{"id":"843dc4b9-a234-4b74-ac45-1a0a3318c9ad","arxiv_id":"2411.12189","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Rescaled generalized Derrida-Retaux dynamics converge in Skorokhod space to a continuous-time process whose semigroup, generator, and martingale problem are characterized.","lead":"This paper proves that a rescaled discrete-time Derrida-Retaux recursive model converges, as the rescaling parameter grows, to a continuous-time limit process, and it characterizes that limit through its generator, martingale problem, and stochastic equation. The result turns a heuristic scaling limit used in the statistical physics of depinning transitions into a theorem.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Tightness proof in Section 5 silently requires m_2 < ∞; Theorem 5.5 as stated also admits q with m_2 = ∞, where the second-moment estimates (5.6) and (5.12) fail.","rationale":"I read the paper as establishing that the generalized DR recursion (1.6) has the generalized CDR equation and SDE as scaling limits, with Theorem 5.5 as the central process-level statement. The reader's verdict is CONDITIONAL, with the weakest assumption identified as m_1 < ∞ and the initial second-moment condition (5.2). I agree those are necessary for the given proof, but I find a more specific and under-appreciated gap: the tightness proof in Section 5 also requires m_2 < ∞, and this is not stated among the hypotheses of Theorem 5.5. The estimates (5.6), (5.12), and Lemma 5.2 all use m_2 as a finite constant; for an offspring law with m_1 < ∞ and m_2 = ∞, even a deterministic rescaled initial state satisfying (5.2) produces infinite second moment after one renewal step, so the L^2 Aldous argument collapses. This does not show the theorem is false; the convergence conclusion may still hold under weaker assumptions, and the one-dimensional convergence (Theorem 2.8) only uses m_1 < ∞. But as written, the proof of Theorem 5.5 is incomplete for q with m_2 = ∞. The minimal fix is to add m_2 < ∞ to the standing Section 5 hypotheses and to Theorem 5.5, or else to supply a different tightness proof. This is a hypothesis/proof mismatch rather than a refutation, so the CONDITIONAL verdict is unchanged; the reader's weakest_assumption is partially correct but misses the m_2 condition. I therefore recommend no change to the verdict, with the explicit request that the authors either state m_2 < ∞ or amend the tightness argument.","tokens_in":22803,"tokens_out":24383,"duration_ms":271440,"concrete_test":"Take q_k = c k^{-3} (so m_1 < ∞ but m_2 = ∞), fix a > 0, and set X^{(k)}_0 = k so that Y^{(k)}_0 = 1 and (5.2) holds. Compute E[(Y^{(k)}_1)^2] directly from the recursion (5.1): the renewal term contributes α Σ_j q_j (1 + j - k^{-1})_+^2, which is at least α Σ_j q_j (1+j)^2 = ∞ because Σ_j q_j j^2 = m_2 = ∞. This shows the second-moment inequality (5.6) has infinite right-hand side for t = 1/k and that the martingale variance bound (5.12) is infinite, so the proof of Lemma 5.2 and the Aldous tightness criterion in Lemma 5.4 do not go through. If the authors add m_2 < ∞ as an explicit hypothesis to Theorem 5.5, the proof is restored; if they intend to allow m_2 = ∞, they must replace the L^2 estimates with a truncation or first-moment argument. This single computation settles whether the theorem's stated hypotheses are sufficient for the supplied proof.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central limit theorem rests on the Aldous-type tightness proof in Lemmas 5.1–5.4. That proof uses the second moment m_2 of the offspring law: equation (5.6) gives E[(Y^{(k)}_{⌊kt⌋})^2] ≤ e^{a(2m_2+1)t}E[(Y^{(k)}_0)^2], and (5.12) bounds the martingale variance by a constant proportional to m_2, which is then used to prove E[sup (Y^{(k)}_{⌊ks⌋})^2] < ∞ in Lemma 5.2. The standing assumptions stated before (5.1) are only m_1 < ∞ (from the earlier sections) and (5.2), the uniform second-moment bound on the rescaled initial states. Neither Theorem 5.5 nor the surrounding assumptions states m_2 < ∞. If q has m_1 < ∞ but m_2 = ∞, e.g. q_k ∼ c k^{-3}, and we take X^{(k)}_0 = k (so Y^{(k)}_0 = 1 and (5.2) holds), then a renewal step with positive probability adds a jump of size jk with probability q_j, so the rescaled variable Y^{(k)}_1 satisfies E[(Y^{(k)}_1)^2] ≥ α Σ_j q_j (1+j)^2 = ∞. Thus the right-hand side of (5.6) is infinite for t ≥ 1/k and the quadratic variation estimate (5.12) is infinite, so the proof does not establish the finiteness needed in Lemma 5.2 and hence does not cover the stated hypotheses. This is a missing hypothesis in the theorem as written, not merely an unpolished estimate: the claimed process-level convergence is proved only under the additional, unstated condition m_2 < ∞, or under a different tightness argument that does not rely on finite second moments of the jump distribution.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies continuous-time generalized Derrida--Retaux (DR) type models. It introduces a generalized CDR model through equations (1.7) and (2.9), proves existence, uniqueness, and Wasserstein stability under the finite-mean condition m_1<\\infty, and shows that rescaled discrete-time generalized DR dynamics converge in Wasserstein distance (Theorems 2.7 and 2.8). It then characterizes the transition semigroup, generator, and martingale problem of the associated Markov process (Theorems 3.2 and 4.7), and proves weak convergence of the rescaled processes in the Skorokhod space D([0,\\infty),\\mathbb{R}_+) (Theorem 5.5). The proofs are based on explicit integral-equation comparisons, Gronwall's inequality, Itô's formula, Lévy--Khintchine uniqueness, and Aldous-type tightness arguments.","tokens_in":23181,"tokens_out":16885,"duration_ms":163979,"significance":"If the results hold, they provide a rigorous scaling-limit justification for the continuous-time DR equation used in the physics literature and extend the exactly solvable CDR model of Hu--Mallein--Pain to general offspring distributions. The paper's strengths are its explicit quantitative bounds, especially (2.19), the self-contained construction of the transition semigroup, and the clean martingale-problem characterization. The main caveat is that Theorem 5.5 is proved only under an additional second-moment condition on the offspring law that is not stated in the theorem; this is repairable but narrows the theorem's scope as written.","major_comments":[{"comment":"The process-level limit theorem is proved only under an unstated second-moment condition on the offspring distribution. Lemma 5.1's second estimate (5.6), the martingale-variance bound (5.12), and Lemma 5.3 all require m_2<\\infty; without it the right-hand sides are infinite. The standing assumptions before (5.1) and the statement of Theorem 5.5 state only m_1<\\infty (from earlier sections) and (5.2). If q has m_1<\\infty but m_2=\\infty, for example q_j \\sim c j^{-3}, and X_0^{(k)}=k so that (5.2) holds, then E[(Y_1^{(k)})^2]=\\infty and the estimates (5.6) and (5.12) fail. Thus the tightness argument in Lemma 5.4 does not apply to the hypotheses as stated. This is a missing hypothesis, not a cosmetic issue; please add m_2<\\infty to the standing assumptions and Theorem 5.5, or replace the Aldous-type tightness proof by one that avoids second moments of the jump distribution.","section":"Section 5, Lemmas 5.1-5.4 and Theorem 5.5"}],"minor_comments":[{"comment":"In the line following Lemma 2.2, the term '\\gamma_t * \\gamma^q_s' should read '\\gamma_s * \\gamma^q_s'; the intended estimate is clear from the context.","section":"Proposition 2.5 proof"},{"comment":"In the equation after 'It follows that', the factor '\\mu^a_s(dz)' should be '\\mu^q_s(dz)', and 'remains trues' should be 'remains true'.","section":"Theorem 3.2 proof"},{"comment":"The displayed boundary term uses the factor e^{a(r-t)}, which is independent of the integration variable s and would not yield the stated \\partial_r formula; it should be the s-dependent atom mass from (4.12). The final backward equation is nevertheless correct, but this displayed step needs repair.","section":"Proposition 4.6 proof"},{"comment":"In the displayed decomposition of Y^{(k)}_{\\lfloor k(\\tau_k+\\delta_k)\\rfloor} - Y^{(k)}_{\\lfloor k\\tau_k\\rfloor}, the final martingale term should be M^{(k)}_{\\lfloor k\\tau_k\\rfloor}, not M^{(k)}_{\\lfloor k\\delta_k\\rfloor}; the subsequent estimate uses the \\tau_k version.","section":"Lemma 5.4 proof"},{"comment":"The theorem's hypotheses should explicitly include the standing second-moment assumption (5.2), since it is used in Lemma 5.4 and is not visible from the theorem statement alone.","section":"Theorem 5.5 statement"},{"comment":"The abstract and introduction state the scaling-limit result without mentioning the additional m_2<\\infty condition needed for Theorem 5.5; once the theorem is amended, these statements should be qualified accordingly.","section":"Abstract and introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a solid contribution and the central estimates are mostly clean. The main reason for major revision is the missing m_2<\\infty hypothesis in Section 5; after adding that condition, or providing a different tightness argument, the paper should be reconsidered. I would not reject: the gap is local and the remainder of the paper appears sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a genuinely useful paper. It makes rigorous the informal scaling from Derrida–Retaux and Hu–Mallein–Pain by proving Wasserstein convergence of the rescaled discrete dynamics to a generalized CDR equation and then going further: a martingale problem, an SDE representation, and a Skorokhod-space process-level limit. The main estimate (2.19) is clean, and the characterization theorems (3.2, 4.7) are argued carefully. This is the first place I know of where the process-level limit is proved for general offspring distribution q, not just the exponential case.\n\nThe soft spots are real but not fatal. The stress-test note is on target: the tightness proof in Section 5 uses the second moment m2 of the offspring law, and m2 = ∞ is not excluded by the standing assumption m1 < ∞. Lemma 5.1's second-moment bound, Lemma 5.2's E[sup Y^2] finiteness, and Lemma 5.3's quadratic variation estimate all blow up if m2 = ∞. So Theorem 5.5 as stated is proven only under the additional, unstated condition m2 < ∞, or under a different tightness argument that does not rely on finite second moments. That is a missing hypothesis, not a cosmetic gap. The authors should either add m2 < ∞ to Theorem 5.5 or fix the proof.\n\nAlso, the introduction says the SDE (1.10) has a pathwise unique solution, but I did not find a proof of that claim anywhere in the body. That is a minor assertion that should be either proved or downgraded to 'weak uniqueness', which is what the martingale problem gives. A few typos and notational slips are scattered around, but nothing that obscures the arguments.\n\nNet: if the moment issue is patched, this is a solid contribution to the recursive-models literature. The Wasserstein bound and the generator characterization will be cited. I would take it for the reading group, and I would send it to a serious referee. My recommendation is to engage with the paper, but condition acceptance of the final version on the Section 5 hypothesis being fixed.","headline":"Solid scaling-limit theorems for generalized Derrida–Retaux models, but the headline process-level theorem needs an extra finite second-moment condition on the offspring law.","tokens_in":23753,"tokens_out":2163,"would_cite":true,"duration_ms":23366,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H20","60J25","60J76"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that generalized Derrida–Retaux recursions, rescaled in time by k and space by 1/k, converge to a unique continuous-time Markov process characterized by an explicit generator, a martingale problem, and a Poisson-driven…","keywords":["Max-type recursive model","Derrida–Retaux model","transition semigroup","generator","martingale problem","weak convergence","Skorokhod space","scaling limit"],"falsifier":"Run the rescaled recursion (2.18) with an infinite-mean offspring law such as $q_k=c k^{-3/2}$: if the marginals $\\gamma^{(k)}_{\\lfloor kt\\rfloor}$ still converge for every $t$ to a solution of (2.9), the finite-mean premise is stronger than needed, while a failure to converge (or several limit points) would show the premise is load-bearing. As a second check, in the solvable case $a=1$, $q_1=1$ with an exponential-type initial law, the simulated marginals should match the closed-form CDR flow with error of order $1/k$; a persistent mismatch would point to a concrete error in Theorem 2.7 or Theorem 5.5.","tokens_in":22552,"feed_emoji":"🎲","tokens_out":22141,"duration_ms":193860,"temperature":0.7,"pith_summary":"The paper establishes that the generalized Derrida–Retaux recursion — a discrete scheme in which a value is replaced by the positive part of itself plus an offspring-distributed jump minus 1 — has a genuine continuous-time limit, and it characterizes that limit completely. With time rescaled by k, space by 1/k, and the renewal rate set to a/k, the marginals converge in Wasserstein distance to the unique solution of the differential equation (2.9) at the explicit rate $W(\\gamma^{(k)}_{\\lfloor kt\\rfloor},\\mu_t)\\le e^{a(m_1+2)t}[\\frac{4}{k}(1+at)+W(\\gamma^{(k)}_0,\\mu_0)]$, and the whole rescaled paths converge weakly in Skorokhod space to a Markov process. That process is described in four equivalent ways: a transition semigroup with generator $A_t$ from (1.8), a martingale problem, an SDE driven by a Poisson random measure (1.10), and the closed entrance law $\\mu_t$ for the semigroup. This matters because the continuous-time model is the regime in which the Derrida–Retaux phase transition — pinning, free-energy asymptotics, and the predicted $4/n^2$ sustainability probability — is exactly solvable, and the theorem supplies the rigorous bridge from the discrete recursions to that regime.","feed_headline":"Rescaled Derrida–Retaux models converge to a Markov limit","feed_subtitle":"The limit is unique: explicit generator, martingale problem, and Poisson-driven stochastic equation.","key_machinery":"The engine of the paper is the pair formed by the generator $A_t$ and the Wasserstein distance built on the truncated metric $\\rho(x,y)=1\\wedge|x-y|$. The generator, $A_t f(x)=a\\int_{\\mathbb R_+}[f(x+z)-f(x)]\\mu_t^q(dz)-f'(x)\\mathbf 1_{\\{x>0\\}}$, has two parts: a pure-jump term that at rate $a$ adds a random amount distributed as $\\mu_t^q$, the offspring mixture $\\sum_k q_k\\mu_t^{*k}$ of the current marginal, and a drift of $-1$ that operates only while the process is positive. The stochastic equation (1.10), driven by a Poisson random measure with intensity $a\\,ds\\,du$ and with $G_s^{-1}$ the right-continuous inverse of $\\mu_s^q$, is the pathwise realization of the same generator, and Theorem 3.2 shows the martingale problem and the SDE describe the same process. Quantitatively, the dual representation (2.2) and the convolution inequality of Lemma 2.2 convert the recursive equation into an integral inequality; Gronwall's inequality then yields both the contraction $W(\\mu_t,\\gamma_t)\\le e^{am_1 t}W(\\mu_0,\\gamma_0)$ between any two solutions and the explicit scaling error (2.19). Existence is produced by a successive-approximation iteration over sub-probabilities (2.14), and path tightness in Lemma 5.4 uses the stopping-time tightness criterion together with the moment bounds of Lemmas 5.1–5.3.","core_discovery":"On the paper's own terms, the central result is that the generalized discrete Derrida–Retaux dynamics (1.6), after the rescaling $\\gamma^{(k)}_n(dx)=\\mu^{(k)}_n(k\\,dx)$ with renewal rate $\\alpha=a/k$, converge as $k\\to\\infty$ to the generalized CDR model: the unique family of probability measures solving $\\partial_t\\langle\\mu_t,f\\rangle=a\\langle\\mu_t*\\mu_t^q-\\mu_t,f\\rangle-\\langle\\mu_t,f'\\mathbf 1_{(0,\\infty)}\\rangle$, equation (2.9). Theorem 2.7 makes this quantitative with the explicit Wasserstein bound above, and Theorem 5.5 lifts it to the process level: whenever the rescaled initial laws converge weakly to $\\mu_0$ and satisfy the second-moment condition (5.2), the rescaled chains $(Y^{(k)}_{\\lfloor kt\\rfloor}:t\\ge 0)$ converge in $D([0,\\infty),\\mathbb R_+)$ to the generalized CDR process $X_t$ with initial law $\\mu_0$, the pathwise unique solution of the stochastic equation (1.10). Along the way the paper proves the process is equivalently a Markov process with inhomogeneous transition semigroup $(P_{r,t})$ generated by $A_t$ in (1.8), a solution of the $(A_t)$-martingale problem, and a weak solution of (1.10), with $\\mu_t$ the closed entrance law of the semigroup.","pith_inferences":["Extension the paper does not state: the $1/k$ rate in (2.19) should be the leading-order average error, and a second-order expansion of the rescaled chains around the CDR path would presumably yield a $k^{-1/2}$ Gaussian fluctuation; proving that would upgrade convergence to a distributional approximation usable for error bars in simulation.","Because the generator (1.8) depends on the offspring law only through the mixture flow $\\mu_t^q$, two laws that generate the same mixture flow would produce identical CDR processes — a reducibility the paper does not exploit and that could simplify simulation by replacing a complicated $q$ with a lighter one.","The second-moment premise (5.2) is used only for tightness, so the process-level theorem may well survive under a first-moment condition; testing initial laws with $E[(X_0^{(k)})^2]\\sim k^{2+\\varepsilon}$, so that (5.2) fails but the rescaled states still converge weakly, would show whether the premise is sharp.","A renormalization reading: iterating the (time $k$, space $1/k$) rescaling composes the discrete dynamics, and the CDR semigroup is the continuous-time flow of that composition; the critical pinning regime would then correspond to a nontrivial fixed point, and the solvable continuous-time setting is a plausible arena for extending the free-energy asymptotics (1.2) to general offspring laws."],"forward_implications":["The rescaled discrete chain becomes a computable approximation of the continuous-time model: once the rescaled starting law is close to $\\mu_0$, the bound (2.19) keeps the Wasserstein error after $k$ time steps at $O(1/k)$ times a factor exponential in $t$, and Theorem 5.5 lifts the approximation from marginals to whole paths in the Skorokhod space.","For every offspring law with finite mean, the limiting process is one well-characterized object: the equivalence of the transition semigroup, the martingale problem, and the SDE (Theorems 3.2 and 4.7) means a result proved in any one representation automatically applies to the others.","The path-space limit transfers convergence to path functionals, so hitting probabilities, excursion statistics, and the quantities entering the pinning–unpinning phase transition of the discrete models converge to those of the continuous-time process.","The moment recursions (5.6) give explicit growth rates — $e^{am_1t}$ for the mean and $e^{a(2m_2+1)t}$ for the mean square — so the qualitative growth of the discrete model is governed by the first two moments of the offspring law exactly as for the limit process."],"supporting_citations":[{"why":"Introduces the discrete max-type recursion (1.1) and the informal continuous-time PDE that this paper derives rigorously as the scaling limit; also the source of the free-energy conjecture (1.2).","marker":"[6]"},{"why":"Introduces the continuous-time DR model (1.4) and proves existence and uniqueness of its weak solutions; the exactly solvable case whose generalization this paper characterizes.","marker":"[9]"},{"why":"Introduces the generalized discrete recursion (1.6) with renewal rate $\\alpha$ and offspring law $q$; the discrete dynamics whose rescaling is the paper's object of study.","marker":"[10]"},{"why":"Supplies the Wasserstein-distance metric facts and the dual representation used in Lemma 2.1 for the paper's quantitative estimates.","marker":"[2]"},{"why":"Supplies the stopping-time tightness criterion used in Lemma 5.4 to prove tightness of the rescaled processes in Skorokhod space.","marker":"[1]"},{"why":"Supplies the Skorokhod-space convergence and representation results (Proposition 5.2) invoked in the proof of Theorem 5.5.","marker":"[8]"},{"why":"Supplies the martingale representation theorem used in Theorem 3.2 to pass from the martingale problem to the Poisson-driven SDE (1.10).","marker":"[11]"},{"why":"Supplies the orthogonal decomposition of martingales used in the proof of Theorem 3.2.","marker":"[7]"}],"fun_headline_variants":["Unique Markov limit for generalized Derrida–Retaux models","Rescaled Derrida–Retaux chains converge to a unique Markov process","Derrida–Retaux scaling limit: unique generator and martingale problem","Explicit generator characterizes the Derrida–Retaux scaling limit","Rescaled Derrida–Retaux models have a unique Markov scaling limit"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The argument stands on the offspring distribution having finite mean, $\\sum_{k\\ge1}kq_k<\\infty$, which buys existence, uniqueness, contraction, and both limit theorems, and the path-level theorem additionally assumes the rescaled initial laws have uniformly bounded second moments.","fun_headline_variants_meta":{"raw":{"variants":["Unique Markov limit for generalized Derrida–Retaux models","Rescaled Derrida–Retaux chains converge to a unique Markov process","Derrida–Retaux scaling limit: unique generator and martingale problem","Explicit generator characterizes the Derrida–Retaux scaling limit","Rescaled Derrida–Retaux models have a unique Markov scaling limit"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000779,"raw_usage":{"total_tokens":3428,"prompt_tokens":915,"completion_tokens":2513,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":531,"completion_tokens_details":{"reasoning_tokens":2420}},"tokens_in":531,"tokens_out":2513,"duration_ms":17519,"temperature":1.0,"reasoning_tokens":2420,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:49:47.537856+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the rescaled recursion (2.18) with an infinite-mean offspring law such as $q_k=c k^{-3/2}$: if the marginals $\\gamma^{(k)}_{\\lfloor kt\\rfloor}$ still converge for every $t$ to a solution of (2.9), the finite-mean premise is stronger than needed, while a failure to converge (or several limit points) would show the premise is load-bearing. As a second check, in the solvable case $a=1$, $q_1=1$ with an exponential-type initial law, the simulated marginals should match the closed-form CDR flow with error of order $1/k$; a persistent mismatch would point to a concrete error in Theorem 2.7 or Theorem 5.5.","supporting_citations":[{"cited_title":"and Retaux, M.: The depinning transition in pr esence of disorder: a toy model","cited_arxiv_id":null,"evidence_quote":"Introduces the discrete max-type recursion (1.1) and the informal continuous-time PDE that this paper derives rigorously as the scaling limit; also the source of the free-energy conjecture (1.2)."},{"cited_title":"and Pain, M.: An exactly solvable conti nuous-time Derrida–Retaux model, Commun","cited_arxiv_id":null,"evidence_quote":"Introduces the continuous-time DR model (1.4) and proves existence and uniqueness of its weak solutions; the exactly solvable case whose generalization this paper characterizes."},{"cited_title":"and Shi, Z.: The free energy in the Derrida–Retaux recursive model","cited_arxiv_id":null,"evidence_quote":"Introduces the generalized discrete recursion (1.6) with renewal rate $\\alpha$ and offspring law $q$; the discrete dynamics whose rescaling is the paper's object of study."},{"cited_title":"World scientific, Sin- gapore, 2004","cited_arxiv_id":null,"evidence_quote":"Supplies the Wasserstein-distance metric facts and the dual representation used in Lemma 2.1 for the paper's quantitative estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the stopping-time tightness criterion used in Lemma 5.4 to prove tightness of the rescaled processes in Skorokhod space."},{"cited_title":"and Kurtz, T.G.: Markov Processes: Characterization and Convergence","cited_arxiv_id":null,"evidence_quote":"Supplies the Skorokhod-space convergence and representation results (Proposition 5.2) invoked in the proof of Theorem 5.5."},{"cited_title":"and Watanabe, S.: Stochastic Differential Equations and Diffusion Processes","cited_arxiv_id":null,"evidence_quote":"Supplies the martingale representation theorem used in Theorem 3.2 to pass from the martingale problem to the Poisson-driven SDE (1.10)."},{"cited_title":"and Meyer, P.A.: Probabilities and Potential","cited_arxiv_id":null,"evidence_quote":"Supplies the orthogonal decomposition of martingales used in the proof of Theorem 3.2."}],"review_version":1}