{"id":"b7d98a38-758b-4c86-9840-710bbf365d05","arxiv_id":"2506.07694","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"On stochastically complete locally finite graphs with p in [2, ∞), fractional Sobolev spaces are complete and reflexive, and the fractional p-Laplace equation has positive and ground state solutions.","lead":"This paper constructs fractional Sobolev spaces and a fractional p-Laplace operator on connected, locally finite graphs using the heat kernel, and proves the spaces are complete, reflexive, and embed into L^q. It then shows that a nonlinear fractional Schrödinger type equation on such graphs has a positive solution and, under an extra monotonicity condition, a ground state solution.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Most load-bearing gap is verification debt: the fractional compact embedding (Lemma 13) is asserted without proof, and Theorems 4–5 depend on it; the heat-kernel hypothesis is explicit and appears sound.","rationale":"The reader's weakest assumption is not wrong: if (1) and (3) failed, W_s would not be finite. But those are explicit hypotheses, and for locally finite stochastically complete graphs the cited regularity facts are standard enough that the finiteness of W_s is not the main risk. The genuinely unresolved part is the compactness machinery. Lemma 13 is cited rather than proved; Lemmas 16 and parts of Theorem 1 are also delegated, but Lemma 13 is the one that directly supports both Theorems 4 and 5. The rest of the argument—the mountain-pass geometry, the vector monotonicity inequality in Lemma 15, the Nehari manifold construction, and the positivity argument in Lemma 14—is internally consistent as far as I can tell. The paper does provide some independent support: the kernel estimate (4) is proved in detail, and Lemma 15's vector inequality is a real, non-obvious step that is argued carefully. Still, the heavy reliance on adapted arguments from [16] and [33] with proofs omitted means a full verification needs those arguments reproduced in the fractional setting. This supports a conditional verdict rather than acceptance or rejection. My recommendation is therefore unchanged: the reader's CONDITIONAL verdict stands, with the explicit caveat that Lemma 13 must be proved in full for the fractional kernel before the central claim is considered established.","tokens_in":17135,"tokens_out":30637,"duration_ms":376280,"concrete_test":"Write out the proof of Lemma 13 in full for the model graph V=Z, μ≡1, unit edge weights, h(x)=1+|x|, p=2, s=1/2, with W_s computed from the heat kernel of Z. Check the two claims separately: (i) any bounded sequence in H^{1/2,2} has a subsequence converging strongly in L^2 (and hence in all L^q by boundedness); (ii) the proof identifies precisely which properties of W_s are used. Then examine whether those properties hold for every connected, locally finite, stochastically complete weighted graph with inf μ>0 and coercive h, in particular when edge weights are not bounded below. If the argument needs W_s to satisfy a uniform local lower bound or a particular decay estimate, that is an extra hypothesis that must be added before Theorems 4 and 5 are unconditional.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Both existence theorems pass through Lemma 13, which asserts that H^{s,p} is compactly embedded in L^q for every q∈[p,∞]. This lemma is the engine of the Palais-Smale condition in Lemma 17 and of attainment of the Nehari minimum in Lemma 19. The paper does not prove it; it says 'adapting a modified argument presented in the proof of ([16], Lemma 2.1)' and omits details. That is the point at which the central claim could fail. The reader's heat-kernel concern, by contrast, is already neutralized by the assumptions: (1) is an explicit hypothesis, and under local finiteness the diagonal derivative (3) follows from standard semigroup regularity, so the finiteness of W_s is not in question. What is genuinely unverified is the nonlocal analogue of the Rellich-Kondrachov argument: boundedness in H^{s,p} gives control of |u(x)-u(y)| only through the weighted kernel W_s, and the adaptation from [16] has to prove both uniform vanishing at infinity (using h→∞) and local convergence on finite balls (using finitely many positive W_s values). If the adaptation requires an extra quantitative bound on W_s, then the theorem's stated hypotheses are insufficient. I do not claim Lemma 13 is false; I claim it is the load-bearing unproven step.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a theory of fractional Sobolev spaces W^{s,p}(V) on connected, locally finite, stochastically complete graphs using a heat-kernel-defined interaction kernel W_s(x,y), and introduces a fractional p-Laplace operator (-Δ)_p^s. It proves that W^{s,p}(V) is a reflexive Banach space, embeds into L^q for q∈[p,∞], and that positive/negative parts preserve membership. The main results, Theorems 4 and 5, assert existence of a strictly positive solution and, under an additional monotonicity condition, a strictly positive ground state solution to the nonlinear Schrödinger-type equation (-Δ)_p^s u + h|u|^{p-2}u = f(x,u), under assumptions (13)-(14) and (A1)-(A4) or (A5). The proofs use the mountain-pass theorem and a Nehari manifold argument.","tokens_in":17427,"tokens_out":4115,"duration_ms":51835,"significance":"If the main theorems are correct, the paper provides a general variational framework for fractional p-Laplace problems on locally finite graphs, going beyond the lattice-graph settings of earlier work. The construction via stochastic completeness and the heat kernel is natural, and the paper explicitly identifies the p∈[2,∞) restriction in Remark 6 rather than hiding it. The paper also includes several explicit estimates, such as the kernel finiteness estimate (4) and the vector inequality (35), which are useful. However, the central existence theorems depend on three lemmas (10, 13, and 16) whose proofs are delegated to earlier papers with only a sentence of adaptation; in particular, the compact embedding Lemma 13 is the engine of the Palais-Smale condition and the attainment of the Nehari minimum. The current manuscript does not give the reader enough detail to verify that the fractional kernel W_s satisfies the hypotheses needed for those arguments, so the central claim is not fully established within the paper.","major_comments":[{"comment":"Lemma 13 is the load-bearing step for both existence theorems: it supplies the compact embedding H^{s,p} ↪ L^q for every q∈[p,∞] that is used in Lemma 17 to prove the Palais-Smale condition and in Lemma 19 to pass to the limit in the Nehari minimization. The proof is omitted with only 'adapting a modified argument presented in the proof of ([16], Lemma 2.1)'. This is not a routine detail: the adaptation must produce both uniform vanishing at infinity via h→∞ and local precompactness on finite balls using only the weighted kernel W_s, and it is not obvious that the estimates in [16] survive the replacement of the discrete gradient by the infinite-dimensional fractional gradient of (5). The authors should either provide a full proof or state precisely which theorem in [16] applies verbatim after the indicated modifications.","section":"§4, Lemma 13"},{"comment":"Lemma 10 asserts W^{s,p}(V) embeds into L^q(V) for all q∈[p,∞] and is proved by 'adapting the argument presented in the proof of ([15], Theorem 7)', with no details. Since this embedding is used in the definition of H^{s,p} and in the argument that H^{s,p} is a closed subspace of W^{s,p}, and since the fractional kernel W_s does not appear in [15], the adaptation is not a formality. Please provide the proof or a precise reference to a statement that covers this specific kernel.","section":"§3, Lemma 10"},{"comment":"Lemma 16, which supplies the mountain-pass geometry (existence of u_0 with E_{s,p}(t u_0)→-∞ and a positive lower bound on the sphere ∥u∥_{H^{s,p}}=r), is stated without proof and attributed to ([16], Lemmas 3.2 and 3.3). These properties depend on the specific form of the fractional energy and on assumptions (A2)-(A4), so they should be proved or at least sketched in the paper. In particular, the verification of the local minimum condition uses the spectral quantity λ_p from (A4), which is new here.","section":"§4.1, Lemma 16"},{"comment":"The paper restricts to p∈[2,∞) and states in Remark 6 that C_c(V)⊆W^{s,p}(V) is unverified for 1<p<2. This limitation is acknowledged, and I do not treat it as an error. However, because Theorem 1(i) is used in the proof of Lemma 9 and in the definition of H^{s,p}, the reader should be told whether the p∈[2,∞) restriction also affects the validity of Lemma 13 or the mountain-pass construction beyond the C_c inclusion. If the restriction is only needed for the truncation argument in Lemma 12, that should be stated explicitly.","section":"Remark 6 and §2.2"}],"minor_comments":[{"comment":"The title reads 'Fractional Sobolev spaces and fractionalp-Laplace equations'; a space is missing between 'fractional' and 'p-Laplace'.","section":"Title"},{"comment":"The interchange of the sum over y and the integral over t in (4) is justified by positivity of the integrand, but this should be stated explicitly, as the reader may otherwise worry about the validity of the equality.","section":"Equation (4)"},{"comment":"The reflexivity of L^p is delegated to [32, Proposition 3.2] without proof. Since this is a standard consequence of uniform convexity of the vector-valued L^p norm, a brief indication of the Clarkson-type inequality would improve self-containedness.","section":"Lemma 8"},{"comment":"In the proof of inequality (35), the function φ(t,m) is minimized over m∈[-1,1], but the case analysis for critical points and endpoints is compressed. The lower bound for critical points should be written out fully, since the constant 1/(2^{p-2}p) is used in Lemma 17.","section":"Lemma 15"},{"comment":"The proof of Lemma 20 follows [2, Lemma 3.5] and is sketched. The argument is plausible, but the choice of v with ⟨E'_{s,p}(u_0),v⟩<0 and the construction of t_m should be given a few more details, as the continuity of φ(t,m) in both variables is essential.","section":"Proof of Theorem 5"}],"recommendation":"major_revision","confidential_remarks":"The paper is a solid contribution but its central existence results rest on several omitted adaptations of arguments from earlier papers by the same group, especially the compact embedding Lemma 13. The authors should be asked to include full proofs of Lemmas 10, 13, and 16, or to state exactly which published results cover the fractional kernel W_s. If those lemmas are indeed correct, the paper will likely be acceptable; the current version is not self-contained enough for a journal publication. The heat-kernel concern raised in the stress-test is not, in my reading, a fatal issue: assumption (1) is explicit, and the finiteness of W_s is proved in (4). The real risk is the unverified compactness."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the paper delivers what the abstract promises—fractional Sobolev spaces and a fractional p-Laplacian on connected, locally finite, stochastically complete graphs, plus existence of positive and ground-state solutions to a nonlinear Schrödinger-type equation. The heat-kernel definition of the interaction kernel is explicit, the kernel estimate (4) is checked, and the vector-valued fractional gradient is a real adaptation rather than a formal copy. I looked at the vector inequality in Lemma 15; the proof is in the text and holds up. There is no parameter fitting and no circularity: the cited results from the same group are independent earlier theorems.\n\nThe soft spots are real but localized. Lemma 13, the compact embedding of H^{s,p} into L^q for every q in [p,∞], is the engine of both the Palais-Smale condition (Lemma 17) and the Nehari attainment (Lemma 19). The paper says it follows by adapting a modified argument from [16] and omits the details. That is genuine verification debt at a load-bearing point. The adaptation could require a quantitative property of W_s that is not in the stated assumptions; I am not claiming the lemma is false, just that the proof is owed. Lemmas 10 and 16 are also delegated, though those are less central. The restriction p≥2 is disclosed in Remark 6 with the reason named, so I do not count it as a hidden flaw, but it does mean the framework does not yet cover 1<p<2.\n\nThe heat-kernel foundation is sound. Stochastic completeness is an explicit hypothesis, positivity is standard, and the diagonal regularity bound (3) comes from known semigroup results, so the finiteness of W_s is not a lurking assumption. The incremental relation to the companion p=2 paper [38] is asserted rather than itemized; the authors should say which estimates for p>2 are genuinely new. The citation pattern is honest.\n\nThis is a useful toolkit paper for people doing variational methods on graphs, not a field reorientation. If Lemma 13 can be proved under the stated assumptions, Theorems 4 and 5 are defensible. My recommendation: send it to a serious referee and make the compact embedding the main focus of the report. I would engage.","headline":"New fractional graph Sobolev framework with plausible existence theorems; the compact embedding is the load-bearing proof debt.","tokens_in":17954,"tokens_out":2674,"would_cite":true,"duration_ms":30612,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A15","35R02","35R11","46E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"Using the graph's heat kernel as the weight for a fractional gradient, this paper constructs fractional Sobolev spaces on locally finite graphs and proves existence of positive and ground-state solutions for a fractional p-Laplace…","keywords":["fractional Sobolev space","fractional p-Laplace operator","variational method","heat kernel","locally finite graph","stochastic completeness","ground state solution","nonlinear Schrödinger equation"],"falsifier":"A concrete check is to take a stochastically incomplete graph, where $\\sum_{y\\in V}p(t_0,x_0,y)\\mu(y)<1$ for some $t_0,x_0$, and evaluate $\\sum_{y\\neq x}W_s(x,y)$ from (4): if the sum diverges, the kernel estimate fails and the fractional objects in the paper are not defined.","tokens_in":16925,"feed_emoji":"🕸️","tokens_out":15233,"duration_ms":138283,"temperature":0.7,"pith_summary":"This paper builds fractional Sobolev spaces and a fractional $p$-Laplace operator on connected, locally finite, stochastically complete graphs, where previous fractional results were mostly confined to lattice graphs. The fractional gradient is defined by weighting vertex differences with a kernel made from the graph's heat kernel, and the paper shows the resulting spaces are reflexive Banach spaces that embed into $L^q$ for every $q\\in[p,\\infty]$. With this toolkit it proves that the nonlinear equation $(-\\Delta)_p^s u + h|u|^{p-2}u = f(x,u)$ has a strictly positive solution, and, under the extra monotonicity assumption that $f(x,y)/y^{p-1}$ is increasing in $y$, a strictly positive ground state solution. The significance is a variational existence theory for nonlocal $p$-Laplace equations on discrete graphs.","feed_headline":"Heat-kernel calculus solves fractional p-Laplace equations on graphs","feed_subtitle":"The heat-kernel fractional gradient yields positive and ground-state solutions on locally finite graphs.","key_machinery":"The load-bearing object is the heat-kernel-weighted kernel $W_s(x,y)=\\frac{s}{\\Gamma(1-s)}\\mu(x)\\mu(y)\\int_0^\\infty p(t,x,y)t^{-1-s}\\,dt$, whose finiteness is guaranteed by stochastic completeness and the on-diagonal heat-kernel bound. It defines the fractional gradient $\\nabla^s u(x)$ whose components are $\\sqrt{\\widetilde W_s(x,y)/(2\\mu(x))}\\,(u(x)-u(y))$, and the fractional $p$-Laplacian $(-\\Delta)_p^s u = -\\operatorname{div}_s(|\\nabla^s u|^{p-2}\\nabla^s u)$. The identity that carries the variational argument is the integration-by-parts formula $\\int_V \\varphi(-\\Delta)_p^s u\\,d\\mu = \\int_V |\\nabla^s u|^{p-2}\\nabla^s u\\cdot\\nabla^s\\varphi\\,d\\mu$, which identifies weak solutions of the equation with critical points of the energy functional $E_{s,p}(u)=\\frac{1}{p}\\int_V(|\\nabla^s u|^p + h|u|^p)\\,d\\mu - \\int_V F(x,u^+)\\,d\\mu$.","core_discovery":"The central claim is that the heat kernel of a graph can define a nonlocal gradient with enough structure to support the full variational machinery. With $W_s(x,y)=\\frac{s}{\\Gamma(1-s)}\\mu(x)\\mu(y)\\int_0^\\infty p(t,x,y)t^{-1-s}\\,dt$, the fractional gradient $\\nabla^s$ and the fractional $p$-Laplace operator $(-\\Delta)_p^s$ are well defined, and the paper proves that $W^{s,p}(V)$ and $W_0^{s,p}(V)$ are reflexive Banach spaces with $W^{s,p}(V)\\hookrightarrow L^q(V)$ for $q\\in[p,\\infty]$. The two main existence theorems state that, under positivity and coercivity of the potential $h$ and the structural conditions (A1)-(A4) on $f$, the equation $(-\\Delta)_p^s u + h|u|^{p-2}u = f(x,u)$ admits a strictly positive solution, and with the additional monotonicity (A5) it admits a strictly positive ground state solution. The proof route is the mountain-pass theorem and the Nehari manifold, both made to work by the integration-by-parts identity and the compact embedding of $H^{s,p}$ into $L^q$.","pith_inferences":["Editorial extension: the same heat kernel could be used to define fractional Laplacian eigenvalue problems on general graphs, extending the lattice eigenvalue estimates that motivated the kernel's form.","Editorial extension: because the proof relies mainly on finiteness and symmetry of $W_s$, graphs that are only locally stochastically complete, or weighted graphs with a different time scale, might support a modified kernel with the same variational structure.","Editorial extension: a concrete test of the $p\\in[2,\\infty)$ restriction is whether a two-point test function on a graph with a single non-zero edge still has finite $W^{s,p}$ energy for $1<p<2$; if it does, the missing inclusion may be provable by a finer estimate.","Editorial extension: if the compact embedding of $H^{s,p}$ into $L^q$ survives under graph perturbations, the mountain-pass and Nehari arguments could adapt to sequences of graphs, yielding discrete-to-continuum convergence for fractional $p$-Laplace problems."],"forward_implications":["If Theorems 4 and 5 hold, the fractional $p$-Laplace equation (12) has at least one strictly positive solution whenever the heat kernel and nonlinearity meet the stated conditions, and a strictly positive ground state when $f(x,y)/y^{p-1}$ is increasing.","The embedding $W^{s,p}(V)\\hookrightarrow L^q(V)$ for every $q\\in[p,\\infty]$ becomes a standard tool, so other variational problems on graphs can be treated with the same spaces.","Because the construction works on every connected, locally finite, stochastically complete graph, lattice results are a special case rather than the whole theory.","The framework applies only for $p\\geq 2$; the inclusion $C_c(V)\\subseteq W^{s,p}(V)$ for $1<p<2$ is not proved, so the existence theorems do not yet cover that range."],"supporting_citations":[{"why":"supplies the kernel formula $W_s$ that the paper modifies to define the fractional gradient on general graphs","marker":"[36]"},{"why":"provides the vector-valued function space $L^p(V)$ and reflexivity argument used to build $W^{s,p}(V)$","marker":"[32]"},{"why":"the variational lemmas for positive solutions on locally finite graphs are adapted here for the mountain-pass geometry and compactness","marker":"[16]"},{"why":"its Sobolev embedding proof is adapted to obtain $W^{s,p}(V)\\hookrightarrow L^q(V)$","marker":"[15]"},{"why":"the fractional divergence definition is improved from this earlier paper by the same authors","marker":"[38]"},{"why":"supplies the direct technique used to show the Nehari ground-state energy is attained","marker":"[33]"},{"why":"the mountain-pass theorem is applied to produce the critical point in Theorem 4","marker":"[3]"},{"why":"with [37], gives $p(0,x,x)\\mu(x)=1$ and $C^1$ regularity of the heat kernel used to bound $W_s$","marker":"[26]"},{"why":"gives heat-kernel regularity used to prove the kernel sums are finite","marker":"[37]"}],"fun_headline_variants":["Heat kernel builds fractional Sobolev spaces on any graph","Fractional p-Laplacian on graphs via heat kernel gradients","Heat kernel yields existence for fractional p-Laplace equations","Graph heat kernel solves nonlocal p-Laplace problems"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the graph is stochastically complete, meaning its heat kernel conserves total mass $\\sum_{y\\in V}p(t,x,y)\\mu(y)=1$ and is smooth enough at $t=0$; on a graph where this fails, the kernel $W_s$ can cease to be finite and the fractional gradient, Sobolev spaces, and $p$-Laplacian are not defined.","fun_headline_variants_meta":{"raw":{"variants":["Heat kernel builds fractional Sobolev spaces on any graph","Fractional p-Laplacian on graphs via heat kernel gradients","Heat kernel yields existence for fractional p-Laplace equations","Graph heat kernel solves nonlocal p-Laplace problems"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000511,"raw_usage":{"total_tokens":2476,"prompt_tokens":926,"completion_tokens":1550,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":542,"completion_tokens_details":{"reasoning_tokens":1484}},"tokens_in":542,"tokens_out":1550,"duration_ms":12195,"temperature":1.0,"reasoning_tokens":1484,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:29:26.830174+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A concrete check is to take a stochastically incomplete graph, where $\\sum_{y\\in V}p(t_0,x_0,y)\\mu(y)<1$ for some $t_0,x_0$, and evaluate $\\sum_{y\\neq x}W_s(x,y)$ from (4): if the sum diverges, the kernel estimate fails and the fractional objects in the paper are not defined.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the vector-valued function space $L^p(V)$ and reflexivity argument used to build $W^{s,p}(V)$"},{"cited_title":"Grigor’yan, Y","cited_arxiv_id":null,"evidence_quote":"the variational lemmas for positive solutions on locally finite graphs are adapted here for the mountain-pass geometry and compactness"},{"cited_title":"Grigor’yan, Y","cited_arxiv_id":null,"evidence_quote":"its Sobolev embedding proof is adapted to obtain $W^{s,p}(V)\\hookrightarrow L^q(V)$"},{"cited_title":"Existence and convergence of solutions to $p$-Laplace equations on locally finite graphs","cited_arxiv_id":"2306.14121","evidence_quote":"supplies the direct technique used to show the Nehari ground-state energy is attained"},{"cited_title":"Ambrosetti, P","cited_arxiv_id":null,"evidence_quote":"the mountain-pass theorem is applied to produce the critical point in Theorem 4"},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"with [37], gives $p(0,x,x)\\mu(x)=1$ and $C^1$ regularity of the heat kernel used to bound $W_s$"},{"cited_title":"Wojciechowski, Heat kernel and essential spectrum of infinite graphs, Indiana Univ","cited_arxiv_id":null,"evidence_quote":"gives heat-kernel regularity used to prove the kernel sums are finite"}],"review_version":1}