{"id":"b8ae4016-f1a5-4b43-a790-349fcc44c533","arxiv_id":"2504.14763","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 and Green function estimates are proved for killed and censored alpha-stable processes in Dini-smooth domains and for non-symmetric stable processes in double-Dini domains via approximate factorization.","lead":"This mathematics paper proves precise formulas for how jumping random processes spread inside regions with mildly rough boundaries, for killed, censored, and non-symmetric stable processes. The results give sharp two-sided bounds on the transition density, including exact boundary decay exponents, in Dini-smooth domains.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 4.5, the approximate-factorization engine behind every application, is stated with proof omitted; the claimed extension beyond [12] is therefore not yet verified.","rationale":"The reader's formal verdict of CONDITIONAL is appropriate, and my independent reading reaches the same verdict for a related reason. The reader's stated weakest assumption focuses on the boundary modulus condition in Lemma 5.5; that concern is real but the paper contains a mechanism, Lemma A.2, that regularizes a Dini modulus into one satisfying polynomial scaling, so the apparent gap in Lemma 5.5 is at least partially closed. The more serious unresolved step is Theorem 4.5, which the reader mentions in the rationale but not as the weakest assumption. The theorem is the bridge from the general survival estimates to the approximate factorization used in every application, and its proof is omitted. Since the paper explicitly claims to go beyond [12] by removing assumptions, a reader cannot verify the central claim without either the full proof or a precise statement of which existing theorem is being imported and why the removed assumptions are unnecessary. A complete proof or a precise citation-plus-verification would settle the matter; if the omitted argument cannot be supplied, the conditional status of the paper is correct. No dishonesty or performative issue is alleged; the concern is purely about verifiability of the central mathematical step.","tokens_in":55951,"tokens_out":12034,"duration_ms":112467,"concrete_test":"Ask the authors to supply a complete proof of Theorem 4.5(i)-(ii), or to state precisely which theorem (with exact assumptions) from [12] is being invoked and to prove that the present setup satisfies those assumptions. A line-by-line check should identify every assumption used in [12, Theorems 2.22-2.23], including the so-called Assumptions A and U, and for each either verify that it holds here or cite the new lemma that replaces it. If no such verification can be provided, the central claim of the paper is unproved and the CONDITIONAL verdict is justified.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central mechanism is Theorem 4.5: it converts the survival estimates (Propositions 3.4, Lemma 4.2, Corollary 4.3) into the two-sided approximate factorization (4.15) for the Feynman-Kac heat kernel. The proof is omitted: 'following the arguments in [12, Theorems 2.22 and 2.23], we obtain the next theorem. We omit the proof.' This would be acceptable if the hypotheses exactly matched [12], but the abstract promises a genuine extension: 'getting rid of Assumptions A and U imposed in [12]'. The paper never states what those assumptions were, which of them are needed at which step of the omitted proof, or how the new survival estimates replace them. Every later result — Theorems 4.8, 4.11, Corollary 4.12, and hence Theorems 5.7, 5.15 and 5.22 — calls Theorem 4.5, so all applications inherit this gap. This is not a matter of disagreement with consensus; it is an unverified load-bearing theorem. The reader's concern about Lemma 5.5 is real but largely mitigated by the regularization Lemma A.2, which produces a larger admissible modulus satisfying the polynomial scale condition; the omitted Theorem 4.5 is the unresolved support.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an approximate factorization for the heat kernel and Green function of a purely discontinuous Markov process with critical killings in η-fat open sets, and then applies it to three families of processes: stable-like processes with critical killings in C^{1,Dini} domains, killed stable-like processes in the setting of Kim–Weidner, and non-symmetric stable processes in C^{1,2-Dini} domains. The main abstract factorization is Theorem 4.5, which is imported from the authors' prior work [12] with the proof omitted; the later boundary estimates are obtained through explicit calculation with barrier functions and a regularized distance. The paper derives explicit boundary exponents, in particular q=α/2 for killed α-stable processes and q=α−1 for censored α-stable processes in C^{1,Dini} domains, and exposes a dependence of the exponent on the normal direction in the non-symmetric case.","tokens_in":56171,"tokens_out":8844,"duration_ms":79049,"significance":"If the results are fully established, this is a substantial contribution: it significantly weakens the regularity assumptions under which sharp two-sided heat kernel and Green function estimates are known for non-local operators, and it covers non-symmetric stable processes with variable boundary exponents. The paper contains many detailed and apparently correct barrier computations, and the regularization lemmas in the appendix are effective tools. The main reservation is that the central engine, Theorem 4.5, is not proved in this manuscript and is inherited from [12] without a detailed verification that the hypotheses used in [12] are indeed dispensed with. The rest of the paper is built on this theorem, so the current version is not fully verifiable as a standalone contribution.","major_comments":[{"comment":"The approximate factorization (4.15) is the central tool for every subsequent application, but its proof is omitted: the text states 'following the arguments in [12, Theorems 2.22 and 2.23], we obtain the next theorem. We omit the proof.' The abstract and Introduction announce that the paper removes Assumptions A and U from [12], yet those assumptions are never stated, and the reader is not told which ingredients of the proof in [12] are replaced by the survival estimates in Section 3, Lemma 4.2, Corollary 4.3, and Proposition 4.4. Since Theorems 4.8, 4.11, Corollary 4.12, and the applications in Section 5 all call on Theorem 4.5, this is a load-bearing gap. Please add at least a detailed proof sketch that explicitly identifies the differences from [12], states what Assumptions A and U are, and verifies that they are not needed under the hypotheses of Theorem 4.5.","section":"Section 4, Theorem 4.5"},{"comment":"The statement of Theorem 4.8 uses Py(ζ>1) in both the upper and lower bounds, but the proof uses bPy(bζ>1) throughout, and the later application in Theorem 4.11 also uses bPy(bζ>1). In the general nonsymmetric setting the transition density is not symmetric, so the theorem as stated appears to be false; the correct statement should involve the survival probability of the dual process at y. Please correct the statement to C−1Px(ζ>1)bPy(bζ>1)e−λ1t ≤ p(t,x,y) ≤ C Px(ζ>1)bPy(bζ>1)e−λ1t, or explain why the two factors coincide under the stated hypotheses.","section":"Section 4.1, Theorem 4.8"}],"minor_comments":[{"comment":"The polynomial-scale condition (5.15) is not satisfied by every Dini modulus, so as stated Lemma 5.5 appears to require an extra assumption. The later text uses the regularized modulus from Lemma A.2, which does satisfy (5.15), but this should be stated explicitly before Lemma 5.5 is used so that the hypothesis is not read as an additional regularity restriction on the domain.","section":"Section 5, Lemma 5.5"},{"comment":"The proof states that lim_{s→0} s^{q−α}ℓ(s) ≥ lim_{s→0} s^{(q−α)/2}ℓ(1) = 0, but for q<α the exponent (q−α)/2 is negative, so the displayed limit is infinite, not zero. The subsequent choice of σ0 is still valid, but the displayed limit should be corrected.","section":"Corollary 5.14, proof"},{"comment":"At the end of part (ii), the text says 'we obtain (5.40)' after proving the estimate for h_{r,σ}; the displayed estimate being proved is (5.39). Please correct the cross-reference.","section":"Proposition 5.13, proof of (ii)"}],"recommendation":"major_revision","confidential_remarks":"For the editor: the main risk is the unverified import of Theorem 4.5 from [12]. I would not reject the paper, because the surrounding arguments are detailed and the gap is local in nature, but I would request a proof sketch or an explicit reduction to [12] before publication. The Theorem 4.8 statement also needs a correction that is easy to make but important for readers."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know before reading. The applications are real and worth attention: sharp two-sided heat kernel estimates for killed stable processes in C^{1,Dini} domains for all alpha, for censored stable processes in C^{1,Dini} for alpha in (1,2), and for non-symmetric stable processes in C^{1,2-Dini} domains. The boundary exponents are explicit and are computed from barrier functions, not fitted. Second, the engine behind all of this, Theorem 4.5, is stated with the proof omitted and a pointer to [12], and the paper never spells out which hypotheses of [12] were dropped or how the new survival estimates plug into that proof. That is the main soft spot, and it is a genuine one because every later theorem calls on Theorem 4.5.\n\nWhat the paper does well: Sections 3 and 4 develop the survival estimates in detail, adapting the box method and replacing the condition that fails under critical killing. The equivalence between heat kernel and Green function estimates in Corollary 4.12 is clean and useful. The barrier constructions in Section 5 are explicit and appear checkable; the non-symmetric stable process case in Section 5.2 is the real technical core and looks like solid work. The reader's worry about Lemma 5.5 requiring a polynomial scale condition is largely mitigated by Lemma A.2, which regularizes a Dini modulus into one satisfying that condition, so that is not where I would focus criticism.\n\nThe citation pattern is a little uncomfortable: Theorem 4.5 comes from [12] with overlapping authors, and the paper also relies on the preprint [13] and a forthcoming [14]. That is not by itself a flaw, but it does mean the novelty of the framework is hard to isolate from the authors' other work. What is needed is a precise statement of the assumptions in [12], a verification that they hold in the present setting, or a proof of Theorem 4.5 adapted to the new conditions. Without that, the central claim is not fully verifiable from this manuscript.\n\nWho this is for: people working on sharp heat kernel estimates for non-local operators in rough domains. A serious referee should see it, but the request to the authors should be specific: either include the omitted proof or make the dependence on [12] exact. If that gets fixed, this is a solid math.PR paper; as it stands, it is close, but the core theorem is on faith.","headline":"Genuinely new heat kernel estimates for stable processes in Dini-type domains, but the proof of the key factorization theorem is omitted and pointed to previous work, so the central mechanism is not fully verifiable in this text.","tokens_in":56731,"tokens_out":2472,"would_cite":true,"duration_ms":26169,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J45","60J50","60J76","47G20","35K08"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under $C^{1,\\mathrm{Dini}}$ boundary regularity, the paper proves sharp two-sided heat kernel estimates for three classes of jump processes, with explicit boundary-decay exponents for killed and censored $\\alpha$-stable processes.","keywords":["heat kernel","Green function","regional fractional Laplacian","stable-like processes","approximate factorization","C^{1,Dini} open sets","censored stable process","non-symmetric stable process"],"falsifier":"Take the $C^{1,\\mathrm{Dini}}$ domain $D=\\{(x_1,x_2):x_2>x_1(1+|\\log x_1|)^{-k}\\}$ with $k>1$, fix $x=(0,t^{1/\\alpha})$ and $y=x$, and estimate the killed $\\alpha$-stable heat kernel $p(t,x,x)$ as $t\\downarrow0$. If it is not comparable to $t^{-d/\\alpha}(\\delta_D(x)/t^{1/\\alpha})^{\\alpha/2}$, the boundary-exponent claim fails.","tokens_in":55721,"feed_emoji":"📐","tokens_out":13629,"duration_ms":112378,"temperature":0.7,"pith_summary":"The paper proves sharp two-sided heat kernel estimates for non-local jump processes in rough open sets, with explicit boundary-decay factors. The tool is an approximate factorization: on short time scales the Dirichlet heat kernel is, up to universal constants, the product of two boundary-decay factors and the free-space heat kernel. The paper extends this factorization to a broader class of Feynman–Kac processes by removing two earlier assumptions, and shows that the heat-kernel factorization is equivalent to a Green-function factorization. The payoff is explicit estimates for killed $\\alpha$-stable processes in $C^{1,\\mathrm{Dini}}$ domains for all $\\alpha\\in(0,2)$, for censored $\\alpha$-stable processes in the same domains for $\\alpha\\in(1,2)$, and for non-symmetric stable processes with direction-dependent boundary exponents.","feed_headline":"Rough-boundary heat kernels factor into boundary decay","feed_subtitle":"Killed α-stable heat kernels get a δ^{α/2} boundary factor; censored ones get δ^{α−1}.","key_machinery":"The load-bearing identity is the approximate factorization $p(t,x,y)\\asymp P_x(\\zeta>t)\\hat P_y(\\hat\\zeta>t)\\bar q(t,x,y)$ for the Dirichlet heat kernel, with the analogous Green-function factorization $G(x,y)\\asymp P_x(\\zeta>\\phi(d(x,y)))\\hat P_y(\\hat\\zeta>\\phi(d(x,y)))\\bar G(x,y)$, and a proof that the two factorizations are equivalent. The boundary exponent is extracted from the half-space identity $\\mathrm{p.v.}\\int_{\\mathbb{R}^d_+}(y_d^q-x_d^q)/|x-y|^{d+\\alpha}\\,dy=C(d,\\alpha,q)x_d^{q-\\alpha}$, which matches the critical killing rate to the boundary decay. A regularized distance $\\rho$ with controlled gradient modulus and Hessian converts this half-space computation into estimates on $D$, and barrier functions built from $\\rho^q$ force the survival probabilities $P_x(\\zeta>t)$ to decay like $(\\delta_D(x)/t^{1/\\alpha})^q$.","core_discovery":"The central claim is Theorem 5.7: if $D$ is a $C^{1,\\mathrm{Dini}}$ open set, the jump coefficient is bounded and Hölder near the diagonal, and the killing measure $\\kappa$ belongs to the critical class $\\mathcal{K}_\\alpha(q)$, then the heat kernel of the Feynman–Kac process satisfies $p(t,x,y)\\asymp (1\\wedge \\delta_D(x)/t^{1/\\alpha})^q(1\\wedge \\delta_D(y)/t^{1/\\alpha})^q(t^{-d/\\alpha}\\wedge t/|x-y|^{d+\\alpha})$ on $(0,T]$. The boundary exponent $q$ is determined by the constant $C(d,\\alpha,q)$ in the killing rate $\\kappa(x)\\approx C(d,\\alpha,q)K(x,x)\\delta_D(x)^{-\\alpha}$. As corollaries, killed stable-like processes have $q=\\alpha/2$ and censored stable-like processes have $q=\\alpha-1$; non-symmetric stable processes instead have exponents $\\gamma(n_{Q_x})$ and $\\alpha-\\gamma(n_{Q_y})$ set by the directional asymmetry of the Lévy measure.","pith_inferences":["The proof regularizes the boundary modulus into one satisfying a power-comparability condition; a natural test is whether the same boundary exponents survive for $C^{1,\\mathrm{Dini}}$ domains whose original modulus decays slower than every power, such as the example in Remark 5.3.","The direction-dependent exponents for non-symmetric stable processes suggest that variable-coefficient versions will have boundary exponents depending on position through the local Lévy measure, not merely through the normal direction.","The factorization route is likely to transfer to other critical-killing Feynman–Kac processes with different jump kernels, producing boundary Harnack principles and Green-function asymptotics whenever the half-space killing identity can be matched."],"forward_implications":["Killed $\\alpha$-stable-like processes in $C^{1,\\mathrm{Dini}}$ domains have explicit heat kernel boundary decay $(\\delta_D(x)/t^{1/\\alpha})^{\\alpha/2}$ for every $\\alpha\\in(0,2)$.","Censored $\\alpha$-stable processes in $C^{1,\\mathrm{Dini}}$ domains have boundary exponent $\\alpha-1$ for every $\\alpha\\in(1,2)$.","For non-symmetric stable processes, heat kernel boundary decay is asymmetric: the factor at $x$ decays like $\\delta_D(x)^{\\gamma(n_{Q_x})}$ while the factor at $y$ decays like $\\delta_D(y)^{\\alpha-\\gamma(n_{Q_y})}$.","In bounded domains the same estimates give Green-function bounds $G(x,y)\\asymp(1\\wedge\\delta_D(x)/|x-y|)^q(1\\wedge\\delta_D(y)/|x-y|)^q|x-y|^{\\alpha-d}$ and large-time spectral decay $e^{-\\lambda_1 t}\\delta_D(x)^q\\delta_D(y)^q$.","Because heat-kernel and Green-function factorizations are equivalent under the paper's conditions, future Green-function estimates in this framework automatically yield heat-kernel estimates."],"supporting_citations":[{"why":"Establishes the approximate factorization for non-local operators with critical killings that this paper extends by removing Assumptions A and U.","marker":"[12]"},{"why":"Introduces censored stable processes and supplies the half-space identity used to match the critical killing constant.","marker":"[5]"},{"why":"Provides the $C^{1,1}$ heat kernel estimates for killed fractional Laplacians that are relaxed here to $C^{1,\\mathrm{Dini}}$.","marker":"[8]"},{"why":"Provides the $C^{1,1}$ heat kernel estimates for censored stable processes extended in Corollary 5.8.","marker":"[9]"},{"why":"Supplies Green-function estimates for killed stable-like operators with low-regularity coefficients that Corollary 4.12 converts into heat kernel estimates.","marker":"[22]"},{"why":"Gives heat kernel bounds for stable-like processes on $d$-sets that verify hypothesis (A) with $\\phi(r)=r^\\alpha$.","marker":"[10]"},{"why":"Provides the analysis of non-symmetric stable operators from which the directional boundary exponents $\\gamma$ and $\\hat\\gamma$ are taken.","marker":"[15]"},{"why":"Supplies the regularized distance construction whose gradient and Hessian estimates drive the boundary error estimates.","marker":"[24]"}],"fun_headline_variants":["Jump process heat kernels factor at rough boundaries","Sharp heat kernel bounds for killed and censored jumps","Boundary factor reveals jump process heat decay","Critical killing gives clean heat kernel factorization","Non-symmetric jumps get sharp heat kernel estimates"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument needs enough control of the region where the domain differs from its tangent half-space, obtained from a Dini boundary modulus that is made power-comparable; if that boundary control is absent, the error integrals in the proof do not close and the boundary exponent would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Jump process heat kernels factor at rough boundaries","Sharp heat kernel bounds for killed and censored jumps","Boundary factor reveals jump process heat decay","Critical killing gives clean heat kernel factorization","Non-symmetric jumps get sharp heat kernel estimates"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000464,"raw_usage":{"total_tokens":2334,"prompt_tokens":974,"completion_tokens":1360,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":590,"completion_tokens_details":{"reasoning_tokens":1292}},"tokens_in":590,"tokens_out":1360,"duration_ms":9462,"temperature":1.0,"reasoning_tokens":1292,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T11:42:05.922799+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the $C^{1,\\mathrm{Dini}}$ domain $D=\\{(x_1,x_2):x_2>x_1(1+|\\log x_1|)^{-k}\\}$ with $k>1$, fix $x=(0,t^{1/\\alpha})$ and $y=x$, and estimate the killed $\\alpha$-stable heat kernel $p(t,x,x)$ as $t\\downarrow0$. If it is not comparable to $t^{-d/\\alpha}(\\delta_D(x)/t^{1/\\alpha})^{\\alpha/2}$, the boundary-exponent claim fails.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the approximate factorization for non-local operators with critical killings that this paper extends by removing Assumptions A and U."},{"cited_title":"Bogdan, K","cited_arxiv_id":null,"evidence_quote":"Introduces censored stable processes and supplies the half-space identity used to match the critical killing constant."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the $C^{1,1}$ heat kernel estimates for killed fractional Laplacians that are relaxed here to $C^{1,\\mathrm{Dini}}$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the $C^{1,1}$ heat kernel estimates for censored stable processes extended in Corollary 5.8."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives heat kernel bounds for stable-like processes on $d$-sets that verify hypothesis (A) with $\\phi(r)=r^\\alpha$."},{"cited_title":"Dipierro, X","cited_arxiv_id":null,"evidence_quote":"Provides the analysis of non-symmetric stable operators from which the directional boundary exponents $\\gamma$ and $\\hat\\gamma$ are taken."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the regularized distance construction whose gradient and Hessian estimates drive the boundary error estimates."}],"review_version":1}