{"id":"f0373646-9df3-4e2c-8e7e-ae74cfd9ab96","arxiv_id":"2505.08449","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Nonnegative solutions to nonlocal parabolic equations with bounded measurable coefficients are represented exactly by a fundamental solution, with sharp two-sided bounds and new Harnack inequalities.","lead":"The paper proves that every nonnegative solution to a broad class of nonlocal diffusion equations can be written as a spreading kernel applied to its starting distribution, and it gives sharp two-sided estimates for that kernel. It introduces fully variational proofs, avoiding probabilistic and semigroup machinery, and yields Harnack-type inequalities that are new even for the fractional heat equation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Uniqueness in Theorem 1.1 is not fully proved in Prop. 5.4: the terminal weighted-L1 integrability is assumed 'without loss of generality' and the duality identity relies on an unperformed time mollification; the central representation therefore has a gap as written.","rationale":"The central claim is a Widder-type representation, and uniqueness is an explicit part of Theorem 1.1(i). The proof of uniqueness is Proposition 5.4. Its first line assumes ∥u(T)∥_{L1(w)}<∞ without justification; Definition 2.3 only guarantees weighted L1 integrability a.e. in time, and terminal-time regularity is not part of the solution class. The final dominated convergence at τ1↗T and the estimate for III_1 both depend on this finiteness. A repair might choose T't and then extending by time-regularity is plausible but is not supplied. Second, the display after 'Modulo a proper time mollification' differentiates the product ξ_R u φ in time. The weak formulations only give ∂t u and ∂t φ as distributions; passing this identity requires a time mollification with uniform control of the error terms, which is not provided. Since this duality identity is exactly how the terminal data ψ is transported back to τo and then to the initial trace, both gaps touch the central claim. The theorem may well be true and the gaps may be routine, but the proof in the manuscript does not establish uniqueness as stated.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper establishes a nonlocal analogue of Aronson's and Widder's theory for parabolic equations with bounded, measurable, symmetric kernels satisfying an upper bound and an integrated coercivity condition. The main results are a Widder-type representation theorem for nonnegative global weak solutions (Theorem 1.1), two-sided polynomial bounds for the fundamental solution (Theorem 1.3), a sharp elliptic-type Harnack estimate (Theorem 1.4), and a growth estimate in terms of the initial measure (Theorem 1.8). The proofs are variational and avoid semigroup, Dirichlet-form, and stochastic tools, with the construction of the fundamental solution and the lower heat-kernel bounds carried out directly from Harnack inequalities. The manuscript is generally well organized and carefully distinguishes the new contributions from the substantial inputs taken from the authors' earlier works.","tokens_in":28675,"tokens_out":4957,"duration_ms":52620,"significance":"If completed, the Widder-type theorem would be a significant advance: it extends a structural representation that was previously known only for translation-invariant operators to general nonlocal operators with measurable coefficients, and it identifies the correct growth class for the initial Radon measure. The two-sided fundamental solution bounds under time-dependent kernels and the short variational proof of the lower bound are also valuable, as is the sharp Harnack-type estimate for global solutions. The paper is transparent about its reliance on earlier results, especially the improved weak Harnack inequality from [LW24] and the upper bounds from [KW23]; this reliance is explicit rather than circular, though it means the central theorem is conditional on those black boxes. The main obstacle is that the uniqueness proof in Proposition 5.4 contains two under-specified steps that are load-bearing for Theorem 1.1(i), and the proof of Theorem 1.1 does not supply the missing arguments.","major_comments":[{"comment":"The assertion 'Without loss of generality we assume \\|u(T)\\|_{L^1(R^d;w)} < \\infty' is not justified. Definition 2.3 only guarantees u_1 and u_2 lie in L^1_loc((0,T); L^1(R^d;w)), so at the fixed terminal time T their weighted L^1 norms need not be finite or even well-defined as Lebesgue points. Moreover, u = u_1 - u_2 is signed, so Lemma 5.1 cannot be applied to u itself. The finiteness of \\|u(T)\\|_{L^1(R^d;w)} is used in the bound on III_1, in the estimate for III_2, and in the final dominated convergence as \\tau_1 \\nearrow T that yields \\int \\psi u(T) = 0. Without a proof that some terminal value of u lies in L^1(R^d;w), or a replacement of T by a Lebesgue point together with a separate continuity argument up to T, the uniqueness conclusion u(T)=0 is not established. This gap directly affects the central claim of Theorem 1.1(i).","section":"Section 5.3, Proposition 5.4"},{"comment":"The display beginning 'Modulo a proper time mollification' differentiates the product \\xi_R u \\phi in time and then uses the weak formulations of u and \\phi. The weak form only provides distributional time derivatives for u and \\phi, so passing to the displayed identity requires a time mollification with uniform control of the resulting error terms as the mollification parameter is removed. This control is not supplied, and it is not immediate because the test functions are products involving the nonsmooth cutoff \\xi_R and because the bilinear form E(t) has time-dependent kernel. Since this duality identity is exactly the mechanism that transports the terminal datum \\psi back to \\tau_o and then to the initial trace, the missing justification is load-bearing for the proof of uniqueness.","section":"Section 5.3, Proposition 5.4, time-mollification step"},{"comment":"The final step 'by Proposition 5.4, we deduce that u = v' applies Proposition 5.4 on the interval (0,T] and concludes equality at time T. However, Proposition 5.4's proof invokes the unjustified WLOG assumption on \\|u(T)\\|_{L^1(R^d;w)}, and Definition 2.3 defines solutions on (\\eta,T) without prescribing regularity or values at the terminal time T. Thus the equality of the two solutions at time T is not actually obtained by the given arguments. The proof could be repaired by working with a terminal time T_o \\in (0,T) that is a Lebesgue point for both solutions and then using local regularity to propagate equality, but this is not carried out in the manuscript.","section":"Section 5.5, proof of Theorem 1.1"}],"minor_comments":[{"comment":"The heading contains a typo: 'intitial data' should be 'initial data'.","section":"Section 5.4, heading"},{"comment":"The definition says 'Let [\\eta,T] \\subset R be an interval' but then refers to solutions on '(\\eta+\\epsilon,T)' and to convergence as t \\searrow \\eta. Please clarify whether the terminal time T is included in the solution interval and whether values such as u(T) are part of the solution concept; this is relevant to Proposition 5.4 and Theorem 1.1.","section":"Definition 2.3"},{"comment":"The statement '\\|u(t)\\|_{L^1(R^d;w)} \\le c\\|u(T)\\|_{L^1(R^d;w)}' is only meaningful when \\|u(T)\\|_{L^1(R^d;w)} < \\infty. Please state explicitly that the estimate is asserted under this finiteness assumption, or clarify the convention when the right-hand side is infinite.","section":"Lemma 5.1"},{"comment":"The sentence 'decomposing the space in the same way' is vague; please spell out the decomposition used to derive the lower bound in [Kan23] from (4.9), or give a precise reference to the relevant part of that paper.","section":"Remark 4.10"}],"recommendation":"major_revision","confidential_remarks":"The heavy self-citation is transparent and the cited inputs are stated as theorems rather than hidden assumptions, so I do not see a novelty-disclosure problem. The central issue is that the uniqueness argument in Proposition 5.4 is not complete as written, and the same gap propagates into the proof of Theorem 1.1. The results are likely salvageable, but the missing terminal-time and mollification arguments must be supplied before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis is a serious piece of work and, if the details are fixed, an important one. The authors prove a Widder-type representation theorem for nonnegative solutions to nonlocal parabolic equations with bounded measurable, time-dependent kernels, under only an upper bound and a coercivity condition on the kernel. Previously such a theorem was known only for translation-invariant operators. They also get sharp two-sided heat kernel bounds for time-dependent kernels using purely variational tools, and Harnack-type estimates that are new even for the fractional heat equation. The writing is clear, and the reliance on earlier results—several by the same authors—is explicit rather than hidden.\n\nThe main soft spot is Proposition 5.4, which is the uniqueness half of the Widder theorem. The proof starts with \"Without loss of generality we assume ||u(T)||_{L1(w)} < ∞.\" That is not a WLOG. The solution class only gives u(t) ∈ L1(w) for almost every t, and T is a fixed endpoint. If T is not a Lebesgue point, the argument as written does not reach u(T)=0. The stress-test note is right that this is a load-bearing gap, not a cosmetic one. The second issue is the \"modulo a proper time mollification\" step when differentiating the product ξ_R u φ. This is probably standard and repairable, but it is not written out; since it is exactly how the terminal datum is transported back to the initial trace, a referee should ask for details.\n\nAll of that said, the central claims look true and the methods are genuinely new. The lower heat kernel bound via the weak Harnack inequality is short and elegant. The paper leans heavily on [LW24], [KW23], [KW24], but those are either published or in press, so that dependence is checkable. I do not see circularity: the Widder theorem and lower bounds are not restated from those papers.\n\nWho is this for? Anyone working on regularity or well-posedness for nonlocal parabolic equations. It deserves a serious referee, and I would send it out rather than desk-reject. The referee should be asked to check Proposition 5.4 carefully and to demand a repaired argument. If the gaps are fixed, this will be a very useful paper.","headline":"Genuinely new Widder-type and heat kernel estimates for nonlocal parabolic equations, but the central uniqueness proof in Proposition 5.4 has an unjustified WLOG assumption and a skipped time-mollification step that a referee should ask to be repaired.","tokens_in":29243,"tokens_out":2965,"would_cite":true,"duration_ms":28501,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47G20","35B65","35R09","31B05","35C15","35K08","35A02"],"pacs":[],"model":"deepseek-v4-flash","headline":"For nonlocal parabolic equations with bounded measurable coefficients, every nonnegative global weak solution is the convolution of a unique Radon initial measure with a fundamental solution, which is pointwise comparable to the…","keywords":["nonlocal parabolic equations","Widder-type theorem","fundamental solution","Harnack inequality","measurable coefficients","initial trace","fractional Laplacian","integro-differential operators"],"falsifier":"Exhibit a kernel satisfying (1.5) and (1.6) for which the improved weak Harnack inequality (3.5) is false, for instance a time-dependent kernel engineered so that a nonnegative supersolution's average over $B_R$ at time $t_o$ decays faster than any constant multiple of its infimum over the later cylinder; since the paper imports this inequality rather than proving it, such a counterexample would invalidate Lemma 5.1 and the Widder-type theorem. A more targeted check is whether the vague limit of the averages $u(t,x)dx$ as $t\\to0$ is unique for every nonnegative global solution in class (1.12): a single solution with two different subsequential limits would contradict Proposition 5.3.","tokens_in":28116,"feed_emoji":"🔥","tokens_out":15804,"duration_ms":127923,"temperature":0.7,"pith_summary":"The paper aims to extend the classical representation theory of parabolic equations, where every nonnegative solution is a convolution of its initial data with a fundamental solution, to a broad class of nonlocal equations whose kernels are only bounded and measurable, not translation invariant. It proves that every nonnegative global weak solution in a natural weighted space has a unique Radon measure as its initial trace and is represented by integrating that measure against a fundamental solution. It then establishes that this fundamental solution is pointwise comparable to the fractional heat kernel, and derives a sharp Harnack-type estimate for all nonnegative global solutions. A sympathetic reader would care because these are structural facts: the whole solution theory of a large class of nonlocal diffusion equations is reduced to one kernel, and the proofs use only variational tools.","feed_headline":"All nonnegative nonlocal heat solutions reduce to one measure","feed_subtitle":"One initial measure determines every nonnegative solution, with sharp Harnack and heat-kernel bounds.","key_machinery":"The load-bearing object is the fundamental solution $p_t(x,y)$, the density that solves the Cauchy problem with a Dirac initial datum at $y$. The engine of the argument is the improved weak Harnack inequality (Theorem 3.3), which bounds the average of a nonnegative supersolution over a ball at early times by its infimum over a later cylinder; this inequality converts local information into the global weighted $L^1$ estimate (Lemma 5.1) that produces the initial trace and later forces uniqueness. The two-sided bounds of $p_t$ are obtained by combining the upper heat-kernel estimate from the previously developed analytic technique with a new lower estimate that uses the weak Harnack inequality with a tail term; this last step is purely variational and does not rely on semigroups, Dirichlet forms, or stochastic analysis.","core_discovery":"The central discovery is a nonlocal analogue of the classical Widder-type representation theorem. For any symmetric kernel satisfying the upper bound (1.5) and the integrated coercivity condition (1.6), every nonnegative global weak solution $u$ of $\\partial_t u - L_t u = 0$ in the class $L^\\infty_{\\mathrm{loc}} L^2_{\\mathrm{loc}} \\cap L^2_{\\mathrm{loc}} H^s_{\\mathrm{loc}} \\cap L^1_{\\mathrm{loc}} L^1(\\mathbb{R}^d;w)$ admits a unique nonnegative Radon measure $\\mu$ with $\\int d\\mu/(1+|x|)^{d+2s} < \\infty$, such that $u(t,x)=\\int p_t(x,y)\\,d\\mu(y)$ and $u(0)=\\mu$ in the vague sense; conversely, any such measure produces a solution by the same formula. Under the additional pointwise lower bound (1.7), the fundamental solution $p_t$ obeys the sharp two-sided estimate $c_1(t^{-d/2s} \\wedge t/|x-y|^{d+2s}) \\le p_t(x,y) \\le c_2(t^{-d/2s} \\wedge t/|x-y|^{d+2s})$, and every nonnegative global solution satisfies the sharp Harnack estimate $u(t,x)/u(\\tau,y) \\le c\\,(t/\\tau)^{-d/2s}(1+|x-y|/\\tau^{1/2s})^{d+2s}$.","pith_inferences":["Because the proof of the lower bound is purely variational, the same two-sided estimates are plausible for nonlinear nonlocal parabolic equations, where semigroup and stochastic tools are unavailable; testing this on fractional porous-medium-type equations would be a natural extension.","The paper leaves open whether the sharp Harnack estimate survives under the weaker coercivity condition (1.6); combining the Widder representation with lower heat-kernel bounds for such kernels would be a concrete test of that possibility.","Since the representation theorem identifies the initial trace without regularity up to $t=0$, it offers a canonical notion of initial data for nonlocal evolutions that may support studying singular limits such as vanishing fractional order or concentrating coefficients.","The growth estimate via initial data singles out the weighted measure space with norm $\\int d\\mu/(1+|x|)^{d+2s}$ as the natural initial-data space; a systematic study of whether condition (1.9) is optimal would test how sharp the theory is."],"forward_implications":["Every nonnegative global weak solution in the stated class is uniquely determined by its initial measure, so the Cauchy problem for nonlocal equations with measure data is well posed and the solution theory reduces to the study of the fundamental solution.","The fundamental solution of any operator whose kernel satisfies (1.5) and (1.7) is pointwise comparable to the fractional heat kernel up to constants depending only on $d$, $s$, $\\lambda$, and $\\Lambda$, including for time-dependent kernels.","The upper bound of the fundamental solution remains valid under the weaker integrated coercivity condition (1.6); only the matching lower bound requires the pointwise lower bound (1.7).","Every nonnegative global solution obeys the sharp Harnack estimate $u(t,x)/u(\\tau,y) \\le c\\,(t/\\tau)^{-d/2s}(1+|x-y|/\\tau^{1/2s})^{d+2s}$ for $0