{"id":"af30997e-3748-4a5d-b973-d7e114787781","arxiv_id":"2411.15485","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Local times of spectrally positive Levy processes with Gaussian components are equal in law to the unique solution of a stochastic Volterra equation, with new comparison, regularity, moment, and Laplace-functional consequences.","lead":"Local times of a spectrally positive Levy process with Gaussian component are shown to solve an explicit stochastic Volterra equation driven by a Gaussian white noise and two Poisson random measures. This turns a non-Markovian object into an equation-based tool with comparison principles, Holder regularity, and Laplace formulas, connecting to branching processes and random trees.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.36 is applied to identify the Lévy-jump term I_4 without verifying the domination condition (4.98); the rescaled intensities have unbounded total mass in the infinite-activity case, so tightness of the infinite-dimensional semimartingales is not established.","rationale":"I read the paper as a serious and largely self-contained proof of a stochastic Volterra representation for local times of spectrally positive Lévy processes with Gaussian components, extending the author's earlier work [68]. The central claim requires a delicate weak-convergence argument through compound-Poisson approximations. The paper is honest about quoted results and open equivalences, and the overall architecture is plausible. My stress-test focused on the internal step where the compensated Poisson term I_4 is identified. The application of Lemma 4.36 in §4.4.3 is the point where the jump measure of the limiting Lévy process is recovered from the approximation, and it is also the step least explicitly checked. The reader's weakest_assumption emphasized the external Lemma 3.4 and the prior SVE lemmas from [68]; I agree that those are important, but the more concrete, internally checkable weakness is the unverified domination condition (4.98), without which the tightness of the infinite-dimensional H#-semimartingales in Lemma 4.36 cannot be concluded. I do not believe this is fatal: the Lévy measure satisfies ∫(y∧y²)ν(dy)<∞, and the approximating measures are built from ν, so a domination by ν(dy)+dy is plausible. But the paper should supply the verification. Because the concern is a gap in a technical lemma rather than a demonstrated falsehood, and the reader's CONDITIONAL verdict already reflects medium risk, I keep the verdict unchanged. A short addition to Section 4.4.2 verifying (4.98) would close the gap and allow full acceptance.","tokens_in":1470,"tokens_out":1182,"duration_ms":307493,"concrete_test":"Verify condition (4.98) for the rescaled intensities μ_n(n·dy)=n³γ_nθ_n(e_c∗dΛ_n)(n·dy) with an explicit dominating measure, e.g., m(dy)=ν(dy)+dy. Concretely: (a) prove sup_n μ_n(n·[a,b]) ≤ C(ν([a/2,b+1]) + (b−a)) for all 0<a<b; (b) prove sup_n ∫_0^δ y² μ_n(n·dy) ≤ C∫_0^{2δ}(y²∧1)ν(dy) + Cδ³; (c) check ∫_0^T |W(t−s)−W(t−s−y)|² m(dy) < ∞ for all T>0. If these bounds hold, the domination condition is satisfied and the proof of the I_4 limit is complete; if any fails—e.g., for ν(dy)=y^{-1−α}dy with α∈(1,2)—the limit characterization must be revisited.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 4.36 is the tool that characterizes the limit of the sequence I^{(n)}_4, which becomes the compensated Poisson integral term in (1.8). The lemma requires, in addition to vague convergence of the intensities, the uniform domination condition (4.98): there exists a σ-finite measure m(dy) such that sup_n ∫ f(y) μ_n(n·dy) ≤ ∫ f(y) m(dy) for every nonnegative measurable f, and sup_{t∈[0,T]} ∫ |G(t,y)|² m(dy) < ∞. In the application in §4.4.3, μ_n(n·dy) = n³γ_n θ_n (e_c ∗ dΛ_n)(n·dy) and the target intensity is ν(dy). For an infinite-activity Lévy measure (ν̄(0)=∞), one has ν̄(η_n)→∞, and the total mass of μ_n(n·) on (0,∞) is ν̄(η_n)→∞, with the mass concentrating near 0. Hence ν alone cannot dominate μ_n(n·) on intervals near 0, and the paper never specifies a dominating measure m that satisfies (4.98). Without this domination, the uniform tightness condition in Lemma 4.36 is unproved, so the claimed weak convergence to ∫_0^t∫_0^{X_ζ(s)}∫_0^∞ (W(t−s)−W(t−s−y)) eN_ν(ds,dz,dy) is not justified. This is load-bearing because the eN_ν term is one of the three independent noises driving (1.8); a missing or incorrectly identified jump term would invalidate Theorem 1.1 and the subsequent comparison, moment, and Laplace-functional results. The gap is probably fixable—for instance, m(dy)=ν(dy)+dy may satisfy the required domination when combined with the estimates in Corollary 4.15 and Proposition 4.14—but the paper should state and prove this explicitly.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper treats a spectrally positive Lévy process ξ with Gaussian component and Laplace exponent (1.1). For ζ≥0, it studies the local-time profile L^ξ_ζ(x)=L^ξ(x,τ^L_ξ(ζ)) conditioned on τ^L_ξ(ζ)<∞. Theorem 1.1 asserts that this profile is equal in law to the unique nonnegative continuous solution of the stochastic Volterra equation (1.8), driven by a Gaussian white noise and two Poisson random measures whose kernel is built from the scale function W. The proof approximates ξ by rescaled compound Poisson processes, establishes C-tightness of the resulting equations, and identifies the limit via Lemmas 4.35–4.36. From (1.8) the paper derives a comparison principle (Theorem 1.5), strong existence and uniqueness under an additional finite-mean-type assumption (Theorem 1.7), uniform moment bounds, Hölder regularity and a maximal inequality (Theorems 1.8–1.9), and an exponential-affine Laplace functional representation (Theorem 1.10).","tokens_in":86142,"tokens_out":18507,"duration_ms":183240,"significance":"If the proof is completed, this would be a substantial extension of the author's earlier stable-process result and would provide a single stochastic Volterra description for local-time profiles of the whole Gaussian-component class, with several nontrivial consequences: pathwise comparison, moment and Hölder estimates, and a Laplace-functional formula. The paper is unusually detailed and contains many explicit estimates, equivalent representations, and standalone lemmas, and it transparently identifies the results imported from the author's previous work [68]. The main risks are concentrated in the compound-Poisson approximation step and in the identification of the jump term in the limit; both are discussed below.","major_comments":[{"comment":"Equation (4.2) claims that the probability law Λ_n has finite mean as a consequence of (1.2), but this is not true in general. Indeed, ∥Λ_n∥_{L^1} = (n/\\barν(η_n)) ∫_0^∞ \\barν(η_n+y)dy = (n/\\barν(η_n)) ∫_{η_n}^∞ \\barν(z)dz, and the last integral equals ∫_0^∞ (z−η_n)_+ ν(dz), which is infinite whenever ∫_0^∞ z ν(dz)=∞. The condition (1.2) allows infinite mean, e.g. ν(dy)∼y^{-2}dy near 0. Consequently the jump law Π_n in (4.5) need not have finite mean, the quantity ∥Π_n∥_{L^1} in (4.7) is not finite, the arrival rate γ_n in (4.4) does not produce a finite-mean compound Poisson process, and Lemma 3.1 cannot be applied. This affects all of Sections 4.2–4.4 for spectra with infinite mean, so the proof of Theorem 1.1 as written covers only finite-mean ν. The manuscript should either restrict the main statement to ∫_0^∞ y ν(dy)<∞ or replace the compound-Poisson approximation by a finite-mean truncation that is then removed in a separate limiting argument.","section":"§4.1, Eq. (4.2)"},{"comment":"The identification of I^{(n)}_4 as the compensated Poisson integral in (1.8) invokes Lemma 4.36 without verifying its domination hypothesis (4.98). The manuscript checks the vague convergence of the intensities n^3 γ_n θ_n (e_c*dΛ_n)(n·dy) to ν(dy), but Lemma 4.36 also requires a σ-finite measure m such that sup_n ∫ f(y) μ_n(n·dy) ≤ ∫ f(y) m(dy) for every nonnegative measurable f and sup_t ∫ |G(t,y)|² m(dy)<∞. No such m is exhibited. In the infinite-activity case the total mass of μ_n(n·dy) is unbounded as n→∞, so domination is not automatic, and vague convergence alone does not yield the uniform tightness of the infinite-dimensional semimartingales used in the proof of the lemma. This is load-bearing because the I_4 limit is one of the three noises in (1.8). The gap is local and probably repairable, but the authors must either construct m satisfying both conditions in (4.98) or replace Lemma 4.36 by a different tightness argument that only uses the available moment and convergence information.","section":"§4.4.3, Eq. (4.105)"}],"minor_comments":[{"comment":"In (1.13) the first term on the right is written ζ(1−bW(x)), but the variable should be t; the same display otherwise uses t throughout.","section":"Remark 1.3, Eq. (1.13)"},{"comment":"The proof cites the bound sup_t W_β(t) ≤ (β+b)^{-1}, but for β>0 the correct scale-function bound from (1.7) applied to W_β is sup_t W_β(t)=1/β. Since the constants in the proposition may depend on β, this does not change the qualitative conclusion, but the displayed inequality should be corrected.","section":"§7, Proposition 7.2"},{"comment":"Lemma 4.37 is the only ingredient that produces the pathwise ordering of the approximating processes in the proof of Theorem 1.5, yet its proof is omitted as 'elementary'. The authors should include a short proof or a precise reference, since the comparison principle depends on it.","section":"§4.4.4, Lemma 4.37"},{"comment":"There are a few typographical slips: 'lake of negative jumps' should be 'lack of negative jumps', and 'limited acknowledge of excursion theory' should be 'limited knowledge of excursion theory'.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"This is a serious and well-written paper, and the main theorem is plausible. The two gaps identified above are independent and both occur in the approximation/compactness argument: the finite-mean claim in (4.2) restricts the current proof to finite-mean Lévy measures, and the domination condition in (4.98) is not checked for the jump-component identification. Both appear repairable either by an additional assumption or by additional technical work in Section 4. I would not reject the paper on this basis, but I would want the revision to address these points before publication. The self-citation to [68] is transparent and appropriate, and the unresolved comparison with the excursion-theoretic representation in [63] is clearly acknowledged and does not bear on the main results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline is that this is a real extension of Xu's own [68]: for every spectrally positive Lévy process with a Gaussian component, the spatial profile of local times stopped at the inverse local time solves one explicit stochastic Volterra equation — (1.8) — driven by independent Gaussian white noise and two Poisson measures. The Gaussian term is genuinely new, and the paper adds a comparison principle, uniform moment bounds, (1/2−ε) Hölder regularity, and an exponential-affine Laplace functional. The paper is unusually transparent: it identifies the lemmas quoted from [68], states that strong uniqueness needs the extra condition ν̄̄(0+) < ∞, and flags the open equivalence with the Rivero–Contreras excursion representation.\n\nThe overall proof strategy is sound and the structure is coherent: compound Poisson approximations, resolvent convergence to the scale function, C-tightness, then a limit characterization; the comparison principle comes from a coupling; the Laplace functional from duality. I find no circularity, and the self-citations are legitimate.\n\nNow the soft spot, and here the stress-test note is right. In §4.4.3, Lemma 4.36 is applied to identify the limit of I_4^{(n)} as the compensated Poisson integral with intensity ν. That lemma requires the uniform domination condition (4.98): a σ-finite m with sup_n ∫ f dμ_n(n·) ≤ ∫ f dm and sup_t ∫ |G(t,y)|² m(dy) < ∞. The paper verifies vague convergence but never checks domination. In the infinite-activity case the rescaled intensities n³γ_nθ_n(e_c∗dΛ_n)(n·dy) have total mass ν̄(η_n) → ∞, concentrated near zero, so ν alone does not obviously dominate them; no dominating m is exhibited. Without (4.98), the uniform tightness part of Lemma 4.36 is unproved, so the convergence to the Ñ_ν term in (1.8) is not justified. It is load-bearing — that term is one of the three noises in the equation — and Theorem 1.1 plus the comparison, moment, and Laplace results inherit the gap. It looks fixable (e.g., m(dy) = ν(dy) + dy with the paper's own Corollary 4.15 and Proposition 4.14), but the argument has to be written out.\n\nThis is a specialist paper — Lévy processes, local times, Volterra equations. A referee should audit Sections 4.2.2–4.4.3. It deserves a serious referee; send it out, and condition acceptance on fixing the domination step.","headline":"Likely-correct SVE representation for local times of spectrally positive Lévy processes with Gaussian component, but the limit characterization in §4.4.3 skips the domination condition in Lemma 4.36 and a referee must check it.","tokens_in":86784,"tokens_out":7818,"would_cite":true,"duration_ms":65799,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G51","60J55","60H20","60G22","60F17","60G55"],"pacs":[],"model":"deepseek-v4-flash","headline":"Local-time profiles obey one explicit Volterra equation","keywords":["local time","Lévy process","spectrally positive Lévy process","stochastic Volterra equation","Ray-Knight theorem","comparison principle","Laplace functional","scale function"],"falsifier":"Compute the Laplace functional (1.22) for a drifted Brownian motion ($\\nu = 0$, $c > 0$, $b \\ge 0$) and compare it with the classical Laplace transform of the squared Bessel / Feller diffusion that Remark 1.2 identifies as the solution of the equation; a mismatch for any measure $\\mu$ would disprove the representation. Alternatively, the paper states that the equivalence between its Laplace-functional formula and the excursion-theoretic representation in [63] remains open; evaluating both sides numerically for a spectrally positive process with Gaussian component and a nontrivial jump measure on a specific test measure $\\mu$ would settle whether the two formulas agree.","tokens_in":85501,"feed_emoji":"📈","tokens_out":12009,"duration_ms":98037,"temperature":0.7,"pith_summary":"The paper proves that the spatial profile of local times of a spectrally positive Lévy process with a Gaussian component—stopped when the local time at zero first exceeds a level $\\zeta$ and conditioned on that time being finite—is, in law, the unique non-negative continuous solution of an explicit stochastic Volterra equation. The equation is driven by a Gaussian white noise and two independent Poisson measures, with a convolution kernel built from the scale function of the process. From this single representation the paper derives a comparison principle for local times under different drifts or stopping levels, uniform moment bounds, $(1/2-\\varepsilon)$-Hölder regularity with all moments of the Hölder coefficient, and an exponential-affine formula for the Laplace functional in terms of a path-dependent nonlinear Volterra equation. If correct, the spatial dynamics of local times for this entire Lévy class are reduced to one equation, and the flow picture suggested by the comparison principle connects the local-time field to branching-type stochastic flows.","feed_headline":"Local-time profiles obey one explicit Volterra equation","feed_subtitle":"Spatial local-time profile solves a single equation; comparison, regularity, and Laplace formulas follow.","key_machinery":"The object that carries the argument is the stochastic Volterra equation (1.8) together with its data: the scale function $W$ of the spectrally positive Lévy process, defined by $W(x)=0$ for $x<0$ and $\\int_0^\\infty e^{-\\lambda x}W(x)\\,dx = 1/\\Phi(\\lambda)$ for $\\lambda>0$, and three mutually independent driving noises—a Poisson random measure $N_0$ with intensity $\\bar\\nu(y)\\,dz\\,dy$ on $(0,\\infty)^2$, a Gaussian white noise $B_c$ with intensity $2c\\,ds\\,dz$, and a compensated Poisson random measure $\\widetilde N_\\nu$ with intensity $ds\\,dz\\,\\nu(dy)$. The four terms of the equation encode, respectively, the average contribution of positive excursions, the randomness of overshoots of jump excursions, the Brownian diffusion part, and the compensated jump part of the local-time profile. The proof machinery is a weak-convergence argument: the Lévy process is approximated by rescaled compound Poisson processes whose jump laws mix an exponential law with a tail-tilted copy of the Lévy measure, their local times are written as Volterra equations, and the rescaled equations are shown to converge to (1.8); the resolvent identity for the renewal equation (3.3) provides the asymptotic link between the compound-Poisson resolvent and the scale function.","core_discovery":"The central claim is that for each $\\zeta \\ge 0$, the process $L^\\xi_\\zeta(x) = L^\\xi(x, \\tau^L_\\xi(\\zeta))$, $x \\ge 0$, conditioned on $\\tau^L_\\xi(\\zeta) < \\infty$, has the same distribution as the unique continuous non-negative solution of the stochastic Volterra equation $$X_\\zeta(t) = \\zeta\\, c\\, W'(t) + \\int_0^\\zeta\\int_0^\\infty \\big(W(t)-W(t-y)\\big) N_0(dz,dy) + \\int_0^t\\$int_0^{{X_\\zeta(s)}}$ W'(t-s)\\, B_c(ds,dz) + \\int_0^t\\$int_0^{{X_\\zeta(s)}}$\\int_0^\\infty \\big(W(t-s)-W(t-s-y)\\big) \\widetilde N_\\nu(ds,dz,dy),$$ where $W$ is the scale function (Laplace transform $1/\\Phi$), $N_0$ has intensity $\\bar\\nu(y)\\,dz\\,dy$, $B_c$ has intensity $2c\\,ds\\,dz$, and $\\widetilde N_\\nu$ is compensated with intensity $ds\\,dz\\,\\nu(dy)$. The paper proves this by rescaling compound Poisson processes with carefully chosen jump laws, passing to the limit in the Volterra equations for their local times, and then reading off the limit equation. Uniqueness in law is obtained as a corollary of the Laplace-functional representation.","pith_inferences":["If the representation extends to the whole real line, the SVE approach would give a unified analytic handle on reflected and non-reflected local-time fields, which the paper only treats on the positive half-line.","The paper leaves open the equivalence of its Laplace-functional formula with the excursion-theoretic formula of [63]; proving that equivalence would tie the Volterra kernel directly to excursion measures and likely simplify the comparison principle.","The comparison principle suggests a pathwise construction of the stochastic flow of local times as a flow of SVE solutions, but the paper does not construct such a flow; that would be the natural next step.","A direct check of the Brownian case ($\\nu = 0$) recovers the classical Feller diffusion via Remark 1.2, so the SVE framework can be calibrated against known results for squared Bessel processes."],"forward_implications":["Local times in the spatial direction for every spectrally positive Lévy process with a Gaussian component are governed by one explicit stochastic Volterra equation, so path properties and distributional identities can in principle be derived from the equation rather than from excursion theory.","The comparison principle (Theorem 1.5) puts the local-time profiles of processes with different drifts or different stopping levels on a common probability space with a pointwise ordering, which yields a stochastic flow with the branching property.","The SVE gives uniform moment bounds in the level variable (Theorem 1.8) and local $(1/2-\\varepsilon)$-Hölder continuity of the spatial profile with all moments of the Hölder coefficient bounded (Theorem 1.9).","The Laplace functional of the local-time profile has the exponential-affine form $\\exp\\{-\\zeta\\, F\\circ V_\\mu(x)\\}$ where $V_\\mu$ solves the nonlinear Volterra equation (1.20); in the subcritical case, the total local-time profile $L^\\xi_\\infty$ is the solution with an exponentially distributed $\\zeta$ (Corollary 1.11).","Under the extra integrability condition $\\bar{\\bar\\nu}(0+) < \\infty$, the SVE has a unique strong solution, which is a semimartingale with an explicit decomposition (Theorem 1.7)."],"supporting_citations":[{"why":"Supplies Lemma 3.4, the finite-dimensional convergence of local times of rescaled compound Poisson processes to those of the Lévy process, which is the bridge needed for the approximation.","marker":"[49]"},{"why":"Provides the stochastic Volterra representation for compound Poisson local times (Lemmas 3.1–3.2), the starting point of the limiting argument.","marker":"[68]"},{"why":"Gives the formula $W''(0+) = (b - \\bar{\\bar\\nu}(0+))/c^2$ and the regularity of the scale function used in the strong-uniqueness and semimartingale representation (Theorem 1.7).","marker":"[12]"},{"why":"Establishes the smoothness of scale functions that the Volterra kernel relies on throughout the proofs.","marker":"[18]"},{"why":"Background for Itô integrals and semimartingale calculus that define the stochastic Volterra integrals in (1.8).","marker":"[39]"},{"why":"Generalized Yamada–Watanabe theorem used in Lemma 5.1 to deduce strong existence and uniqueness from pathwise uniqueness.","marker":"[44]"},{"why":"The classical pathwise-uniqueness method that Theorem 1.7 adapts for the Hölder-continuous coefficients of the SVE.","marker":"[69]"},{"why":"Provides the exponential formula for Poisson integrals and the excursion classification used in the Laplace-functional representation and the interpretation of the four terms of (1.8).","marker":"[14]"},{"why":"Offers an alternative excursion-theoretic representation of the Laplace functional that the paper compares its formula against, with the equivalence left open.","marker":"[63]"}],"fun_headline_variants":["A single Volterra equation dictates local-time law","Local-time profiles fit one stochastic Volterra equation","One equation, all local times: Volterra representation","Explicit Volterra equation for Lévy local times","Local times solved by a unified Volterra equation"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The representation stands only if the approximating compound Poisson processes converge to the target Lévy process in the correct distributional sense—the argument needs both convergence of their Laplace exponents and finite-dimensional convergence of their local times—and the separate strong-uniqueness claim additionally assumes the jump measure has finite mean.","fun_headline_variants_meta":{"raw":{"variants":["A single Volterra equation dictates local-time law","Local-time profiles fit one stochastic Volterra equation","One equation, all local times: Volterra representation","Explicit Volterra equation for Lévy local times","Local times solved by a unified Volterra equation"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000206,"raw_usage":{"total_tokens":1483,"prompt_tokens":1121,"completion_tokens":362,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":737,"completion_tokens_details":{"reasoning_tokens":289}},"tokens_in":737,"tokens_out":362,"duration_ms":3292,"temperature":1.0,"reasoning_tokens":289,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:15:19.925691+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Laplace functional (1.22) for a drifted Brownian motion ($\\nu = 0$, $c > 0$, $b \\ge 0$) and compare it with the classical Laplace transform of the squared Bessel / Feller diffusion that Remark 1.2 identifies as the solution of the equation; a mismatch for any measure $\\mu$ would disprove the representation. Alternatively, the paper states that the equivalence between its Laplace-functional formula and the excursion-theoretic representation in [63] remains open; evaluating both sides numerically for a spectrally positive process with Gaussian component and a nontrivial jump measure on a specific test measure $\\mu$ would settle whether the two formulas agree.","supporting_citations":[{"cited_title":"Lambert and F","cited_arxiv_id":null,"evidence_quote":"Supplies Lemma 3.4, the finite-dimensional convergence of local times of rescaled compound Poisson processes to those of the Lévy process, which is the bridge needed for the approximation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the stochastic Volterra representation for compound Poisson local times (Lemmas 3.1–3.2), the starting point of the limiting argument."},{"cited_title":"Behme, D","cited_arxiv_id":null,"evidence_quote":"Gives the formula $W''(0+) = (b - \\bar{\\bar\\nu}(0+))/c^2$ and the regularity of the scale function used in the strong-uniqueness and semimartingale representation (Theorem 1.7)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the smoothness of scale functions that the Volterra kernel relies on throughout the proofs."},{"cited_title":"Ikeda and S","cited_arxiv_id":null,"evidence_quote":"Background for Itô integrals and semimartingale calculus that define the stochastic Volterra integrals in (1.8)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Generalized Yamada–Watanabe theorem used in Lemma 5.1 to deduce strong existence and uniqueness from pathwise uniqueness."},{"cited_title":"Yamada and S","cited_arxiv_id":null,"evidence_quote":"The classical pathwise-uniqueness method that Theorem 1.7 adapts for the Hölder-continuous coefficients of the SVE."},{"cited_title":"Rivero and J","cited_arxiv_id":null,"evidence_quote":"Offers an alternative excursion-theoretic representation of the Laplace functional that the paper compares its formula against, with the equivalence left open."}],"review_version":1}