{"id":"147a09d0-0de8-4b10-b5c0-89930b37e5d0","arxiv_id":"1908.07655","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For symmetric pure jump Dirichlet forms, two-sided heat kernel estimates, jumping kernel bounds, and Sobolev/Faber-Krahn/Poincaré inequalities are mutually stable under the two-scale assumptions.","lead":"The paper proves that for a broad class of symmetric jump processes, sharp two-sided heat kernel bounds are equivalent to a set of standard analytic inequalities (Faber-Krahn, Poincaré, cut-off Sobolev) plus two-sided jumping kernel bounds. It extends earlier stability results to processes with light tails whose large jumps are rare, so the process can become diffusive at large scales.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main equivalences are conditional on an externally supplied scale function φc that condition (1.13) does not uniquely determine; without a construction or uniqueness test, the stability claim is not a property of the Dirichlet form alone.","rationale":"The reader's weakest assumption correctly identifies the free choice of φc as the most load-bearing unresolved point. I read the full manuscript and did not find a specific internal contradiction in the main chain of implications; the proof of Proposition 3.5, the principal new technical step, is given in detail, and the surrounding implications are sensibly delegated to the companion papers [CKW1], [CKW2], and [CKW4]. However, the theorems are explicitly conditional on the user supplying a correct scale function φc, and the paper states both that no universal formula for φc is known and that a bad choice can make all statements fail. That admission confirms that the characterization is not a property of the Dirichlet form alone; rather, it characterizes equivalence inside a family of pairs (φj,φc). This warrants a conditional verdict, which is exactly what the reader gave. I do not see grounds to move the verdict to ACCEPT or REJECT: the mathematical claim, read as a conditional equivalence, is not refuted by the manuscript text, and the deferred proofs are a verification burden rather than a demonstrated error. The concrete test above would settle whether the φc-dependence can produce inconsistent truth values, and would sharpen the recorded limitation into either a counterexample or a confirmation that the free input only affects applicability, not logical soundness.","tokens_in":37890,"tokens_out":16863,"duration_ms":156704,"concrete_test":"Take the subordinate process of Example 1.1 with a fixed pair (β,α1,α2) and VD/RVD manifold. Enumerate all admissible φc satisfying (1.10) and (1.13), including the true scale φc(r)=r^β on [1,∞) and at least one deliberately larger and one deliberately smaller admissible choice. For each φc, evaluate the six conditions in Theorem 1.11 directly from the known heat kernel. If some admissible φc gives mixed truth values (e.g., PI(φ), Jφj and Eφ true while HK−(φj,φc) fails), the equivalence is false as stated. If all admissible φc choices give the same all-true or all-false pattern, then the free choice of φc does not affect the stability claim, and the remaining issue is only how to identify an admissible φc for a new process.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.11 and its companions are stated for arbitrary pairs (φj,φc) satisfying (1.10) and (1.13), but (1.13) only imposes the one-sided bound φc≤c0φj, not an intrinsic determination by (M,d,μ,J). The authors explicitly disclaim a universal formula for φc and warn that a bad selection of φc can make every statement in Theorems 1.11 and 1.12 false. Consequently, for a given symmetric pure-jump Dirichlet form, the theorems do not give a decision procedure: one must guess the correct φc before the equivalences can be applied. The proofs show equivalence within the subset of admissible φc for which one of the six or seven conditions holds, but they do not show that this subset is non-empty, unique, or algorithmically identifiable from J and μ. This is not an internal inconsistency, but it is a genuine soft spot in the central claim: the characterization is not closed under the data of the Dirichlet form, and the rôle of φc is load-bearing rather than decorative.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies symmetric pure-jump Dirichlet forms on metric measure spaces whose jumping kernel may have different scaling at small and large jumps, including light-tailed kernels with possible diffusive large-scale behavior. Two scale functions φj and φc are fixed, satisfying regularity conditions (1.10) and the comparability condition (1.13), and φ is built from them via (1.14). The main results, Theorems 1.11, 1.12 and 1.15, assert stability/equivalence: two-sided heat kernel estimates HK−(φj,φc) and HK(φj,φc), upper estimates UHK(φj,φc), diagonal upper bounds, Faber–Krahn inequalities, cut-off Sobolev inequalities, Poincaré inequalities, generalized capacity inequalities, jumping-kernel bounds, and parabolic Harnack inequalities are mutually equivalent under VD, RVD and (1.10)–(1.13). The proofs are largely reductions to the authors' earlier works CKW1, CKW2 and CKW4, with a detailed proof of Proposition 3.5 as the main new analytic step. A motivating example of a subordinate diffusion with light tails is worked out in the appendix.","tokens_in":38120,"tokens_out":4852,"duration_ms":142153,"significance":"If the stated equivalences hold, the paper substantially extends the stability program for non-local Dirichlet forms: it covers jumping kernels whose lower scaling index can be larger than 1 on the large-scale part, it accommodates cases where the heat-kernel scaling differs from the jumping-kernel scaling, and it provides stable characterizations of parabolic Harnack inequalities with the UJS condition. The motivating example is instructive and the detailed proof of Proposition 3.5 is a genuine technical contribution. The paper also explicitly delineates the relation to, and differences from, the parallel work of Bae–Kang–Kim–Lee (BKKL1, BKKL2). However, the central equivalences are conditional on a freely chosen scale function φc, and several load-bearing proofs are omitted with references to unpublished or hard-to-verify companion papers; these points limit the present form of the manuscript.","major_comments":[{"comment":"The scale function φc is a free functional input that is not intrinsically determined by the Dirichlet form data (M,d,μ,J). Condition (1.13) only imposes the one-sided bound φc≤c0φj on the relevant intervals, and the authors explicitly state that they have no universal formula for φc and that a bad selection of φc can make every statement in Theorems 1.11 and 1.12 false. Consequently, the equivalences are not closed under the data of the Dirichlet form: for a concrete process, one must guess an admissible φc before the characterization can be applied. This is not an internal contradiction, but it weakens the claim of a 'stable characterization' of heat kernel estimates. The manuscript should either provide a constructive or algorithmic determination of φc in the cases covered, or reformulate the theorems as equivalences quantified over admissible pairs (φj,φc) and discuss the non-uniqueness and its consequences explicitly.","section":"§1.3, Theorems 1.11 and 1.12, and the paragraph after Theorem 1.12"},{"comment":"Several load-bearing results are not proved in the text but are deferred to earlier papers, some of which are unpublished or lack verifiable identifiers. Specifically, Proposition 2.2 (UHK + conservativeness implies Jφj,≤, and HK− implies Jφj), Proposition 3.2 (FK + Eφ + Jφ,≤ implies UHKD), Lemma 3.3 (heat kernel upper bound for the truncated form), Lemma 3.4 (tail estimate implies off-diagonal upper bound), Lemma 3.6 (subexponential off-diagonal decay), and Proposition 4.3 (HK− implies HK under connectedness and chain condition) are essential inputs to Theorems 1.11 and 1.12. The reference [CKW4] is listed as 'available at arXiv' without an arXiv number, so the reader cannot verify the quoted arguments. Since Proposition 3.5, the paper's main new proof, relies on Lemmas 3.4 and 3.6, the omitted proofs are not merely cosmetic. The authors should either include full proofs or provide exact statements with complete references to published or identifiable sources.","section":"§2.1, §3.2, §4.2"},{"comment":"The first assertion of Theorem 1.15, namely PHI(φ) ⇔ PI(φ) + Jφ,≤ + CSJ(φ) + UJS, is one of the three main results of the paper, yet its proof is dismissed in a single sentence: 'the first assertion of Theorem 1.15 can be established by the same arguments in [CKW2, Subsection 4.3]'. Given that the setting here is more general than CKW2 (light tails, φc not necessarily equal to φj, and the new UJS condition), the proof should be at least outlined, and the precise modifications needed for the present case should be stated. As written, the main characterization of parabolic Harnack inequalities is not independently verifiable from this manuscript.","section":"§5.1, Theorem 1.15"}],"minor_comments":[{"comment":"The phrase 'measure metric space' should be 'metric measure space'.","section":"§1.2, first sentence"},{"comment":"The first condition in (1.13) should presumably read φc(r)≤c0φj(r) on [0,1] if β∗>1 (with the small-scale index), while the text currently repeats β∗>1 in both conditions; please check and correct the notation.","section":"Equation (1.13)"},{"comment":"In the definition of c∗ = (1 + (2c1c5/c2))^{1/(β1,φc−1)}, the constants c1, c5 and c2 are not clearly identified before use; please define them or refer to the displayed inequalities where they first appear.","section":"§3.3, proof of Proposition 3.5, step (ii)"},{"comment":"There is a typo: 'Propisition' should be 'Proposition'.","section":"Proposition 4.1"},{"comment":"'in the sprit of' should be 'in the spirit of'.","section":"Remark 1.18(i)"},{"comment":"The reference [CKW4] is cited as 'available at arXiv' but no arXiv identifier is given; this makes the extensive reliance on it impossible to check.","section":"Reference [CKW4]"}],"recommendation":"major_revision","confidential_remarks":"The paper is a substantial sequel in a well-established line of work, and the conditional stability results are plausible and important. The main concerns are (i) the free and non-unique scale function φc, which the authors themselves acknowledge, and (ii) the unusually large number of proofs deferred to unpublished or incompletely identified companion papers, including [CKW4]. For a journal referee, this makes verification difficult. I would ask the authors to either include the missing proofs or cite only published sources, and to give a more precise discussion of what it means for φc to be 'intrinsically determined by φj and the metric measure space'. I do not see an internal inconsistency in the main derivation; the issues are about scope and verifiability, so major revision rather than rejection seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Chen, Kumagai and Wang have written a real sequel to their CKW stability program. The genuinely new territory is pure jump Dirichlet forms with light tails, where the local scaling indices β* and β* are allowed to be ≤1, so the small-jump and large-jump scales can differ. That case is not covered by CKW1/2/4 or by BKKL2, and the paper's main equivalences (Theorems 1.11, 1.12, 1.15) are plausible and carefully delimited. The load-bearing new step is Proposition 3.5, and they give it a detailed proof. The paper also deserves credit for being upfront about its limitations, including the warning that a bad choice of φc makes the statements false.\n\nThe soft spot is the one flagged in the stress test: the main theorems are conditional on a scale function φc supplied by the user. Condition (1.13) is only a one-sided comparability on one interval, not a determination. The authors state they have no universal formula, so for a given (M,d,μ,J) the characterization is not intrinsic. This is a genuine limitation, and it is load-bearing, but it is not an internal flaw—the proofs show equivalence inside the class of admissible φc. A referee will want the role of φc clarified, ideally with a discussion of when it is unique or how to test candidate choices.\n\nA lesser soft spot is the number of deferred proofs. Several supporting results (Propositions 2.2 and 3.2, Lemmas 3.3, 3.4, 3.6, Proposition 4.3) are stated as having the same proof as earlier work. That is acceptable for a paper in a series, but a referee should check the transfers, especially where the two-scale form changes the argument. The heavy self-citation is legitimate—CKW1/2/4 are independent published results—but it makes the paper hard to read in isolation.\n\nThis is a paper for specialists in Dirichlet forms and heat kernel estimates. It will become a standard toolbox reference if the φc caveat is handled honestly, and it deserves a serious referee. I would recommend acceptance after revision, not desk rejection.","headline":"A substantial, mostly solid extension of the CKW program to light-tailed jump kernels; the free functional input φc is a real caveat, but the paper deserves a serious referee.","tokens_in":38692,"tokens_out":3287,"would_cite":true,"duration_ms":31082,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J35","35K08","60J75","31C25","60J25","60J45"],"pacs":[],"model":"deepseek-v4-flash","headline":"Stability theorem pins down heat kernels of jump processes","keywords":["symmetric non-local Dirichlet form","pure jump process","heat kernel estimate","jumping kernel","cut-off Sobolev inequality","Faber-Krahn inequality","Poincaré inequality","parabolic Harnack inequality"],"falsifier":"Exhibit a metric measure space with volume doubling and a jump kernel with light tails for which two different admissible φc both satisfy (1.10)-(1.13) but lead to different heat kernel regimes (e.g., one diffusive, one not), or for which no admissible φc exists; this would show that the equivalence is not an intrinsic property of the jump process alone.","tokens_in":37680,"feed_emoji":"⚡","tokens_out":8683,"duration_ms":229581,"temperature":0.7,"pith_summary":"The paper tries to determine exactly when a symmetric pure-jump Markov process on a doubling metric measure space has a two-sided heat kernel estimate, including processes whose large jumps are so light that long-time behavior becomes diffusive. Its central result is a set of equivalences: the upper-and-lower heat kernel bound holds if and only if the jumping kernel is comparable to a prescribed rate φj, the Poincaré inequality holds with rate φ, and a cut-off Sobolev inequality (or equivalently a Faber-Krahn or generalized capacity inequality) holds. Because large jumps can be light, the formulation needs two scale functions: φj for the jump kernel and an intrinsic φc for the large-scale motion, and the claims are only asserted relative to a correct choice of φc. The paper also gives a stable characterization of the parabolic Harnack inequality by the same Poincaré and Sobolev conditions plus an averaging condition on the jump kernel.","feed_headline":"Stability theorem pins down heat kernels of jump processes","feed_subtitle":"Equivalence with Poincaré, jump-kernel bounds and cut-off Sobolev inequalities covers light-tailed jumps.","key_machinery":"The central object is the two-scale heat kernel ansatz: p^(j)(t,x,y)=1/V(x,$φj^{{-1}}$(t)) ∧ t/(V(x,d(x,y))φj(d(x,y))) for the jump part, and p^(c)(t,x,y)=1/V(x,$φc^{{-1}}$(t)) exp(-d(x,y)/φ̄$c^{{-1}}$(t/d(x,y))) for the diffusive part, where φ̄c(r)≈φc(r)/r. The function φ is defined by switching between φj and φc according to the lower scaling indices β_* and β_* from (1.14), and the cut-off Sobolev inequality CSJ(φ) controls the energy of cut-off functions at scale r by the jump energy in an enlarged annulus plus φ(r) times the L² mass. The load-bearing mechanism is Proposition 3.5, which uses truncation at scale ρ and Meyer's decomposition to split the heat kernel into a truncation term and a tail term, then estimates the crossing point r*(t) where the exponential 'diffusive' term and the jump term are comparable; this yields the full upper bound UHK(φj,φc) from the diagonal bound, the jump-kernel upper bound, and the expected exit time bound.","core_discovery":"On a metric measure space satisfying volume doubling and reverse volume doubling, take a symmetric pure jump Dirichlet form with jump measure J and scale functions φj, φc satisfying (1.10)-(1.13), and write φ(r)=φj(r) on the scale where β_*≤1 and φc(r) where β_*>1 (and similarly for large r). The main theorem states that the following are equivalent: the heat kernel upper bound with near-diagonal lower bound HK-(φj,φc); the upper heat kernel bound plus near-diagonal lower bound and two-sided jump kernel bounds Jφj; the diagonal upper bound plus Dirichlet near-diagonal lower bound and Jφj; and any of the combinations PI(φ)+Jφj+Eφ, PI(φ)+Jφj+Gcap(φ), or PI(φ)+Jφj+CSJ(φ); with connectedness and the chain condition, all of these are also equivalent to the full two-sided heat kernel estimate HK(φj,φc). A second theorem characterizes the parabolic Harnack inequality PHI(φ) as exactly PI(φ)+Jφ,≤+CSJ(φ)+UJS, and consequently HK-(φj,φc) ⇔ PHI(φ)+Jφj. The heat kernel form itself is the minimum of a jump-kernel term p^(j)(t,x,y)=1/V(x,$φj^{{-1}}$(t)) ∧ t/(V(x,d)φj(d)) and, on diffusive scales, a term p^(c) with exponential decay driven by the derivative of φc.","pith_inferences":["Because the paper offers no construction of φc, the practical content of the equivalence is 'for each admissible φc'; a natural next step would be to prove that φc is uniquely determined by φj and the metric measure space, for instance as the smallest rate for which the diagonal heat kernel satisfies UHKD(φ).","The UJS condition in the PHI characterization is an averaging condition rather than a pointwise one; this suggests that PHI may hold even when the jump kernel is not pointwise two-sided, and one could test whether a one-sided average bound suffices in the light-tail regime.","The switching construction of φ in (1.14) depends on the indicators of β_*≤1 and β_*≤1, but the paper's own examples show the real transition is at α2=β, so the formulation might be made coordinate-free by replacing the indicator functions with a threshold defined by the heat kernel itself."],"forward_implications":["To prove a two-sided heat kernel estimate it is enough to verify a Poincaré inequality, two-sided jump-kernel bounds, and a cut-off Sobolev inequality at the same rate φ; conversely, any process with such an estimate satisfies all of these.","The parabolic Harnack inequality holds for a pure jump process exactly when Poincaré, upper jump-kernel bounds, the cut-off Sobolev inequality, and the UJS averaging condition hold, so PHI can be checked without knowing the heat kernel.","Heat kernel estimates and PHI transfer between any two jump processes whose jumping kernels are comparable, so a single model process can serve as a base for a whole class.","Processes with light large jumps (large-scale lower index >1) can have diffusive long-time behaviour, and their heat kernels then contain a 'super-Gaussian' exponential term governed by φc rather than by the jump kernel.","The results recover earlier stability theorems for pure jump processes (where φ=φj) and for symmetric diffusions with jumps as special cases, with the time intervals (0,1] and (1,∞) interchanged."],"supporting_citations":[{"why":"Supplies the stability framework and the cut-off Sobolev inequality CSJ(φ) used throughout Theorems 1.11-1.12; most proofs in Sections 2-4 follow its arguments.","marker":"[CKW1]"},{"why":"Its Proposition 4.6 and Lemmas 4.5/4.7 are the model for Proposition 3.5 and Lemma 3.6, the core step deriving upper heat kernel estimates.","marker":"[CKW4]"},{"why":"Provides the characterizations of parabolic Harnack inequalities and the implication NDL(φ)⇒PI(φ) used in Theorem 1.15 and Proposition 4.1.","marker":"[CKW2]"},{"why":"Lemma 4.3 is the source of the tail-estimate lemma 3.4; also used in Example 5.2 for PI(φ) and exit-time bounds.","marker":"[BKKL1]"},{"why":"Used for regularization of transition densities and for Lemma 3.6-type heat kernel bounds, and in Example 1.1.","marker":"[BBCK]"},{"why":"Supplies the generalized capacity inequality Gcap(φ) and the lemmas connecting Gcap(φ), CSJ(φ) used in Propositions 2.4-2.5.","marker":"[GHH]"},{"why":"Independent simultaneous work covering the case β_*∧β_*>1; Remark 1.18 uses it to argue that its setting requires β_*>1 and to compare heat kernel formulations.","marker":"[BKKL2]"},{"why":"Used for the chaining argument proving the full off-diagonal lower bound HK(φj,φc) from HK-(φj,φc) under the chain condition, and for heat-kernel comparison in Remark 1.18.","marker":"[BGK]"}],"fun_headline_variants":["Jump process heat kernels pinned by Sobolev, Poincaré bounds","Equivalence theorem for heat kernels of pure jump Dirichlet forms","Heat kernel estimates stable under jump measure conditions","Pure jump heat kernels: stability via cut-off Sobolev and Poincaré","Jump processes: heat kernel bounds characterized by analytic inequalities"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The user must supply a scale function φc satisfying (1.10) and (1.13), and the paper states that there is no universal formula for φc; a bad choice of φc can make every equivalent statement fail.","fun_headline_variants_meta":{"raw":{"variants":["Jump process heat kernels pinned by Sobolev, Poincaré bounds","Equivalence theorem for heat kernels of pure jump Dirichlet forms","Heat kernel estimates stable under jump measure conditions","Pure jump heat kernels: stability via cut-off Sobolev and Poincaré","Jump processes: heat kernel bounds characterized by analytic inequalities"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000692,"raw_usage":{"total_tokens":3188,"prompt_tokens":1054,"completion_tokens":2134,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":670,"completion_tokens_details":{"reasoning_tokens":2048}},"tokens_in":670,"tokens_out":2134,"duration_ms":16919,"temperature":1.0,"reasoning_tokens":2048,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:00:12.716532+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a metric measure space with volume doubling and a jump kernel with light tails for which two different admissible φc both satisfy (1.10)-(1.13) but lead to different heat kernel regimes (e.g., one diffusive, one not), or for which no admissible φc exists; this would show that the equivalence is not an intrinsic property of the jump process alone.","supporting_citations":[],"review_version":1}