{"id":"bedf2801-876e-442a-94c1-61c688d02725","arxiv_id":"1908.01255","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A unified L^q(L^p) theory establishes weak differentiability of SDE flows, a Bismut-Elworthy-Li derivative formula, and endpoint weak well-posedness for singular locally integrable coefficients.","lead":"This paper proves that solutions of stochastic differential equations with rough, locally integrable coefficients are weakly differentiable in the starting point, and it gives an explicit derivative formula. It also proves uniqueness of weak solutions in a critical borderline case for the drift.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.4's uniqueness proof assumes every martingale solution satisfies the Krylov estimate (5.4), although Definition 1.3 does not impose it; as written, uniqueness is shown only in that subclass.","rationale":"The reader's weakest assumption focused on the L^q(L^p) maximal regularity machinery in Theorem 3.1, which is indeed the engine of both main theorems. I did not find a concrete flaw in that machinery strong enough to overturn the reader's assessment. However, the proof of Theorem 1.4 contains a more specific and checkable logical gap: uniqueness is proved only in the class of martingale solutions satisfying the Krylov estimate (5.4), whereas Definition 1.3 and the theorem's statement concern all martingale solutions. Because this gap directly affects one of the two headline results, the appropriate verdict is CONDITIONAL rather than unconditional acceptance: the paper should either supply the missing lemma that every martingale solution satisfies (5.4), or weaken the uniqueness statement to the class in which it is proved. This is consistent with the reader's moderate confidence and does not require rejecting the paper's likely-correct mathematical programme.","tokens_in":21730,"tokens_out":44647,"duration_ms":450976,"concrete_test":"Prove or disprove the missing lemma: for every P ∈ M^{σ,b}_{s,x}, the Krylov estimate (5.4) holds for all p,q with d/p + 2/q < 2. Concretely, for nonnegative f ∈ C_c^∞ supported in [t0,t1], solve the backward parabolic PDE with source f using Theorem 3.1, approximate the solution by smooth functions, and check whether passing to the limit under an arbitrary martingale solution can be justified without first assuming (5.4). If the limit cannot be justified, Theorem 1.4 should be restated with uniqueness restricted to the class of martingale solutions satisfying (5.4), and the current assertion of uniqueness for all martingale solutions is unsupported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The endpoint well-posedness claim in Theorem 1.4 is central to the paper, but the uniqueness proof contains an unstated restriction. In the proof of Theorem 1.4, after obtaining existence from the tightness argument, the authors write: 'Let P^(i)_x ∈ M^{σ,b}_{0,x}, i=1,2 be any two martingale solutions of SDE (1.2) so that for any T>0, there is a constant C>0 such that ... (5.4).' Definition 1.3 of a martingale solution only requires the C^2_b martingale property; it does not include the Krylov-type bound (5.4). The subsequent argument solves a backward PDE with coefficients (σ,b) and applies Itô's formula under P^(i), then passes to the limit in the commutator term (L^{σ,b}u)*ρ_n − L^{σ,b}(u*ρ_n). That limiting step uses precisely the estimate (5.4). Thus the proof establishes uniqueness only among martingale solutions satisfying (5.4), while the theorem asserts uniqueness in the full class M^{σ,b}_{s,x}. Since the existence construction supplies one solution satisfying (1.6), the written proof does not rule out additional martingale solutions that might fail (5.4). Unless an additional lemma shows that every element of M^{σ,b}_{s,x} automatically satisfies (5.4), the stated uniqueness—and hence Theorem 1.4 as phrased—is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops an L^q(L^p) maximal regularity theory for second-order parabolic equations with uniformly continuous, uniformly elliptic coefficients, including the previously open case p > q via a duality argument in negative Sobolev spaces (Theorem 3.1 and Theorem 3.3). This estimate is then applied to SDEs (1.2) with diffusion coefficient sigma that is bounded, uniformly continuous, and nondegenerate, with grad sigma in ~L^{p1}_{q1} and drift b in ~L^{p2}_{q2}, where d/p_i + 2/q_i < 1. For this subcritical regime, the paper proves existence and uniqueness of strong solutions, weak differentiability of the flow, and the Bismut-Elworthy-Li derivative formula (Theorem 1.1). In the endpoint case b in ~L^{d;uni}_infty, the paper claims weak well-posedness of martingale solutions (Theorem 1.4). The main technical novelty is the extension of Krylov-type maximal regularity to localized spaces and to all p,q in (1,infty), with the p>q case handled by duality.","tokens_in":22014,"tokens_out":7666,"duration_ms":75341,"significance":"If correct, the localized L^q(L^p) maximal regularity estimate of Theorem 3.1/3.3 is a substantial improvement over earlier results that were restricted to p <= q, and it is of independent interest for PDEs with rough coefficients. The subcritical flow theorem (Theorem 1.1) unifies and extends several earlier results on Sobolev differentiable stochastic flows and derivative formulas, and the localized spaces allow global-in-space conclusions from local integrability assumptions. The endpoint weak well-posedness result (Theorem 1.4) would be new for multiplicative noise with critical drift. The paper is generally well structured, and the subcritical part is convincingly reduced to the maximal regularity estimate and standard stochastic tools. However, the endpoint uniqueness proof has a restrictive assumption that is not present in the definition of martingale solution, and the core duality argument in Theorem 3.3 contains a potentially circular use of an estimate whose proof is only announced as 'similar'. These issues affect load-bearing claims and require attention before the paper can be accepted as stated.","major_comments":[{"comment":"The uniqueness proof in Theorem 1.4 assumes that any two martingale solutions P^(i)_x in M^{sigma,b}_{0,x} satisfy the Krylov bound (5.4). However, Definition 1.3 of a martingale solution only requires the C^2_b martingale property; it does not include (5.4). No lemma is provided showing that every element of M^{sigma,b}_{s,x} automatically satisfies (5.4). The passage to the limit in the commutator term (L^{sigma,b}u)*rho_n - L^{sigma,b}(u*rho_n) uses (5.4) essentially. Consequently, the proof establishes uniqueness only within the subclass of martingale solutions satisfying (5.4), not in the full class M^{sigma,b}_{s,x} as the theorem states. Since the existence construction supplies at least one solution satisfying (1.6), the written argument does not rule out additional martingale solutions that fail (5.4). The theorem should be weakened to assert uniqueness in the class of solutions satisfying (5.4), or an additional argument must be supplied showing that (5.4) holds for every martingale solution.","section":"5, proof of Theorem 1.4, Eq. (5.4)"},{"comment":"In Step (ii) the authors state that they will prove (3.10) and (3.12) for p <= q and that (3.9) and (3.11) are 'similar'. However, in Step (v), inequality (3.21) explicitly invokes (3.9) to bound the term involving integral of ||nabla^2 w||^q_{-2,p}. Since (3.9) for p <= q has not been proved before this point and its proof is not displayed anywhere, the proof of (3.10) is circular as written. The authors should either provide the proof of (3.9) for p <= q before using it, or restructure the argument so that the bound on nabla^2 w in Step (v) does not depend on (3.9).","section":"3.1, proof of Theorem 3.3, Step (v), Eq. (3.21)"}],"minor_comments":[{"comment":"The notation ~L^d_infty is used in Lemma 5.2 and in the proof of Theorem 1.4, but the paper only defines ~L^{p;uni}_infty in (2.3); please clarify whether ~L^d_infty means the same space or the localized space with q = infinity.","section":"2, Eq. (2.3) and Section 5"},{"comment":"Lemma 5.1 is stated for f in C^infty_c, but Lemma 5.2 applies it to |b_n|, which is not compactly supported; please add a truncation/approximation argument or extend the statement of Lemma 5.1.","section":"5, Lemma 5.1 and Lemma 5.2"},{"comment":"The proofs rely heavily on results from the same group's preprints or to-appear papers, especially [18, Theorem 1.1] and [19, Theorem 5.7]; the authors should ensure these references are publicly available with precise theorem numbers, and update reference [19] if it has appeared.","section":"References"},{"comment":"The extracted text contains many LaTeX artifacts such as '/greaterorequalslant' and broken symbols; the final version should be carefully proofread for these typographical issues.","section":"General presentation"}],"recommendation":"major_revision","confidential_remarks":"The paper has substantial value in Theorem 3.3 and Theorem 1.1, but the endpoint uniqueness claim of Theorem 1.4 is not established as written. If the authors cannot prove that all martingale solutions satisfy (5.4), the theorem should be weakened; the paper may still be publishable with the subcritical results. The heavy reliance on companion papers from the same group is acceptable but warrants extra scrutiny by the editor, and the potentially circular step in Theorem 3.3 should be resolved before acceptance."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: the subcritical part of this paper is solid and genuinely useful; the endpoint weak well-posedness theorem has a real gap. The uniqueness proof for Theorem 1.4 only works for martingale solutions that already satisfy the Krylov estimate (5.4), while Definition 1.3 does not require it. The limiting step that kills the commutator uses (5.4) explicitly. So as written, Theorem 1.4 does not rule out martingale solutions outside that subclass. The existence argument produces one solution with (1.6), but nothing shows every element of M^{σ,b} has that bound. This is not a nitpick: the endpoint case is one of the two advertised results. The theorem needs either a new argument that every martingale solution satisfies (5.4), or a reformulation that states uniqueness within the class satisfying the Krylov bound.\n\nWhat is actually new and good: the localized ~L^p_q setup is a real extension beyond global L^p_q spaces, and it lets the authors get global bounds (1.4) and (1.5) with locally singular coefficients. Theorem 3.3, the duality argument for maximal regularity when p>q, looks like a genuine technical contribution and is interesting in its own right. The proof of Theorem 1.1 is clean: Zvonkin's transformation, Krylov's estimate from the PDE regularity, and the stochastic Gronwall inequality from prior work. The paper is well-organized and the machinery is used honestly. The dependence on earlier papers by the same group ([18,19,22,23]) is not a problem; those are published or available arXiv preprints with their own proofs.\n\nSoft spots beyond Theorem 1.4: the proof of Theorem 3.3 has several interpolation and freezing steps where constants are claimed but not fully pinned down. I did not find an obvious error, and the reader's check was consistent, but this is the kind of thing that wants a careful referee. Minor point: the paper says the proof of weak differentiability is 'much simpler than [10]' while relying on some heavy analytic machinery; that is a matter of taste.\n\nBottom line: the paper deserves a serious referee, but not acceptance in its current form. The subcritical theorems are presumably correct and valuable. The endpoint theorem is the headline claim for critical drifts, and it is not established as stated. I would send it back for a major revision: either prove the missing automatic Krylov estimate or state the uniqueness result for the smaller class. The technical content is strong enough that the fix is likely straightforward.","headline":"Subcritical part is solid and extends known results; the endpoint weak well-posedness theorem has an unproven uniqueness claim because the proof assumes a Krylov estimate not imposed in Definition 1.3.","tokens_in":22596,"tokens_out":2953,"would_cite":true,"duration_ms":28577,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The authors prove that strong solutions of multiplicative SDEs with locally integrable drift and diffusion gradient are weakly differentiable in the initial point, with a Bismut-Elworthy-Li derivative formula, and that the martingale…","keywords":["stochastic differential equations","weak differentiability","Bismut-Elworthy-Li formula","L^q(L^p) maximal regularity","Zvonkin transformation","Krylov's estimate","martingale problem","singular drift"],"falsifier":"Exhibit a uniformly continuous, uniformly elliptic $a=\\sigma\\sigma^*/2$ and data $f\\in L^q([0,T];L^p)$ with $p>q$ for which the solution of $\\partial_t u=a^{ij}\\partial_i\\partial_j u+f$ has $\\|\\nabla^2 u\\|_{L^p_q(T)}=\\infty$, or for which the constant in the duality bound (3.19) diverges as the localization radius shrinks; either counterexample would break the Krylov estimate and the Zvonkin step on which Theorems 1.1 and 1.4 are built.","tokens_in":21526,"feed_emoji":"🎲","tokens_out":12521,"duration_ms":112507,"temperature":0.7,"pith_summary":"This paper shows that stochastic differential equations with multiplicative noise, whose drift and diffusion gradient are only locally integrable, still have solutions that depend weakly differentiably on the starting point. The main theorem proves that the unique strong solution $X_t(x)$ is weakly differentiable in $x$, satisfies $\\sup_x \\mathbb{E} \\sup_{t\\le T} |\\nabla X_t(x)|^p < \\infty$ for every $p\\ge 1$, and obeys Bismut-Elworthy-Li's derivative formula expressing $\\nabla \\mathbb{E}\\varphi(X_t(x))$ as a normalized stochastic integral. The assumptions allow locally integrable coefficients in the subcritical range $d/p_i + 2/q_i < 1$, which includes bounded drifts that earlier Zvonkin-transformation arguments could not handle. In the critical endpoint case where the drift lies in a localized $L^d$-type space, the paper proves uniqueness of martingale solutions. The new ingredient is an $L^q(L^p)$-maximal regularity estimate for parabolic equations with variable coefficients that works for all $p,q\\in(1,\\infty)$, not only $p\\le q$.","feed_headline":"SDE solutions with singular coefficients are weakly differentiable","feed_subtitle":"A parabolic PDE estimate extends smooth-flow results to locally integrable coefficients and bounded drifts.","key_machinery":"The load-bearing mechanism is the $\\widetilde{\\mathbb{L}}^p_q$-maximal regularity estimate of Theorem 3.1 for the parabolic equation $\\partial_t u = a^{ij}\\partial_i\\partial_j u + b\\cdot\\nabla u - \\lambda u + f$: for all $p,q\\in(1,\\infty)$ and $\\alpha\\in[0,2-2/q)$, $$\\$lambda^{{1-\\alpha/2-1/q}}$|||u|||_{\\widetilde{H}^{\\$\\alpha$,p}_\\infty(T)} + |||\\partial_t u|||_{\\widetilde{L}^p_q(T)} + |||u|||_{\\widetilde{H}^{2,p}_q(T)} \\le C|||f|||_{\\widetilde{L}^p_q(T)}.$$ The novelty is that the estimate holds when $p>q$, not only $p\\le q$; this is achieved by a duality method that studies the adjoint heat equation in negative Sobolev spaces $H^{-2,p}$, with the freezing-coefficient product estimate (3.13) and the interpolation bound (3.19). From this estimate the paper derives Krylov's inequality along the solution path, a generalized Itô formula for functions in $\\widetilde{H}^{2,p'}_{q'}$, and the Zvonkin transformation that converts the singular-coefficient SDE into one with bounded Lipschitz coefficients.","core_discovery":"The central claim is that under (Hσ), $\\nabla\\sigma\\in\\widetilde{\\mathbb{L}}^{p_1}_{q_1}$ and $b\\in\\widetilde{\\mathbb{L}}^{p_2}_{q_2}$ with $d/p_i+2/q_i<1$, the unique strong solution $X_t(x)$ of (1.2) is weakly differentiable in the initial point, with $\\sup_x \\mathbb{E} \\sup_{t\\in[0,T]} |\\nabla X_t(x)|^p$ finite for every $p\\ge 1$. For every $\\varphi\\in C^1_b$ and Lebesgue-almost every $x$, $$\\nabla \\mathbb{E}\\varphi(X_t(x)) = \\frac{1}{t}\\mathbb{E}\\left(\\varphi(X_t(x))\\int_0^t \\$sigma^{{-1}}$(s,X_s(x))\\nabla X_s(x)\\,dW_s\\right).$$ The proof runs through Zvonkin's transformation $\\Phi(t,x)=x+u(t,x)$, where $u$ solves a backward parabolic equation; the transformed SDE has bounded, continuous coefficients, and the estimate on the difference of two solutions implies weak differentiability of the original flow. In the endpoint regime $b\\in\\widetilde{\\mathbb{L}}^{d;\\mathrm{uni}}_\\infty$, the martingale problem is well posed, giving weak well-posedness beyond the reach of the subcritical strong-solution theorem.","pith_inferences":["The paper does not develop it, but the same maximal regularity estimate should yield strong well-posedness and flow differentiability for drifts in critical or Lorentz-type spaces whenever the localized norm is finite, since the Zvonkin step only needs boundedness of the transformed coefficients.","A consequence left implicit is that the derivative formula gives a pathwise expression for gradient estimates of semigroups generated by singular-coefficient diffusions; combined with the uniform gradient bound it should produce Bismut-type gradient and coupling estimates.","One testable extension would be to relax the condition on $\\nabla\\sigma$ from a subcritical localized space to a critical one; the proof as written needs Theorem 3.1 with the drift term to control $\\sigma$, so this does not follow immediately.","The duality method for negative Sobolev spaces is transferable: it should prove $L^q(L^p)$ maximal regularity for non-divergence parabolic operators with lower-order terms beyond the heat-type equation considered here."],"forward_implications":["For each initial point $x$, the SDE has a unique strong solution whose dependence on $x$ is weakly differentiable, with $\\sup_x \\mathbb{E} \\sup_{t\\le T}|\\nabla X_t(x)|^p$ finite for every $p\\ge 1$.","The Bismut-Elworthy-Li formula holds for almost every starting point, expressing the gradient of $\\mathbb{E}\\varphi(X_t(x))$ as a normalized stochastic integral along the path; this is the basic tool for gradient estimates and for studying the associated semigroup.","Krylov's estimate holds along the solution path for functions in the localized spaces $\\widetilde{\\mathbb{L}}^p_q$, uniformly in the starting point, so path integrals of singular functions are controlled even with multiplicative noise.","In the critical endpoint case $b\\in\\widetilde{\\mathbb{L}}^{d;\\mathrm{uni}}_\\infty$, the martingale problem is well posed, giving weak well-posedness that earlier strong-solution results did not cover.","Because the coefficients are only required to lie in localized spaces, bounded drifts and locally integrable diffusion gradients are included, extending the regime in which stochastic flows are Sobolev differentiable."],"supporting_citations":[{"why":"supplies the $L^p_q$ estimates for the associated Kolmogorov equation and the strong well-posedness result for singular time-dependent drift that this paper extends to multiplicative noise with localized coefficients.","marker":"[9]"},{"why":"introduces Zvonkin's transformation for SDEs with Sobolev coefficients and the derivative formula in the globally integrable case; the present paper removes the global integrability restriction.","marker":"[22]"},{"why":"establishes the $L^q((0,T),L^p)$ estimate for the heat equation with $x$-independent coefficients, the prototype for the $p\\ne q$ maximal regularity used here.","marker":"[8]"},{"why":"proves $L^q(L^p)$ estimates for variable coefficients only for $p\\le q$; this is the restriction the present paper removes by duality.","marker":"[5]"},{"why":"shows weak differentiability and Malliavin differentiability for bounded drift; it is one of the results unified and extended here.","marker":"[10]"},{"why":"provides the Krylov estimate, generalized Itô formula, and stochastic Gronwall machinery used to transfer differentiability from the transformed SDE back to $X_t$.","marker":"[19]"},{"why":"establishes Sobolev differentiable flows for SDEs with local Sobolev and superlinear growth coefficients; it supplies the step from the two-solution estimate to the uniform gradient bound.","marker":"[18]"},{"why":"supplies the localization spaces, the freezing lemma, and the weak-well-posedness framework used in the critical endpoint case.","marker":"[23]"}],"fun_headline_variants":["SDE solutions weakly differentiable for singular coefficients","Bismut-Elworthy-Li formula for SDEs with rough coefficients","Weak differentiability of SDE flows with singular drifts","Endpoint well-posedness for SDEs with unbounded drift"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on the maximal regularity estimate (3.2) holding for every $p,q\\in(1,\\infty)$, including $p>q$, with constants independent of the localization centers and of $\\lambda$; if the duality bounds (3.13), (3.17), or (3.19) fail for the uniformly continuous coefficient $a=\\sigma\\sigma^*/2$, the differentiability and endpoint uniqueness results lose their key input.","fun_headline_variants_meta":{"raw":{"variants":["SDE solutions weakly differentiable for singular coefficients","Bismut-Elworthy-Li formula for SDEs with rough coefficients","Weak differentiability of SDE flows with singular drifts","Endpoint well-posedness for SDEs with unbounded drift"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001123,"raw_usage":{"total_tokens":4718,"prompt_tokens":1039,"completion_tokens":3679,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":655,"completion_tokens_details":{"reasoning_tokens":3608}},"tokens_in":655,"tokens_out":3679,"duration_ms":25493,"temperature":1.0,"reasoning_tokens":3608,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T15:18:58.316622+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Exhibit a uniformly continuous, uniformly elliptic $a=\\sigma\\sigma^*/2$ and data $f\\in L^q([0,T];L^p)$ with $p>q$ for which the solution of $\\partial_t u=a^{ij}\\partial_i\\partial_j u+f$ has $\\|\\nabla^2 u\\|_{L^p_q(T)}=\\infty$, or for which the constant in the duality bound (3.19) diverges as the localization radius shrinks; either counterexample would break the Krylov estimate and the Zvonkin step on which Theorems 1.1 and 1.4 are built.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the $L^p_q$ estimates for the associated Kolmogorov equation and the strong well-posedness result for singular time-dependent drift that this paper extends to multiplicative noise with localized coefficients."},{"cited_title":"Zhang, Stochastic differential equations with Sobol ev coefficients and applications","cited_arxiv_id":null,"evidence_quote":"introduces Zvonkin's transformation for SDEs with Sobolev coefficients and the derivative formula in the globally integrable case; the present paper removes the global integrability restriction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes the $L^q((0,T),L^p)$ estimate for the heat equation with $x$-independent coefficients, the prototype for the $p\\ne q$ maximal regularity used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"proves $L^q(L^p)$ estimates for variable coefficients only for $p\\le q$; this is the restriction the present paper removes by duality."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"shows weak differentiability and Malliavin differentiability for bounded drift; it is one of the results unified and extended here."},{"cited_title":"Reconstruction of Missing Big Sensor Data","cited_arxiv_id":"1705.01402","evidence_quote":"provides the Krylov estimate, generalized Itô formula, and stochastic Gronwall machinery used to transfer differentiability from the transformed SDE back to $X_t$."},{"cited_title":"Xie and X","cited_arxiv_id":null,"evidence_quote":"establishes Sobolev differentiable flows for SDEs with local Sobolev and superlinear growth coefficients; it supplies the step from the two-solution estimate to the uniform gradient bound."}],"review_version":1}