \\le \\frac{p}{2}\\big(\\Theta_\\alpha(\\frac{p}{p-1}\\beta) - \\frac{(n-1)^2}{\\max(2,r)}K_1\\big)$, with $\\Theta_\\alpha$ the inverse of a Laplace-transform kernel coming from the fractional Laplacian of the heat kernel. Theorem 22 asserts that, if the sectional curvature is also bounded below by $-K_2$, then in the limit $\\beta\\to\\infty$ the moment growth matches, up to constants, the powers of $\\beta^2$ predicted by the upper bound—so the bound is sharp at high noise. The paper further claims that for small $\\beta$ the moments decay exponentially when $r<\\infty$ and have zero Lyapunov exponent for $r=\\infty$, a curvature-induced phase transition. Finally, combining upper and lower bounds yields intermittency in the sense of diverging normalized moment ratios.","pith_inferences":["The same renewal-and-hypercontractivity machinery should transfer verbatim to any manifold whose heat kernel satisfies the Davies–Mandouvalos upper bound (3), such as the asymptotically hyperbolic manifolds cited by the authors; testing the Lyapunov exponent on such an example would separate the role of curvature from the role of the heat-kernel decay rate.","The exact value of $\\Theta_\\alpha(\\beta)$ away from $\\beta\\to\\infty$ is not identified; a natural conjecture is that the second-moment Lyapunov exponent equals the spectral radius of the renewal operator (the infimum of $\\rho$ such that the Laplace transform $\\widehat{\\Psi}(\\rho)<(C\\beta^2)^{-1}$), which could be checked numerically on the hyperbolic plane for intermediate $\\beta$.","Because the lower bound uses only the small-distance singularity of $G_{2\\alpha}$ and the Dirichlet eigenvalue bound $\\lambda(x,R)\\le c/R^2 + C$, the intermittency threshold in $p$ is essentially determined by the local regularity of the noise; one prediction is that on manifolds where the heat kernel has slower decay (e.g., asymptotically flat manifolds) the intermittency threshold shifts toward "],"forward_implications":["Well-posedness: for every $\\beta>0$ and every noise regularity $\\alpha>(n-2)/4$, the mild solution exists and is unique; the threshold is exactly Dalang's condition and is sharp in the sense that the chaos series diverges for $\\alpha \\le (n-2)/4$.","Small-noise decay: if $\\beta$ is below the threshold $\\beta_c$ defined by $\\Theta_\\alpha(\\beta_c) = \\frac{(n-1)^2}{\\max(2,r)}K_1$, the $p$-th moment Lyapunov exponent is zero for $r=\\infty$ and strictly negative for $r< \\infty$, so negative curvature makes moments decay even with multiplicative noise.","Sharpness at high noise: in the regime $\\beta\\to\\infty$, the lower bound of Theorem 22 grows with the same power of $\\beta^2$ (up to logarithmic corrections at $\\alpha=n/4$) as the upper bound, so the curvature-corrected exponent is asymptotically exact.","Intermittency: for any fixed $\\beta>0$ and $q\\ge 2$, there is $p_0>q$ such that for $p>p_0$ the ratio $\\mathbb{E}[|u|^{p}]^{1/p}/\\mathbb{E}[|u|^{q}]^{1/q}$ tends to infinity in time, so the solution develops increasingly high peaks."],"supporting_citations":[{"why":"Supplies the heat kernel comparison $P_t(x,y)\\le P^{K_1}_t(d(x,y))$ that is the starting point of Theorem 3.","marker":"[CY81]"},{"why":"Gives the two-sided Davies-Mandouvalos heat kernel bounds on hyperbolic space whose scaled form is used throughout.","marker":"[DM88]"},{"why":"Provides the chaos-expansion, renewal-inequality, and Feynman-Kac methodology that the present paper adapts to manifolds, including the interpolation argument for $|P_t u_0|$ decay.","marker":"[BCH+24]"},{"why":"Supplies the renewal procedure (Laplace transform of the renewal kernel) used to bound the chaos series.","marker":"[KK15]"},{"why":"Gives the lower bound $(n-1)^2 K_1/4$ on the bottom of the $L^2$ spectrum, the constant that the curvature correction is compared with.","marker":"[McK70]"},{"why":"Defines the Itô-Walsh stochastic integral used in the mild formulation of the equation.","marker":"[Wal86]"},{"why":"Gives the integrability condition that motivates the threshold $\\alpha>(n-2)/4$.","marker":"[Dal01]"},{"why":"Supplies the Wiener chaos expansion and hypercontractivity estimate used for $p$-th moments.","marker":"[Nua95]"},{"why":"Gives the Dirichlet eigenvalue bound $\\lambda(x,R)\\le c/R^2+C$ used in the lower bound for moments.","marker":"[Ber23]"}],"fun_headline_variants":["Negative curvature tames stochastic heat moments","Curvature-induced cap on heat moment growth","Heat moments feel the pull of negative curvature","Moment estimates show curvature slows heat growth","Stochastic heat moment growth slowed by geometry"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything in the upper-bound half rests on the heat kernel bound $P_t(x,y)\\le C h(K_1 t,\\sqrt{K_1}d(x,y))$ delivered by the curvature assumption $\\sec M\\le -K_1<0$; if that exponential decay factor $e^{-(n-1)^2K_1t/4}$ is absent or weaker, the renewal kernel need not be integrable and the curvature correction in the Lyapunov exponent collapses.","fun_headline_variants_meta":{"raw":{"variants":["Negative curvature tames stochastic heat moments","Curvature-induced cap on heat moment growth","Heat moments feel the pull of negative curvature","Moment estimates show curvature slows heat growth","Stochastic heat moment growth slowed by geometry"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000245,"raw_usage":{"total_tokens":1534,"prompt_tokens":941,"completion_tokens":593,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":557,"completion_tokens_details":{"reasoning_tokens":527}},"tokens_in":557,"tokens_out":593,"duration_ms":5891,"temperature":1.0,"reasoning_tokens":527,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:29:08.873434+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Invert the function $F_1(\\rho)$ from Lemma 11 and compare the resulting $\\Theta_\\alpha(\\beta) = F_1^{-1}(1/(C\\beta^2))$ with the asymptotics stated in Theorem 13; the exponents differ (the inverse gives $(C\\beta^2)^{1/(2\\alpha - n/2 +1)}$, while Theorem 13 states $(C\\beta^2)^{1-2\\alpha+n/2}$), so a direct check of this identity settles whether the large-$\\beta$ sharpness claim is correctly stated.","supporting_citations":[],"review_version":1}