{"id":"d53675ef-62e9-4bdd-a2c2-34d8b3ad9886","arxiv_id":"1908.07650","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors characterize two-sided heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms with both local and non-local parts under volume doubling and mild scale-function assumptions.","lead":"This paper proves that two-sided heat kernel estimates and parabolic Harnack inequalities for symmetric diffusions with jumps are equivalent to a short list of functional inequalities on metric measure spaces. It provides stable criteria that work even when the walk dimension exceeds 2 and no Ahlfors regularity is assumed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 2.9's omitted proof is the load-bearing soft spot: if self-improvement of CS(φ) fails for mixed local+nonlocal forms, the upper-bound half of Theorems 1.13/1.14 and hence (1.35) collapses.","rationale":"The reader's weakest_assumption identified exactly this cluster: heavy reliance on self-cited prior works with several 'details omitted' statements. My stress-test pass confirms that the most load-bearing of these is Proposition 2.9, because it is the step that produces the sharp self-improvement of CS(φ) used in the Caccioppoli and mean-value machinery. Without a written proof of Proposition 2.9 in the mixed local+nonlocal setting, the chain from functional inequalities to upper heat kernel bounds (Theorem 1.14) and then to the two-sided characterizations (Theorem 1.13) and (1.35) is incomplete. This does not mean the theorem is false; the imported arguments are plausible and likely adaptable. It does mean the preprint's central claim is not fully verified as written. Since the authors have a strong track record and the referenced works are close in spirit, the appropriate editorial action is to require the omitted proof or a precise published pointer before unconditional acceptance. The reader's CONDITIONAL verdict already reflects this, so my read does not alter the verdict; the concern reinforces it. I did not find an internal inconsistency or a counterexample; the main issue is rigor/completeness of an essential proof step.","tokens_in":50124,"tokens_out":5470,"duration_ms":125387,"concrete_test":"Write out the complete proof of Proposition 2.9 for a Dirichlet form of the form (1.1) with both the strongly local and the pure-jump terms nonzero, following the strategies of [AB, Lemma 5.1] and [CKW1, Proposition 2.4]. Verify that the function ϕ defined in (2.7) satisfies (2.6) for every ε>0 with C0=1, checking in particular: (i) the telescoping series over the annuli U_n converges in F; (ii) the local-energy term ∫ f² dΓc(ϕ_n,ϕ_n) is controlled after inserting ϕ² in front of dΓc(f,f); and (iii) the enlarged ball B(x0,R+2r) is handled consistently for both the local and nonlocal parts. If this verification cannot be completed, the upper-bound claims of Theorems 1.13 and 1.14 remain unproven.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central equivalence HK−(φc,φj) ⇔ PHI(φ)+Jφj in Theorem 1.17 rests on Theorem 1.13. Every upper-bound path in that theorem (e.g., (v)⇒(i) and Theorem 1.14) uses the Caccioppoli inequality (Lemma 2.14) and the mean-value inequalities (Propositions 2.15–2.16, 4.2). Those results rely on Corollary 2.10, whose 1/8 coefficient comes from the self-improvement of CS(φ) stated in Proposition 2.9. In Section 2.4 the authors explicitly say 'The details are omitted here' and refer to combining [AB, Lemma 5.1] with [CKW1, Proposition 2.4]. This is not a routine transcription: the form (1.1) has two energy measures of different scaling, and the cut-off ϕ constructed in (2.7) must simultaneously control the strongly local energy Γc and the jump energy over the enlarged ball B(x0,R+2r). The omitted proof of (2.6) with arbitrary ε>0 is what makes the coefficient 1/8 in (2.8) possible; any hidden assumption in that adaptation—about convergence of the telescoping sums, about the Leibniz rule for Γc, or about the cross terms between local and nonlocal energies—would invalidate the Caccioppoli step and the entire upper-bound half of the main theorems. Since Proposition 2.9 is not proved in the preprint and the referenced proofs are for pure-jump or strongly-local cases separately, the central claim is not fully verified as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies symmetric regular Dirichlet forms on metric measure spaces whose Beurling-Deny decomposition contains both a strongly local part and a pure-jump part, with no killing term. Under volume doubling, reverse volume doubling, and power-type scale-function assumptions, it establishes stable characterizations of upper heat kernel bounds (Theorem 1.14), two-sided heat kernel estimates (Theorem 1.13), and parabolic Harnack inequalities (Theorem 1.17). The main structural result is the equivalence HK^-(φc,φj) ⇔ PHI(φ) + Jφj, together with intermediate equivalences involving cut-off Sobolev inequalities, generalized capacity inequalities, Faber-Krahn inequalities, Poincaré inequalities, and near-diagonal lower bounds.","tokens_in":50503,"tokens_out":4943,"duration_ms":495695,"significance":"If the omitted proofs are supplied, this is a substantial contribution: it unifies and extends the earlier stability results for pure-jump Dirichlet forms and strongly local Dirichlet forms to the mixed local/nonlocal case, under the weak VD/RVD hypotheses rather than an Ahlfors d-set condition. The paper is honest about what is new, gives concrete applications (Example 1.1 for reflected diffusions with jumps, Example 7.2 for d-sets via subordination), and provides a valuable counterexample showing that PHI(φ) alone does not imply the sharp jumping-kernel upper bound. The logical architecture is transparent and there are no fitted free parameters. However, the manuscript as written leaves several keystone analytic steps to prior papers or to 'details omitted here', and these steps are load-bearing for the main theorems.","major_comments":[{"comment":"Proposition 2.9 is load-bearing: its ε-improvement of the cut-off Sobolev inequality produces the 1/8 coefficient in Corollary 2.10, which is used in the Caccioppoli inequality (Lemma 2.14) and hence in the upper-bound half of Theorems 1.13 and 1.14. The proof says only that the function ϕ = Σ (b_{n-1}-b_n)ϕ_n satisfies (2.6) by combining [AB, Lemma 5.1] with [CKW1, Proposition 2.4], and that 'the details are omitted here'. This is not a routine transcription: the form (1.1) has two energy measures with different scalings, and the omitted verification must control Γ^c simultaneously with the jump energy over the enlarged ball B(x0,R+2r). The full proof, or a precise statement of the adapted lemma with all hypotheses, should be included.","section":"Section 2.4, Proposition 2.9 and Corollary 2.10"},{"comment":"The proof that Gcap(φ) plus Jφ,≤ implies CS(φ) is only a sketch: after deriving a key inequality, it says to follow the proof of [GHH, Lemma 2.4] from the corresponding display to the end, replacing the balls B and Ω by B2 and B3. The displayed calculation gives a bound on integrals over B2 and B3×B3, but the final passage to the full Γ(ϕ,ϕ) includes a tail term over B1×B3^c and requires Lemma 2.1. Since Proposition 2.5 supplies the implication (iv)⇒(v) in Theorem 1.13 and (iii)⇒(iv) in Theorem 1.14, the mixed-energy adaptation should be written out explicitly rather than left as an instruction to re-run a prior proof.","section":"Section 2.2, Proposition 2.5"},{"comment":"Several propositions that are central to the main equivalences are deferred with no proof or with only a statement that the proof is 'the same as' in [CKW1] or [CKW2]. These include Proposition 2.16 (L2/L1 mean-value inequalities for truncated forms), Proposition 4.1 (NDL implies PI and Eφ), Proposition 4.2 (FK + Jφ,≤ + CS implies Eφ), Proposition 4.3 (FK + Eφ + Jφ,≤ implies UHKD), and Proposition 6.2 (NDL + Eφ,≤ + Jφ,≤ implies PHR and EHR). These results are used in the proofs of Theorems 1.14 and 1.17. The manuscript should either provide complete proofs or identify precisely which statements are being imported and what modifications are needed because the form has both a local and a nonlocal part.","section":"Sections 2.5, 4.1, 4.2, 4.3 and 6"},{"comment":"The headline equivalence HK^-(φc,φj) ⇔ PHI(φ) + Jφj depends on the implication (v)⇒(i) in Theorem 1.13, proved in Proposition 5.5. That proof in turn uses Proposition 5.4 (NDL follows from PI + Jφ,≤ + CS) and Proposition 5.3 (PI + Jφ,≤ + CS implies EHR), both of which are asserted to follow by adapting [CKW2]. Because the lower-bound estimate (1.31) requires the off-diagonal term t/(V φj), the adaptation is not purely formal. As written, the chain of equivalences is not fully self-contained, and the reader cannot verify the main theorem without reconstructing several long arguments from earlier papers.","section":"Theorem 1.17 and its proof"}],"minor_comments":[{"comment":"The abstract contains a typo: 'symmetric Random measure' should be 'symmetric Radon measure'. The phrase 'reps.' should be 'resp.', and in the proof of Proposition 2.4 the reference '[GHH, Leamm 2.8]' should read '[GHH, Lemma 2.8]'.","section":"Abstract and Introduction"},{"comment":"The notation PHR(φ) and EHR is used in Theorem 1.17 before the definitions are given in Definition 5.1. A forward reference would improve readability.","section":"Section 1.3 and Definition 5.1"},{"comment":"The notation UHKweak(φ) is introduced in Remark 4.9 but is used as item (ii) of Theorem 1.17; it would be clearer to give this a numbered definition together with UHK(φc,φj) in Definition 1.11.","section":"Remark 4.9"},{"comment":"In the construction of ϕ, the coefficients (b_{n-1}-b_n) with b_n = e^{-nλ} sum to 1, but the text does not explicitly justify that the series converges in F_b. This is presumably straightforward because the terms are truncated cut-off functions and the coefficients decay geometrically, but a one-line justification would help.","section":"Section 2.4, equation (2.7)"},{"comment":"The diagram in Figure 2 is informative but the arrow labels are small and dense; in the published version, a larger or simplified diagram with all referenced proposition numbers would be easier to read.","section":"Figure 2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the authors' own previous works, which is not itself a defect, but combined with the omitted proofs it makes independent verification difficult. The editor may wish to request a supplementary appendix containing, at minimum, a complete proof of Proposition 2.9 and explicit statements of the imports used in Propositions 2.16, 4.1, 4.2, 4.3 and 6.2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper extends the heat-kernel stability program to symmetric Dirichlet forms with a strongly local part plus a jump part. The mixed scaling is handled carefully, and Theorems 1.13, 1.14 and 1.17 are real advances: no Ahlfors regularity, walk dimensions larger than 2, and a clean statement of the parabolic Harnack / heat-kernel discrepancy. Example 1.1 (reflected diffusion with jumps on Lipschitz domains) and Example 7.2 (diffusions with jumps on d-sets) are useful illustrations, and the PHI-counterexample in Example 7.1 is convincing. The architecture is sound: capacity inequality, cut-off Sobolev inequality, self-improvement, Caccioppoli, mean-value inequalities.\n\nThe soft spot is where the stress-test points. Proposition 2.9 is the self-improvement of the cut-off Sobolev inequality, and its proof is omitted. The 1/8 coefficient in Corollary 2.10 comes out of it, and that coefficient feeds into the Caccioppoli inequality (Lemma 2.14) and the mean-value inequalities, so it carries the upper-bound half of Theorems 1.13/1.14. The authors refer to combining [AB, Lemma 5.1] with [CKW1, Prop 2.4], but the present form has two energy measures of different scaling; that is not obviously a routine transcription. Several other propositions are deferred by 'same proof' (2.16, 4.1, 4.2, 4.3, 6.2). If the prior papers are truly applicable, a pointer to the exact theorem in the published version may suffice; if the adaptation is not verbatim, this is a gap.\n\nI see no circularity or fitted parameters; the theorems are implications among independent analytic conditions. Self-citations are to established results, not to restatements of the conclusion. My view: the central claims are plausible and likely correct, but the paper as written is not fully verifiable until Proposition 2.9 is proved or pinned to a published argument.\n\nWho should read it: researchers working on heat kernels for jump processes, Dirichlet forms, and stability theory. It deserves a serious referee; I would send it out, but I'd ask the authors to include the missing proof or a precise reference before acceptance.","headline":"Genuinely new stability results for mixed local-and-jump Dirichlet forms, probably correct, but a load-bearing self-improvement proof is 'omitted' and needs to be supplied.","tokens_in":51022,"tokens_out":4163,"would_cite":true,"duration_ms":146154,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J35","35K08","60J60","60J75","31C25","60J25","60J45"],"pacs":[],"model":"deepseek-v4-flash","headline":"For symmetric diffusions with jumps, two-sided heat kernel estimates are equivalent to a parabolic Harnack inequality plus an upper bound on the jumping kernel.","keywords":["symmetric Dirichlet form","heat kernel estimates","parabolic Harnack inequality","cut-off Sobolev inequality","jumping kernel","metric measure space","stability","diffusions with jumps"],"falsifier":"Take the reflected diffusion with jumps on a Lipschitz domain from Example 1.1, with $\\varphi_c(r)=r^2$ and $\\varphi_j(r)=r^\\alpha$, and compute the on-diagonal heat kernel $p(t,x,x)$ at small times. The theorem predicts $p(t,x,x)\\asymp 1/V(x,\\varphi^{-1}(t))$ with $\\varphi(r)=r^2\\wedge r^\\alpha$; if for some $t,x$ the ratio $p(t,x,x)V(x,\\varphi^{-1}(t))$ grows without bound while the Poincaré, cut-off Sobolev and jumping-kernel conditions all hold, the implication (v)$\\Rightarrow$(i) of Theorem 1.13 fails. A more direct check is to verify the self-improved cut-off Sobolev inequality (2.6) on $\\mathbb{R}$ with a local $u''$-type energy and jump kernel $|x-y|^{-1-\\alpha}$, since the upper-bound chain depends on that inequality holding with an arbitrarily small leading constant.","tokens_in":49917,"feed_emoji":"🔥","tokens_out":10547,"duration_ms":97014,"temperature":0.7,"pith_summary":"This paper proves that for symmetric Markov processes whose energy form has both a diffusive (strongly local) part and a jumping part, on metric measure spaces satisfying volume doubling, the sharp two-sided heat kernel estimate is equivalent to a parabolic Harnack inequality together with an upper bound on the jumping kernel. It establishes a full circle of equivalences: heat-kernel upper bound plus near-diagonal lower bound, or a Poincaré inequality plus a cut-off Sobolev (or generalized capacity) inequality, all characterize the same two-sided estimate. These characterizations matter because they are stable under rough isometries and are checkable from the Dirichlet form data, rather than from the unknown heat kernel itself. They also cover spaces with walk dimension larger than 2, where purely local Gaussian estimates fail.","feed_headline":"Two-sided heat kernel bounds equal Harnack plus jump-kernel control","feed_subtitle":"A parabolic Harnack inequality plus an upper bound on jumps gives sharp heat kernel estimates on doubling spaces","key_machinery":"The workhorse is the cut-off Sobolev inequality $\\mathrm{CS}(\\varphi)$: for every pair of concentric balls $B(x_0,R)\\subset B(x_0,R+r)$ there is a cut-off function $\\phi$ whose energy, weighted by $f^2$, is bounded by the local and jump energies of $f$ on a slightly enlarged ball plus an $L^2$-term with constant $1/\\varphi(r)$. Together with the jumping-kernel upper bound $J_{\\varphi_j}$ and the generalized capacity inequality $\\mathrm{Gcap}(\\varphi)$ (a capacity bound $\\mathrm{cap}^{(\\kappa)}_f(B(x_0,R),B(x_0,R+r))\\le C\\varphi(r)^{-1}\\int f^2$), the self-improving form of $\\mathrm{CS}(\\varphi)$ yields Caccioppoli and mean-value inequalities, which in turn produce exit-time estimates $E_\\varphi$ and the diagonal heat kernel upper bound. The weak Poincaré inequality $\\mathrm{PI}(\\varphi)$ supplies the lower bounds, and the parabolic Harnack inequality $\\mathrm{PHI}(\\varphi)$ is characterized through these same ingredients together with the upper jumping-kernel bound $J_{\\varphi,\\le}$ and the uniform jump-size condition $\\mathrm{UJS}$.","core_discovery":"The central claim is a stability theorem: under volume doubling, reverse volume doubling, and mild comparability assumptions on the scale functions $\\varphi_c$ (the diffusive scale) and $\\varphi_j$ (the jump scale), the two-sided heat kernel estimate $\\mathrm{HK}^-(\\varphi_c,\\varphi_j)$ is equivalent to each of the following: the upper heat kernel bound plus a near-diagonal lower bound plus the jumping-kernel upper bound $J_{\\varphi_j}$; the diagonal upper bound plus a Dirichlet near-diagonal lower bound plus $J_{\\varphi_j}$; the weak Poincaré inequality plus $J_{\\varphi_j}$ plus the generalized capacity inequality; and the weak Poincaré inequality plus $J_{\\varphi_j}$ plus the cut-off Sobolev inequality. The paper further proves that $\\mathrm{HK}^-(\\varphi_c,\\varphi_j)$ holds if and only if the parabolic Harnack inequality $\\mathrm{PHI}(\\varphi)$ holds and $J_{\\varphi_j}$ holds, where $\\varphi=\\varphi_c\\wedge\\varphi_j$; on connected spaces satisfying the chain condition, the full two-sided estimate $\\mathrm{HK}(\\varphi_c,\\varphi_j)$ is also equivalent to the same pair of conditions.","pith_inferences":["An implication the authors leave implicit: the equivalence $\\mathrm{HK}^-(\\varphi_c,\\varphi_j)\\Leftrightarrow \\mathrm{PHI}(\\varphi)+J_{\\varphi_j}$ gives practitioners a two-step receipt—prove or assume the parabolic Harnack inequality, then check the upper density of the jump kernel; the lower density needed for the full estimate is then automatic from the Harnack side.","The subordination route in Example 7.2 suggests a transfer principle: any diffusion with sub-Gaussian heat kernel, when subordinated and then compared through the stability theorems, yields mixed heat kernel estimates; testing whether the same holds for general subordinators whose Laplace exponent satisfies (1.15) is a direct extension the authors flag in Remark 7.3.","A natural next question is whether the chain condition can be weakened for the full estimate $\\mathrm{HK}(\\varphi_c,\\varphi_j)$: the paper uses it only to chain small-time off-diagonal lower bounds, so one could look for examples of connected, non-chain spaces where $\\mathrm{HK}^-$ holds but $\\mathrm{HK}$ fails."],"forward_implications":["On any doubling, reverse-doubling metric measure space, the two-sided heat kernel estimate $\\mathrm{HK}^-(\\varphi_c,\\varphi_j)$ is determined by the combination $\\mathrm{PI}(\\varphi)+\\mathrm{CS}(\\varphi)+J_{\\varphi_j}$; verifying these three Dirichlet-form conditions is as good as knowing the heat kernel.","A parabolic Harnack inequality $\\mathrm{PHI}(\\varphi)$ alone is strictly weaker than two-sided heat kernel estimates when jumps are present: the extra datum $J_{\\varphi_j}$ (an upper bound on the jump kernel) is essential, and without it only the weak upper estimate $\\mathrm{UHK}_{\\mathrm{weak}}(\\varphi)$ follows.","On connected spaces satisfying the chain condition, the full two-sided estimate $\\mathrm{HK}(\\varphi_c,\\varphi_j)$ follows from $\\mathrm{PHI}(\\varphi)+J_{\\varphi_j}$, so sharp off-diagonal decay (Gaussian for the local part, stable-like for the jump part) is a consequence of the same hypotheses.","The equivalent conditions are stable under rough isometries, so heat kernel estimates proved for one diffusion with jumps transfer to every rough isometric process with comparable scale functions and comparable jumping kernels.","The framework covers walk dimensions larger than 2, giving heat kernel control on fractal-like spaces where the diffusive part alone cannot produce Gaussian estimates."],"supporting_citations":[{"why":"Supplies the cut-off Sobolev framework, the mean-value inequalities, and the truncation arguments that the paper adapts from the pure-jump case to mixed local-nonlocal forms.","marker":"[CKW1]"},{"why":"Supplies the characterization strategy for parabolic Harnack inequalities in the pure-jump case that Theorem 1.17 extends to forms with both a local and a jumping part.","marker":"[CKW2]"},{"why":"Supplies the generalized capacity inequality and the argument that Gcap(φ) together with a jumping-kernel upper bound yields the cut-off Sobolev inequality CS(φ).","marker":"[GHH]"},{"why":"Supplies the self-improvement argument for cut-off Sobolev inequalities that Proposition 2.9 adapts to the mixed setting.","marker":"[AB]"},{"why":"Supplies the chaining argument used in Proposition 5.6 to upgrade HK−(φc,φj) to the full HK(φc,φj) under connectedness and the chain condition.","marker":"[BGK]"}],"fun_headline_variants":["Heat kernel iff Harnack plus jump bound","Two-sided heat kernel from Harnack and jumps","Stability of heat kernels via jumps and Harnack","Sharp heat kernel bounds: Harnack + jump control"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the analytic arguments imported from the purely local and purely jump settings—particularly the self-improvement of the cut-off Sobolev inequality and the mean-value inequalities, several of which are deferred to earlier papers with details omitted—continue to work unchanged when the Dirichlet form has both a strongly local and a jumping part.","fun_headline_variants_meta":{"raw":{"variants":["Heat kernel iff Harnack plus jump bound","Two-sided heat kernel from Harnack and jumps","Stability of heat kernels via jumps and Harnack","Sharp heat kernel bounds: Harnack + jump control"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000147,"raw_usage":{"total_tokens":1202,"prompt_tokens":978,"completion_tokens":224,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":594,"completion_tokens_details":{"reasoning_tokens":160}},"tokens_in":594,"tokens_out":224,"duration_ms":3195,"temperature":1.0,"reasoning_tokens":160,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T12:00:54.797366+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the reflected diffusion with jumps on a Lipschitz domain from Example 1.1, with $\\varphi_c(r)=r^2$ and $\\varphi_j(r)=r^\\alpha$, and compute the on-diagonal heat kernel $p(t,x,x)$ at small times. The theorem predicts $p(t,x,x)\\asymp 1/V(x,\\varphi^{-1}(t))$ with $\\varphi(r)=r^2\\wedge r^\\alpha$; if for some $t,x$ the ratio $p(t,x,x)V(x,\\varphi^{-1}(t))$ grows without bound while the Poincaré, cut-off Sobolev and jumping-kernel conditions all hold, the implication (v)$\\Rightarrow$(i) of Theorem 1.13 fails. A more direct check is to verify the self-improved cut-off Sobolev inequality (2.6) on $\\mathbb{R}$ with a local $u''$-type energy and jump kernel $|x-y|^{-1-\\alpha}$, since the upper-bound chain depends on that inequality holding with an arbitrarily small leading constant.","supporting_citations":[],"review_version":1}