{"id":"5e4375c1-83e5-4214-9ae5-a3bc1dbb812d","arxiv_id":"2412.05030","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The jump kernel of a subordinated diffusion-with-jumps process is characterized by an integrability condition on the scale function, extending a known result for pure diffusions.","lead":"This math paper characterizes the jumping behavior of Markov processes that are created by time-changing a diffusion-with-jumps process. The result is a simple integral condition on a scale function that decides when the new process has a prescribed jump kernel.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.2's proof omits the argument for (a) => (c); as written the equivalence is not established.","rationale":"The paper's central claim is the equivalence in Theorem 1.2. The substantive content is Proposition 3.3, which estimates the jump kernel of the subordinate process. That proposition appears largely correct: the upper and lower bounds follow the pattern of the pure-jump and diffusion cases, and the volume-doubling uses are standard. However, the proof of Theorem 1.2 contains an unproved implication, (a) => (c). The sentence \"(a) implies \\int_{(0,1)} dt/\\psi(\\phi^{-1}(t)) < \\infty\" is not derived. It is likely true, via finiteness of the kernel and Lemma 3.8, but the argument is absent, and the reader cannot check whether a hidden assumption is needed. This is more load-bearing than the volume-doubling hypothesis, which is explicitly assumed and used throughout, or the abstract's transferring-method promise, which concerns an advertised application rather than the theorem itself. The gaps in Lemma 3.5 and Lemma 3.6 concerning constants c1 and c2 are also real but appear repairable by replacing c1 with c1 \\wedge 1 or c1 \\vee 1 and using the doubling property of scale functions. Given these omissions, the verdict CONDITIONAL is appropriate: the theorem is plausible and likely correct, but as written the proof is incomplete. The reader's weakest_assumption (VD) is not the main issue; the main issue is an unproved step in the proof of the central equivalence.","tokens_in":13970,"tokens_out":21919,"duration_ms":277766,"concrete_test":"Formally verify the omitted chain: (i) show that any pure-jump regular Dirichlet form whose kernel satisfies (1.5) has J(x,y) < \\infty for \\mu\\times\\mu-a.e. (x,y), so the right-hand side of (1.5) must be finite; (ii) show that finiteness of \\int_0^\\infty (1-e^{-\\lambda t}) dt/(t\\psi(\\phi^{-1}(t))) at one \\lambda > 0 is equivalent to \\int_0^1 dt/\\psi(\\phi^{-1}(t)) < \\infty using the LU bounds on \\phi and \\psi; (iii) conclude (c). If (i) fails because a Radon jumping measure may have infinite density, or (ii) fails, Theorem 1.2's (a) => (c) is false and the equivalence collapses; if both succeed, the gap is only an omitted proof and the theorem is salvageable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In Section 3.3, the proof of Theorem 1.2 states \"(a) implies \\int_{(0,1)} dt/\\psi(\\phi^{-1}(t)) < \\infty\" and then invokes Lemma 3.8. This is the only route from the existence of a pure-jump Dirichlet form whose jump kernel is comparable to (1.5) to the integrability condition (1.7), but no justification is supplied. The natural completion is that (1.5) contains the term \\phi(\\phi_j(r)^{-1}); if the Bernstein-type function in (1.6) were infinite at \\lambda = \\phi_j(r)^{-1}, no finite Radon jump density could be comparable to the right-hand side, and finiteness for one \\lambda > 0 is equivalent to \\int_0^1 dt/\\psi(\\phi^{-1}(t)) < \\infty by the same change-of-variable used in Lemma 3.8. However, this chain is not written, and the implication is asserted without proof. Since (a) => (c) is one half of the claimed equivalence, the central theorem is not fully proved as written. Separately, the abstract promises a \"transferring method\" for non-subordinate processes that never appears in the body; this is an unsupported advertised application, though it does not affect Theorem 1.2 itself.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies subordinate symmetric Markov processes associated with regular non-killing Dirichlet forms of diffusion+jump type on a metric measure space satisfying the volume doubling property. For a given scale function ψ, it defines a Bernstein function φ by (1.6) and shows in Proposition 3.3 that the jump kernel of the subordinated process satisfies the two-scale estimate (1.5), namely J(x,y) ≃ V(x,r)^{-1}(1/ψ(r)+φ(φ_j(r)^{-1})). The main theorem, Theorem 1.2, states an equivalence between (a) existence of a pure-jump Dirichlet form whose jump kernel satisfies (1.5), (b) existence of a subordinator whose subordinate process has such a jump kernel, and (c) the integrability condition ∫_0^1 φ(s)/(sψ(s)) ds < ∞. A pure-jump analogue is stated as Corollary 3.9. The paper also gives examples of processes satisfying the heat kernel estimates and discusses the non-comparability of the two scales in the jump kernel.","tokens_in":14222,"tokens_out":23429,"duration_ms":216801,"significance":"If the proof is completed, the result is a natural and useful generalization of Liu-Murugan's comparison theorem from the diffusion case to the diffusion+jump case. The identification of the second scale φ(φ_j(r)^{-1}) in (1.5) is a genuine contribution, and Example 3.7 indicates that this term is not comparable to 1/ψ(r) in general. The paper is also careful to work without reverse volume doubling, and it connects the abstract estimates to concrete examples on Euclidean spaces and fractals. However, the advertised 'transferring method' for non-subordinate processes does not appear in the body, and the proof of Theorem 1.2 has a missing implication; the significance is therefore conditional on a successful revision.","major_comments":[{"comment":"The implication (a) ⇒ (c) is asserted without proof. The text states '(a) implies ∫_{(0,1)} dt/ψ(φ^{-1}(t)) < ∞' and then invokes Lemma 3.8, but no argument connects the existence of a pure-jump Dirichlet form with jump kernel satisfying (1.5) to this integrability. Since (a) ⇒ (c) is one half of the claimed equivalence, this is load-bearing. A natural completion is to observe that if the integral in (c) diverged, then by Lemma 3.8 and the definition (1.6) the Bernstein function φ(λ) would be infinite for every λ>0, in particular for λ=φ_j(r)^{-1}, so the right-hand side of (1.5) would be infinite and no finite jump kernel could be comparable to it; this argument should be written out.","section":"Section 3.3, proof of Theorem 1.2"},{"comment":"The reduction of the jump-type contribution to Proposition 3.1 is not immediate and is not justified as written. The term (II) in (3.18) is integrated against dt/(tψ(φ^{-1}(t))) with φ=φ_c∧φ_j, whereas Proposition 3.1 estimates the analogous integral with φ_j in place of φ. The needed comparison 1/V(x,φ^{-1}(t)) ≤ C/V(x,φ_j^{-1}(t)) and 1/ψ(φ^{-1}(t)) ≤ C/ψ(φ_j^{-1}(t)) follows from φ≤φ_j, but it is omitted. Similarly, in (3.25) the estimate of (I) replaces the upper limit φ_c(c_1r) by φ_c(r); this requires the doubling property of φ_c and should be stated explicitly. These are local gaps in the proof of the central estimate, though they appear repairable.","section":"Lemma 3.5, equations (3.18)–(3.20) and (3.25)"},{"comment":"In the lower-bound estimate for the truncated Bernstein function, the integral over 0<t<φ(r) is bounded by an integral against the heat kernel lower bound p(t,x,y) ≥ C t/(V(x,r)φ_j(r)), which is only available for 0<t<φ(c_1r). If c_1<1, the interval [φ(c_1r),φ(r)] is not covered, and the phrase 'by using the doubling property of φ and ψ' does not by itself close the gap. The argument should show that φ(r) ≤ C φ(c_1r) uniformly in r, or else use the alternative lower bound p(t,x,y) ≥ C/V(x,φ^{-1}(t)) on the remaining interval. Without this, the lower bound in Proposition 3.3 is not fully established as written.","section":"Lemma 3.6, equation (3.37)"}],"minor_comments":[{"comment":"The same symbol φ is used both for the scale function φ_c∧φ_j and for the Bernstein function defined in (1.6); this makes statements such as (1.5), (1.7), and (3.20) genuinely ambiguous. In Theorem 1.2(c), φ(s) in the integrand is the scale function, while in (1.6) φ(λ) is the Bernstein function. Recommend using a separate symbol, for example φ for the Bernstein function.","section":"Throughout, especially Section 1"},{"comment":"The abstract promises a 'transferring method' for non-subordinate processes, but no such method appears in the body; Sections 2 and 3 contain only the direct subordination calculation and Theorem 1.2. Please either add the transferring method or revise the abstract to remove this claim.","section":"Abstract"},{"comment":"The claimed asymptotics (3.45)–(3.46) are stated without derivation, and φ_j is not specified in the example; please include the computation or a precise reference, since the example is used to illustrate the non-comparability of the two scales.","section":"Example 3.7"},{"comment":"There are several typographical errors: 'appliable' in the abstract, 'sence' in Example 2.6(iii) and Remark 2.8, 'insatnce' in Remark 2.5(i), and 'Bernstetin' in Lemma 3.6, Case 2.","section":"Various"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a serious extension of known subordination estimates and, once the missing comparability arguments and the (a) ⇒ (c) step are supplied, is likely publishable. The abstract's transferring-method claim should be reconciled with the content. I recommend major revision rather than rejection because the gaps are local and the central strategy is sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper has one real new result, Proposition 3.3, and it looks mostly right. The advertised Theorem 1.2, however, is not fully proved as written: the (a)=>(c) step is asserted, not argued. The gap is probably fillable, but a referee should ask for the argument before accepting the equivalence.\n\nWhat is new: the two-scale jump kernel estimate for subordinated diffusion+jump processes, with the term phi(phi_j(r)^{-1}) alongside 1/psi(r). Prior work by Liu-Murugan covered pure diffusions and Bae-Kang-Kim-Lee treated jump processes, so this genuinely extends the characterization to mixed processes. The non-comparability example (Example 3.7) is useful and makes the new scale concrete. The proof of Proposition 3.3 follows the standard heat-kernel split, and the lower bound argument with a truncated Bernstein function is a reasonable adaptation. The paper also correctly notes the pure-jump counterpart as a corollary.\n\nThe soft spots are real but mostly addressable. Most seriously, in Section 3.3 the proof says \"(a) implies \\int_{(0,1)} dt/psi(phi^{-1}(t)) < \\infty\" without justification. That is one direction of the claimed equivalence. A natural completion exists: if a finite Radon jump density is comparable to the right-hand side of (1.5), then phi(phi_j(r)^{-1}) must be finite, and by the change of variables in Lemma 3.8 this yields (1.7). But the chain is not written, and the implication is asserted. Second, the abstract promises a \"transferring method\" to non-subordinate processes that never appears in the body; that is over-advertisement. Third, several displayed estimates in Lemmas 3.5 and 3.6 are abbreviated to the point of obscuring the argument. For instance, (3.20) jumps from phi(phi(r)^{-1}) to phi(phi_j(r)^{-1}) without saying that for r >= 1 the condition phi_c <= phi_j on (0,1] and >= on (1,infty) makes phi = phi_j; and (3.27) hides several volume-doubling ratio steps. These are routine, but they should be spelled out.\n\nThe VD hypothesis is structural and fine as a background assumption; the paper honestly flags that it is always assumed. I did not find circularity in Proposition 3.3, and the integrability condition is not manufactured.\n\nBottom line: this deserves peer review. The core estimate is plausible and useful, and the missing (a)=>(c) argument is likely repairable. I would send it to a competent referee with explicit instructions to check that implication. If the author supplies the missing argument and tightens the cryptic displays, it should be publishable. If the gap turns out to hide a real obstruction, the verdict changes.","headline":"The new jump-kernel estimate for subordinated diffusion+jump processes is plausible and worth refereeing, but Theorem 1.2's proof omits the (a)=>(c) argument.","tokens_in":14775,"tokens_out":1882,"would_cite":true,"duration_ms":19926,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60J76","31C25","31E05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A single integrability condition decides when a subordinate process has a two-scale jump kernel.","keywords":["symmetric Markov processes","Dirichlet forms","heat kernel estimates","subordination","Bernstein functions","volume doubling","pure jump processes","scale functions"],"falsifier":"On $M=\\mathbb{R}^d$, take the sum of Brownian motion and an independent $\\alpha$-stable process, so $\\phi_c(r)=r^2$, $\\phi_j(r)=r^\\alpha$, and $\\phi(r)=r^2\\wedge r^\\alpha$, with an admissible $\\psi$, for example $\\psi(r)=r^{\\gamma\\beta}$ with $\\beta=2$. Compute the subordinate process's jump kernel directly from $J(x,y)=\\frac12\\int_0^\\infty p(t,x,y)\\,dt/(t\\psi(\\phi^{-1}(t)))$; if for some $r$ the ratio $J(x,y)V(x,r)/\\left(1/\\psi(r)+\\Phi(\\phi_j(r)^{-1})\\right)$ leaves a fixed constant range, Proposition 3.3 and Theorem 1.2 would be false, and the integral $\\int_0^1 \\phi(s)/(s\\psi(s))\\,ds$ would fail to predict the existence boundary.","tokens_in":13752,"feed_emoji":"🔄","tokens_out":12707,"duration_ms":115442,"temperature":0.7,"pith_summary":"This paper asks when a symmetric Markov process with both a diffusion part and a jump part can be obtained from another such process by subordination, meaning time is reparameterized by an independent increasing Lévy process. For a process whose heat kernel has the two-sided bounds called HK-($\\phi_c,\\phi_j$), the paper shows the subordinate process's jump kernel is always comparable to $J(x,y)\\asymp \\frac{1}{V(x,r)}\\left(\\frac{1}{\\psi(r)}+\\Phi(\\phi_j(r)^{-1})\\right)$ with $r=d(x,y)$, where the first term comes from the diffusion component and the second, new term comes from the jump component. The main theorem states that a pure jump Dirichlet form with such a kernel exists, and is in fact the Dirichlet form of a subordinated process, exactly when the scale functions satisfy $\\int_0^1 \\frac{\\phi(s)}{s\\psi(s)}\\,ds<\\infty$. This gives a single, checkable integrability condition that separates subordinate processes from arbitrary jump processes, and it extends a diffusion-only comparison theorem to the mixed setting.","feed_headline":"One integral decides when a jump process comes from subordination","feed_subtitle":"For processes mixing diffusion and jumps, one integral on scale functions decides if the kernel has the two-scale form.","key_machinery":"The machinery has three pieces. First, subordinator calculus: a subordinator with Bernstein function $\\Phi(\\lambda)=\\int_0^\\infty(1-e^{-\\lambda t})\\frac{dt}{t\\psi(\\phi^{-1}(t))}$ has Lévy measure $d\\nu(t)=dt/(t\\psi(\\phi^{-1}(t)))$, and Theorem 2.10 writes the subordinate process's jump kernel as $J(x,y)=\\frac12\\int_0^\\infty p(t,x,y)\\,d\\nu(t)$, where $p$ is the heat kernel of the original process. Second, heat-kernel sieving: the two-sided bounds in $HK^{-}(\\phi_c,\\phi_j)$ split the integral at the time scale $\\phi(r)=\\phi_c(r)\\wedge\\phi_j(r)$, and at $\\phi_c(c_1 r)$ for small $r$; the diffusion part produces $1/\\psi(r)$ and the jump part produces $\\Phi(\\phi_j(r)^{-1})$. Third, volume doubling as bookkeeping: the property $V(x,R)/V(x,r)\\le C(R/r)^{d_2}$ converts the volume ratios that appear when the heat kernel is integrated into power-law bounds, used for instance in displays (3.13), (3.27), (3.33), and (3.35). The proof of Theorem 1.2 then reduces existence of the subordinate representation to the integral in condition (c) via Lemma 3.8.","core_discovery":"The paper's central discovery is Theorem 1.2: for a diffusion+jump type regular Dirichlet form satisfying $HK^{-}(\\phi_c,\\phi_j)$ on a volume-doubling metric measure space, three statements are equivalent: (a) there is a pure jump Dirichlet form whose jump kernel satisfies $$J(x,y)\\asymp \\frac{1}{V(x,r)}\\left(\\frac{1}{\\psi(r)}+\\Phi(\\phi_j(r)^{-1})\\right),\\quad r=d(x,y),$$ (b) there is a subordinator $S_t$ such that the subordinated process $X_{S_t}$ has such a jump kernel, and (c) the integral $\\int_0^1 \\frac{\\phi(s)}{s\\psi(s)}\\,ds$ is finite, where $\\Phi$ is the Bernstein function defined from $\\psi$ in (1.6) and $\\phi=\\phi_c\\wedge\\phi_j$ is the scale function used in the heat kernel estimates. Proposition 3.3, which estimates the subordinate jump kernel, is the engine: it shows the kernel is always comparable to that two-scale expression, with the term $\\Phi(\\phi_j(r)^{-1})$ arising from the jump part of the original process and not present in the diffusion-only case. The same result holds for pure jump processes with $\\phi=\\phi_j$ (Corollary 3.9). The author emphasizes that the two scales are generally not comparable, and Example 3.7 shows $1/\\psi(r)$ and $\\Phi(\\phi_j(r)^{-1})$ can scale with different power-law exponents.","pith_inferences":["Beyond the paper: the integral $\\int_0^1 \\frac{\\phi(s)}{s\\psi(s)}\\,ds$ can be viewed as a criticality boundary, with a two-scale subordinate kernel on one side and no such kernel on the other; this boundary could be compared with phase transitions observed in simulated subordinate walks on fractals.","Beyond the paper: on Sierpinski-type spaces with $\\phi_c(r)=r^\\beta$ and $\\phi_j(r)=r^\\alpha$, the theorem gives a constructive recipe: pick $\\psi$ with finite integral, take a Brownian-plus-stable-like process, and subordinate it to obtain a jump process with prescribed kernel $J\\asymp (1/V)(1/\\psi(r)+\\Phi(r^{-\\alpha}))$.","Beyond the paper: since volume doubling is used only through power-law bounds on volume ratios, a plausible extension is to spaces satisfying only a one-sided or local doubling condition, though the paper does not pursue this."],"forward_implications":["For any $HK^{-}(\\phi_c,\\phi_j)$ process, the pure jump kernels of the form (1.5) are subordinate kernels precisely when $\\int_0^1 \\frac{\\phi(s)}{s\\psi(s)}\\,ds<\\infty$; otherwise no subordinator produces them.","The jump part contributes a genuinely new scale $\\Phi(\\phi_j(r)^{-1})$ that is generally not comparable to the diffusion scale $1/\\psi(r)$; Example 3.7 shows the two scales can have different power-law exponents.","The pure-jump analogue in Corollary 3.9 gives the same equivalence with $\\phi=\\phi_j$ and the simpler kernel estimate $J\\asymp \\Phi(\\phi_j(r)^{-1})/V(x,r)$.","Because the criteria rest only on scale-function integrability, they transfer to non-subordinate processes through stability of Dirichlet forms under perturbation, as the abstract indicates."],"supporting_citations":[{"why":"Supplies the diffusion-only comparison theorem this paper generalizes and provides Lemma 3.8, which converts condition (c) into the subordinator's integrability condition.","marker":"[23]"},{"why":"Gives the diffusion-type jump kernel estimate used as Proposition 3.4, which contributes the $1/\\psi(r)$ term in the two-scale formula.","marker":"[2, Lemma 4.2]"},{"why":"Defines the $HK^{-}(\\phi_c,\\phi_j)$ heat kernel estimates and supplies the small-time bound used in the upper-bound proof of Lemma 3.5.","marker":"[13]"},{"why":"Provides the jump-type heat kernel estimates and stability machinery that underlie the pure jump case and the transfer to non-subordinate processes.","marker":"[14]"},{"why":"Produces Theorem 2.10, the formula expressing the subordinate process's Dirichlet form and jump kernel as an integral against the heat kernel.","marker":"[26]"},{"why":"Supplies the foundational Beurling-Denny decomposition, regularity theory, and Hunt process correspondence used throughout the setup.","marker":"[16]"}],"fun_headline_variants":["One integral decides subordination for jump processes","Integral test reveals two-scale jump kernels","Subordination characterized by a single integral","Jump kernel scale decided by one integral","Two-scale jump kernels from one integral test"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument assumes the underlying metric measure space has the volume doubling property, meaning the volume of a ball grows at most polynomially in its radius; the proofs repeatedly replace volume ratios by such power-law bounds, so if this property fails the jump-kernel estimates and the equivalence are not established.","fun_headline_variants_meta":{"raw":{"variants":["One integral decides subordination for jump processes","Integral test reveals two-scale jump kernels","Subordination characterized by a single integral","Jump kernel scale decided by one integral","Two-scale jump kernels from one integral test"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000252,"raw_usage":{"total_tokens":1559,"prompt_tokens":939,"completion_tokens":620,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":555,"completion_tokens_details":{"reasoning_tokens":556}},"tokens_in":555,"tokens_out":620,"duration_ms":7303,"temperature":1.0,"reasoning_tokens":556,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:02:27.781424+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On $M=\\mathbb{R}^d$, take the sum of Brownian motion and an independent $\\alpha$-stable process, so $\\phi_c(r)=r^2$, $\\phi_j(r)=r^\\alpha$, and $\\phi(r)=r^2\\wedge r^\\alpha$, with an admissible $\\psi$, for example $\\psi(r)=r^{\\gamma\\beta}$ with $\\beta=2$. Compute the subordinate process's jump kernel directly from $J(x,y)=\\frac12\\int_0^\\infty p(t,x,y)\\,dt/(t\\psi(\\phi^{-1}(t)))$; if for some $r$ the ratio $J(x,y)V(x,r)/\\left(1/\\psi(r)+\\Phi(\\phi_j(r)^{-1})\\right)$ leaves a fixed constant range, Proposition 3.3 and Theorem 1.2 would be false, and the integral $\\int_0^1 \\phi(s)/(s\\psi(s))\\,ds$ would fail to predict the existence boundary.","supporting_citations":[{"cited_title":"Liu and M","cited_arxiv_id":null,"evidence_quote":"Supplies the diffusion-only comparison theorem this paper generalizes and provides Lemma 3.8, which converts condition (c) into the subordinator's integrability condition."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the $HK^{-}(\\phi_c,\\phi_j)$ heat kernel estimates and supplies the small-time bound used in the upper-bound proof of Lemma 3.5."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the jump-type heat kernel estimates and stability machinery that underlie the pure jump case and the transfer to non-subordinate processes."},{"cited_title":"ˆOkura, Recurrence and transience criteria for subordinated symmetric Markov processes, Forum Math","cited_arxiv_id":null,"evidence_quote":"Produces Theorem 2.10, the formula expressing the subordinate process's Dirichlet form and jump kernel as an integral against the heat kernel."},{"cited_title":"Fukushima, Y","cited_arxiv_id":null,"evidence_quote":"Supplies the foundational Beurling-Denny decomposition, regularity theory, and Hunt process correspondence used throughout the setup."}],"review_version":1}