{"id":"2c7ea711-61fd-4d0c-949d-3e34e5550e1a","arxiv_id":"2512.24807","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Sharp two-sided heat-kernel estimates are established for symmetric jump Markov processes with jump kernels that blow up at the boundary of a κ-fat domain, under a strict bound on the blow-up's Matuszewska index.","lead":"This paper proves matching upper and lower bounds for the heat kernel — the probability density of a random jump process moving from x to y in time t — for a wide class of processes whose jump rate blows up near the boundary of the state space. It is the first result of this sharpness for such processes, and it covers stable-process traces, nonlocal Neumann processes, and resurrected half-space processes.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's extension from t<R0^α to all t<T∨R0^α in §5.3 rests on an unproved, potentially t-dependent comparability of ep(t/n), ep(t/n^2), and ep(t); as written this leaves the claimed T-uniformity unverified.","rationale":"The reader's verdict is CONDITIONAL, and my stress-test supports that: no fatal flaw in the t<R0^α proof emerged from checking the bootstrap lemmas, Meyer decomposition, Hardy estimates, or the lower-bound arguments. The most load-bearing concern is the semigroup extension in §5.3, which the reader also flagged as an unproved ≍ chain. I do not regard the strict index hypothesis β<γ∧α as a defect: it is part of the stated assumptions, and all three applications verify it. The §5.3 gap is concrete and fixable, but until it is closed the theorem is not fully verified for t≥R0^α. I therefore keep the reader's CONDITIONAL verdict; no adjustment is needed.","tokens_in":60120,"tokens_out":51010,"duration_ms":457469,"concrete_test":"Write out and verify the missing comparison lemma for a fixed integer N≥1: for all t>0 and z,w∈D, ep(t/N,z,w) ≤ C_N ep(t,z,w) ≤ C_N' ep(t/N,z,w), with C_N,C_N' depending only on N, the constants in (2.5), and the parameters in (1.7). Then redo §5.3 with N=N(T)=⌊T/R0^α⌋+1 and check that the semigroup product chain p(t,x,y) ≍ ∫∏p(t/N,z_{i-1},z_i)dz_i gives constants independent of t∈[R0^α,T]. If the comparison lemma fails (for example, if the Φ-ratio cannot be controlled uniformly in z,w for fixed N), then (1.7) is not established for large t.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim (1.7) is stated for every T>0 with a constant C depending only on T, but the proof for t≥R0^α is contained in the six-line §5.3. The argument says: 'Since ep(t/n2,z,w)≍ep(t/n,z,w)≍ep(t,z,w)...' and then uses the semigroup property to compare p(t) with products of p(t/n) and p(t/n2). Three things are unproved there. First, ep(t,x,y) contains the factor Φ(((r∧A0)^2)/(((δD(x)∨t^{1/α})∧A0)((δD(y)∨t^{1/α})∧A0))); changing t to t/n changes this argument by an amount controlled only by the upper Matuszewska bound (2.5), and the resulting constant depends on n. Second, the proof chooses n∈N depending on t ('Choose n∈N such that t/n<R0^α'), so if the comparability constants grow with n, the final constant in (1.7) may depend on t, contradicting the theorem's 'for every T there exists C' uniformity. Third, the semigroup chain requires replacing each of the n factors p(t/n) by ep(t/n) and then by ep(t/n2); even if the pointwise comparison is true for fixed n, the product of n comparison constants must be uniformly bounded for all t≤T, which requires fixing N=N(T) once and for all. A repair is likely possible by taking N=⌊T/R0^α⌋+1 and proving a fixed-N comparison lemma using (2.5), but this lemma is absent. Because the theorem's full time range depends on this step, it is the most load-bearing unresolved point. The strict index assumption β<γ∧α is a genuine hypothesis, not an inconsistency: all three applications in §6 satisfy it, so the theorem does not overreach there.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes sharp two-sided heat kernel estimates for symmetric pure-jump Markov processes on κ-fat open sets D⊂R^d whose jump kernels are comparable to |x−y|^{-d−α} Φ((|x−y|∧A_0)^2 / ((δ_D(x)∧A_0)(δ_D(y)∧A_0))), where Φ is a blow-up weight of upper Matuszewska index β satisfying β < γ∧α with γ = d − dim_A(∂D). The main theorem (Theorem 1.2, eq. (1.7)) gives, for every T>0 and t<T∨R_0^α, two-sided bounds of the form p(t,x,y) ≍ t^{-d/α} ∧ [t |x−y|^{-d−α} Φ(...)] with a T-dependent constant. The proof develops weighted integral estimates via the framework of [14], a Hardy inequality adapted to Assouad codimension, truncated Dirichlet forms, Meyer-type decompositions via Mosco convergence, and a bootstrap argument to remove polynomial decay. Applications are given to the nonlocal Neumann problem, traces of isotropic α-stable processes in C^{1,Dini} sets, and resurrected processes in the closed upper half-space.","tokens_in":60533,"tokens_out":8243,"duration_ms":72549,"significance":"If correct, this is the first sharp heat-kernel result for the class of jump kernels that blow up on part of the state space, and it unifies several natural examples. The paper is not circular: the two-branch min-structure in (1.7) is derived, not assumed, and the proofs check the hypotheses of the imported weighted inequalities. The strict index condition β<γ∧α is explicit and is satisfied by all three applications, so the theorem does not overreach its assumptions. The main proof is long, but I found no internal contradiction in the part treating t<R_0^α. The treatment of t≥R_0^α, however, contains a genuine gap that must be repaired before the claimed T-uniform statement is justified.","major_comments":[{"comment":"The extension from t<R_0^α to t∈[R_0^α,T) is not proved. The proof asserts 'Since ep(t/n²,z,w)≍ep(t/n,z,w)≍ep(t,z,w)' and then uses the semigroup chain p(t)=p(t/n)^{*n}≍ep(t/n)^{*n}≍ep(t/n²)^{*n}=p(t/n)≍ep(t). Three things are missing. (i) The comparability ep(t/n^k)≍ep(t) is not uniform: the definition of ep contains (δ_D(·)∨(t/n^k)^{1/α})∧A_0 in the argument of Φ, and changing t to t/n^k changes this quantity by a factor controlled only through (2.5); the constants depend on n and possibly on t. (ii) Since n is chosen after t, any growth of these constants in n would make the final constant in (1.7) depend on t, contrary to the theorem's 'for every T there exists C' statement. (iii) The semigroup chain multiplies n comparison constants, so one needs a fixed N=N(T) and a uniform comparability lemma for all s∈[R_0^α,T] and k=0,...,N, with constants depending only on T. A repair is likely","section":"§5.3"},{"comment":"The Green-function comparability for C^{1,Dini} sets is imported by saying that the argument of [12] for C^{1,1} sets extends and 'we omit the details'. This is not a routine extension, and it is used essentially in Proposition 7.10(i) and therefore in Proposition 6.6 (the trace-process application). Either supply a proof of Proposition 7.9 from the available estimates, or state the C^{1,Dini} Green-function estimate as an explicit assumption and adjust the claims in Section 6 accordingly. The main theorem does not depend on this step, but the application's validity as written does.","section":"§7"}],"minor_comments":[{"comment":"The statement has a typo: 'f(·,·) : (0,∞)× ∞)→(0,1)' should be 'f(·,·) : (0,∞)×(0,∞)→(0,1)'.","section":"§4"},{"comment":"The parameter a appears in the integrand and on the right-hand side but is not quantified in the statement. It should read 'for all a>0, A>0, x,z∈D, r>0 and 0<u≤s'.","section":"§2"},{"comment":"The abstract says 'closed subsets F of R^d' while the body consistently works with an open set D and its closure D; please align the terminology.","section":"§1"},{"comment":"In the semigroup chain, the notation 'p(t/n, x, y_1)...p(t/n, y_{n−1}, y)' should make explicit that there are n convolution factors; this is clear from context but can be written more cleanly.","section":"§5.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is technically substantial and the main t<R_0^α argument appears sound, but §5.3 must be reworked; the current proof of the full time range is not sufficient. The Green-function extension in §7 is also asserted rather than proved. These are repairable, and I would not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real advance and the proof architecture holds up, but the paper needs one more pass on the time-extension step.\n\nThe new thing is real. Prior heat-kernel work on jump processes assumed uniformly bounded jump-measure tails. This paper removes that assumption for a substantial class — jump kernels that blow up at the boundary — and delivers matching two-sided bounds of the expected shape. The path is coherent: weighted inequalities from [14], Meyer decomposition via Mosco convergence to handle unbounded tails, and a two-stage bootstrap. The three applications (nonlocal Neumann, trace of stable processes, resurrected half-space processes) genuinely fit the framework, and they all satisfy the strict index condition β<γ∧α. That condition is real but not overreach.\n\nThe main soft spot is exactly the one flagged in the stress test. Section 5.3 extends the estimates from t<R0^α to all t<T∨R0^α by claiming, without proof, that ep(t/n^2) ≍ ep(t/n) ≍ ep(t) uniformly. The chosen n depends on t, and the comparability constants from the Matuszewska bound (2.5) will depend on n. So as written, the theorem's “for every T there exists C” uniformity is not established. This is not hard to repair — fix N=N(T), prove a lemma controlling ep(t/N^2) vs ep(t/N) vs ep(t) via (2.5), then run the semigroup chain with N fixed — but the authors need to do it. Until then, I'd treat the t<R0^α statement as fully proved and the full-range statement as conditional.\n\nTwo smaller things. First, Proposition 7.9 extends Green-function estimates to C^{1,Dini} sets by “we omit the details”; for the applications section that is a real omission, though not for the main theorem. Second, the metadata abstract promises a broader closed-set/Borel-Σ setting than the theorem's boundary blow-up for κ-fat open sets; the paper should align the abstract with what is proved.\n\nThe reliance on the authors' own previous papers [14] and [30] for key inputs is fine — they are published and the framework is consistent — but it means a referee cannot fully verify without those papers at hand.\n\nFor a serious referee: yes, send it. The gap is localized and likely fixable, and the result is important enough to justify the referee's time. If the authors fix §5.3 and tighten Prop 7.9, I'd be comfortable citing it as the definitive treatment for this class.","headline":"A genuine, mostly sound first sharp heat-kernel result for blow-up jump kernels that merits refereeing, but the proof of the full-time-range statement in §5.3 needs an explicit uniform comparability argument.","tokens_in":61131,"tokens_out":3719,"would_cite":true,"duration_ms":37959,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J35","60J45","31C25","35K08","60J46","60J50","60J76"],"pacs":[],"model":"deepseek-v4-flash","headline":"Sharp two-sided heat kernel estimates hold for jump processes whose kernels blow up at the boundary, under a strict index bound and a geometric condition on the boundary.","keywords":["heat kernel","Markov processes","Dirichlet forms","jump kernels blowing up at the boundary","stable processes","fractional Laplacian","trace process","nonlocal Neumann problem"],"falsifier":"Take the simplest domain D = R^d_+ with d≥2 and α∈(1,2), so γ=1 and γ∧α=1. Construct a process in the paper's framework with Φ(r)=1∨r (so β=1=γ∧α) and compute its heat kernel explicitly (or via high-precision simulation). The paper's bound (1.7) predicts p(t,x,y) ≍ min{t^{-d/α}, t|x−y|^{-d−α} (|x−y|^2/((x_d∨t^{1/α})(y_d∨t^{1/α})))}. A direct calculation showing an extra logarithmic factor or a different power of x_d∨t^{1/α} would falsify the theorem exactly at the threshold.","tokens_in":59874,"feed_emoji":"💥","tokens_out":5396,"duration_ms":52335,"temperature":0.7,"pith_summary":"The paper proves the first sharp two-sided heat kernel estimates for purely discontinuous symmetric Markov processes whose jump kernels blow up at a boundary subset. Under a geometric condition on the domain (κ-fat, with boundary Assouad dimension < d) and a strict index bound on the blow-up weight, it gives a single closed-form bound that interpolates between the usual t^{-d/α} diagonal term and a t|x-y|^{-d-α} term with a boundary-dependent blow-up factor. This matters because three canonical families — traces of α-stable processes, the nonlocal Neumann process, and resurrected processes in half-spaces — fit the framework, so their heat kernels are obtained as corollaries. The key novelty is that the jump-measure tails are not uniformly bounded, so standard off-diagonal techniques fail; the proof replaces them with weighted integral estimates tuned to both the blow-up profile and the domain geometry.","feed_headline":"Blowing-up jump kernels still give sharp heat kernel bounds","feed_subtitle":"First two-sided estimates for processes whose jump-measure tails are unbounded, covering three major examples.","key_machinery":"The load-bearing object is the blow-up weight function Φ and its upper Matuszewska index β, entering through the comparison (1.4). The new machinery is a chain of weighted integral estimates that replace the missing uniform tail bound: a Hardy-type inequality (Prop 3.2) that uses dim_A(∂D)<d, an admissibility check for Φ(r/δ_D(x)) in the weighted functional-inequality scheme, and two bootstrap lemmas (5.4, 5.10) that iteratively improve the off-diagonal decay of the truncated heat kernel until it reaches the sharp Φ-weighted form. Truncation at scales ρ, combined with Meyer's decomposition of the process, controls the large jumps that are no longer uniformly integrable.","core_discovery":"The central claim, Theorem 1.2, states: for a κ-fat open set D with dim_A(∂D) < d and a jump kernel of the form J(x,y) ≍ |x-y|^{-d-α} Φ(((|x-y|∧A0)^2)/((δ_D(x)∧A0)(δ_D(y)∧A0))), where Φ is a blow-up weight of upper Matuszewska index β < γ∧α with γ = d − dim_A(∂D), the associated regular Dirichlet form admits a jointly continuous heat kernel p(t,x,y) satisfying two-sided bounds of exactly the same form, with t^{-d/α} ∧ t|x-y|^{-d-α} Φ(...), uniformly on compact time intervals. The bounds are sharp up to multiplicative constants and remain valid for all times t < T∨R_0^α through a semigroup argument. The result is new because the unbounded tails of the jump measures preclude the uniform-tail a","pith_inferences":["The threshold case β = γ∧α is left open; at exactly this index, all the proof's positive exponents (α−β_1, d−β_1, γ−β_1) vanish, so the closed form (1.7) cannot follow from the present bootstrap. A plausible extension is that an extra logarithmic factor in time appears at criticality.","The method suggests a transfer principle: any jump process whose kernel comparison satisfies the single index condition β < γ∧α on a κ-fat set inherits full two-sided heat kernel bounds, regardless of the specific form of Φ, so future examples can be verified by checking only (1.4).","A concrete testable extension is to compute or simulate the heat kernel of the resurrected process on the half-space with Φ(r)=1∨r^{γ} (β=γ); a deviation from (1.7) would precisely identify the failure at the boundary case.","Since the estimates hold for all t up to T∨R_0^α, the same proof should extend to global time in any domain with localization constant R_0=∞ (e.g., half-spaces and complements of bounded sets), giving unbounded-time two-sided bounds."],"forward_implications":["The heat kernel of the nonlocal Neumann process on Lipschitz domains (bounded, half-space-like, or exterior) is now known in closed two-sided form: t^{-d/α} on the diagonal, and t|x-y|^{-d-α} log(e + (|x-y|∧A)^2/((δ∧t^{1/α})^2)) off diagonal (Theorem 6.4).","The trace of the α-stable process on C^{1,Dini} sets has heat kernel estimates with Φ(r)=1∨r^{α/2} (Theorem 6.7).","The resurrected process in the closed upper half-space, including the trace process and the Neumann process as special cases, has heat kernel estimates with Φ = Ψ_1, an integrated weak-scaling function, global in time (Theorem 6.9).","The estimates are sharp: upper and lower bounds match up to a multiplicative constant, including the boundary-sensitive correction in the Φ-argument, so no improvement is possible within the assumed class.","Because the form is conservative, the semigroup preserves total mass; the heat kernel estimates therefore also describe the long-time behavior of the processes in unbounded domains."],"fun_headline_variants":["First sharp heat estimates for blow-up jump kernels","Sharp heat kernel bounds for processes with blowing-up jumps","First two-sided heat estimates despite unbounded jump tails","Blow-up jump kernels: first sharp heat estimates","Sharp heat kernel bounds when jump kernels blow up"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The entire proof rests on the strict index bound β < γ∧α: the blow-up weight must grow more slowly than the geometric parameter γ = d − dim_A(∂D) and than the stability index α; if it blows up at or faster than that rate, the closed-form estimate collapses and the imported weighted inequalities stop applying.","fun_headline_variants_meta":{"raw":{"variants":["First sharp heat estimates for blow-up jump kernels","Sharp heat kernel bounds for processes with blowing-up jumps","First two-sided heat estimates despite unbounded jump tails","Blow-up jump kernels: first sharp heat estimates","Sharp heat kernel bounds when jump kernels blow up"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00097,"raw_usage":{"total_tokens":4027,"prompt_tokens":874,"completion_tokens":3153,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":3089}},"tokens_in":618,"tokens_out":3153,"duration_ms":20256,"temperature":1.0,"reasoning_tokens":3089,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T13:15:32.719912+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the simplest domain D = R^d_+ with d≥2 and α∈(1,2), so γ=1 and γ∧α=1. Construct a process in the paper's framework with Φ(r)=1∨r (so β=1=γ∧α) and compute its heat kernel explicitly (or via high-precision simulation). The paper's bound (1.7) predicts p(t,x,y) ≍ min{t^{-d/α}, t|x−y|^{-d−α} (|x−y|^2/((x_d∨t^{1/α})(y_d∨t^{1/α})))}. A direct calculation showing an extra logarithmic factor or a different power of x_d∨t^{1/α} would falsify the theorem exactly at the threshold.","supporting_citations":[],"review_version":1}