{"id":"bfb60c69-abd9-4437-923b-b93ea3ca7b2e","arxiv_id":"2412.10836","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Coupling methods yield sharp Lp regularity estimates for SDEs with path-dependent Lipschitz or scalar Holder diffusion coefficients, plus a counterexample showing Malliavin differentiability fails for general Holder diffusions.","lead":"This mathematics paper proves new estimates for how solutions of stochastic differential equations respond when the Brownian driving noise is partially copied (coupled). It shows that Holder continuous diffusion coefficients in one dimension still give sharp Besov regularity bounds, and that in general such solutions can fail to be Malliavin differentiable.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the main external dependency is the transference step, but the paper's Proposition 3.10 covers non-Lipschitz coefficients and the central estimates are internally consistent.","rationale":"The reader's weakest_assumption correctly identifies the transference identities (7.12)-(7.13) as the hinge connecting the pathwise estimates to the Lp Besov estimates. I examined whether this is a real weakness. The transference theory in [14] is designed for general Wiener-space couplings and stochastic integral representations, not specifically for Lipschitz SDEs; Proposition 3.10 in this paper is an adaptation of [14, Theorem 3.3] and does not impose Lipschitz regularity on σ. For the Holder setting of Section 7, the conditions of Proposition 3.10 are satisfied: the solution is a continuous adapted semimartingale, the coefficient processes are predictable and continuous in the spatial variable, and the finite-moment condition holds. Therefore the distributional identities (7.12)-(7.13) are justified, provided [14, Theorem 3.3] is as general as the authors state. This is a dependence on prior work, not an internal flaw. I also checked the counterexample in Section 6 for internal consistency: the Schauder-function construction has the stated Holder and Lipschitz constants, the Malliavin chain-rule step is valid for the piecewise-linear coefficient, and the lower bound in (6.3) contradicts the assumed finiteness of ||DX_1||. The proof of Theorem 7.3 has a minor notational collision involving pθ in Lemma 7.7, but the intended argument (interpolate between the p=1 estimate and the p=3-2θ estimate, then apply Lenglart) is clear and does not affect the result. Overall the central claim survives scrutiny, and the reader's ACCEPT verdict need not be adjusted.","tokens_in":46515,"tokens_out":41272,"duration_ms":375584,"concrete_test":"Reproduce the proof of [14, Theorem 3.3] for the cut-off coupling φ=1_(a,c] and a deterministic bounded θ-Holder σ, approximating the transferred Ito integral by dyadic Riemann sums and checking that the L2 limit is preserved by the transference operator C_T; if the L2 convergence of the approximating sums does not require Lipschitz continuity of σ, the identities (7.12)-(7.13) are sound.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is the use of the transference identities (7.12)-(7.13) to convert the pathwise estimates of Lemma 7.7 into the Lp estimates of Theorem 7.3. This dependence is real, but it is not an unresolved gap inside the paper: Proposition 3.8 and Proposition 3.10 are stated for processes satisfying SDE representations without a Lipschitz restriction on the diffusion coefficient, and they are justified by the authors' Memoirs [14, Theorem 3.3]. Under Assumption 7.1, σ is deterministic and θ-Holder in x, so it transfers to itself, while b is P_{t,T}-measurable and Lipschitz in x, so the transference (P3) supplies b^φ. Proposition 3.8 then yields the C^P relations used for (7.12)-(7.13). No internal contradiction or missing assumption was found in Theorem 6.1 or Section 7; the only nontrivial verification is that [14, Theorem 3.3] indeed applies to coefficients with merely Holder spatial regularity and to the cut-off coupling function 1_(a,c].","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper develops a coupling-and-transference method on the Wiener space and applies it to the regularity of stochastic differential equations. In the Lipschitz setting with random, path-dependent coefficients and a BMO drift, Theorem 5.3 gives Lp estimates for the difference between a solution and its coupled copy; Corollaries 5.11 through 5.13 then yield Malliavin differentiability and Besov regularity obtained by real interpolation. Section 6 constructs a scalar SDE with bounded, theta-Holder diffusion coefficient whose terminal value is not in D_{1,2}, for every starting point. Section 7 treats one-dimensional SDEs with theta in [1/2,1)-Holder diffusion and proves Lp estimates for cut-off couplings (Theorem 7.3, Corollary 7.11); Proposition 7.12 supplies a matching lower bound showing the exponent 1/(2p) is optimal. Section 9 applies these estimates to the Lp-variation of BSDEs, including a CIR-process example.","tokens_in":1335,"tokens_out":1482,"duration_ms":287350,"significance":"The paper's central two-fold message is valuable and nontrivial: general Holder diffusion coefficients can destroy Malliavin differentiability, while a sharp order of Besov-type regularity survives. The construction in Section 6 is elegant and the matching lower bound in Proposition 7.12 makes the upper estimates credible. The manuscript is also unusually explicit about its lineage: the transference machinery, Besov-space characterizations, and several BSDE estimates are imported from the authors' Memoirs [14]. The stress-test concern about the identities (7.12) and (7.13) does not land: Proposition 3.8, Remark 3.9(1), Proposition 3.10, and Lemma 3.7 are exactly the tools needed, and under Assumption 7.1 the diffusion coefficient is deterministic while the drift is Lipschitz in the state variable, so the distributional identities are justified by the stated results. The central estimates appear internally consistent and are supported by detailed proofs.","major_comments":[],"minor_comments":[{"comment":"In the proof of Lemma 7.5, the statement that the first three terms converge in L2 is not justified under the stated assumptions: the lemma only assumes E times the integral of |b_s| ds is finite and A is in L2, and the drift term involving Phi'_n(D_u) b_u need not converge in L2 without square-integrability of the drift. The conclusion of the lemma is still correct, and in the later applications the stronger integrability with the integral of |b_s| ds in L^{p or 2} is available, so this is a local proof repair rather than a substantive gap. Please either add the needed integrability assumption to Lemma 7.5 or replace the L2-convergence statement by convergence in probability with a dominated-convergence argument for the drift term.","section":"Lemma 7.5"},{"comment":"The case split in (9.4) appears to omit the range p in [2, 2/alpha) when alpha is less than 1: for example with alpha = 0.4 and p = 3, none of the three listed intervals applies. The intended division should presumably be p < 1/alpha, 1/alpha <= p < 2/alpha, and p >= 2/alpha, because Corollary 7.11 is applied with exponent alpha p. In the omitted range alpha p lies in [2 alpha, 2), and Corollary 7.11(3) gives an estimate with exponent alpha/4, which is no worse than the stated exponent 1/(2p) - epsilon; please clarify the case boundaries and the proof for that range.","section":"Corollary 9.4, equation (9.4)"},{"comment":"There is a recurring spelling 'Cieselski' in the introduction; it should be 'Ciesielski'. Also, in the displayed constants of Section 7, expressions such as 'e^{L_b T} p^p sqrt(1-p)' are typeset ambiguously; the intended constant appears to be (p^p/(1-p))^{1/p} or similar, and should be written unambiguously.","section":"Section 1 and Section 6"},{"comment":"The displayed definition of the quantity with parameters B eta, beta, gamma has a malformed brace structure and an unclear case for the second alternative. Please rewrite the definition with separate cases so that the admissible infimum over kappa is displayed consistently with the q = infinity case.","section":"Definition 5.9"},{"comment":"In the proof of Theorem 7.3, the paper states that Proposition 3.8 and Remark 3.9(1) give the transference relations (7.12) and (7.13). This is correct, but it would improve readability to note explicitly that the transferred coefficient b^phi may be chosen to satisfy the same linear-growth and Lipschitz bounds as b, since those properties are used in the pathwise estimates immediately after the identities.","section":"Section 7.1, identities (7.12) and (7.13)"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is a solid contribution in the authors' established line of research. I did not find a gap in the central arguments; the transference concern flagged in the stress-test is handled by Proposition 3.10 and Lemma 3.7. The issues I found are local: a small proof repair in Lemma 7.5, a case-split typo in Corollary 9.4, and several presentation ambiguities. These do not undermine the main theorems, so minor revision is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The two things to know: this paper contains real new mathematics, and its main caveat is that the verification burden is pushed onto the authors' previous Memoirs [14]. If you are deciding whether to referee it, the answer should be yes.\n\nThe genuinely new results are Theorem 6.1, which constructs bounded uniformly Hölder diffusion coefficients for which X_1 is never in D_{1,2}, and the Besov regularity theorem Corollary 7.11 with the matching lower bound Proposition 7.12. The sharp exponent 1/(2p) for cut-off couplings is exactly the kind of result that makes the Besov-space framework credible rather than merely convenient. The Lipschitz-case extension to path-dependent random coefficients with BMO drift (Section 5) is also broader than what was previously available, and the BSDE application in Section 9 is a natural payoff. The proofs are detailed and I found no internal contradiction.\n\nThe soft spots are real but not disqualifying. The main load-bearing step is the transference identities (7.12)–(7.13), which turn pathwise estimates into Lp estimates. These rely on Proposition 3.10 being valid for merely Hölder diffusion coefficients and for the discontinuous cut-off coupling indеed. The reader's stress-test note is right: Proposition 3.10 is stated without a Lipschitz restriction and derives from [14, Theorem 3.3], but the paper does not re-prove that transference for non-smooth coefficients, so a referee needs to check that citation carefully. This is a lineage issue, not circularity, but it does make the proof \"as good as [14]\" in a way that is slightly unsatisfying for a standalone paper. The generalized-derivative identity in Theorem 6.1 is also used in a way that deserves a bit more justification; I could not fully verify it, though nothing suggests it is wrong. Minor textual issues in the Introduction and the direct-consequences framing of Section 9 do not affect the mathematics.\n\nWho is this for? Anyone working on Malliavin regularity of SDEs, Besov spaces on Wiener space, or BSDE numerics that depend on Lp-variation of the forward process. A serious referee should be assigned, and the report should focus on the transference step and the generalized-derivative identity.\n\nMy recommendation: send it to a good referee. It is not a desk reject, and it is not ready for blind trust either. The referee needs time to check the [14] dependencies.","headline":"A genuinely new extension of the Geiss–Ylinen coupling machinery: a sharp Malliavin non-differentiability counterexample for Hölder diffusions plus matching Besov regularity bounds, worth serious refereeing despite heavy reliance on the authors' Memoirs.","tokens_in":47259,"tokens_out":1200,"would_cite":true,"duration_ms":15137,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H07","60H10","46E35","46B70"],"pacs":[],"model":"deepseek-v4-flash","headline":"For Hölder-continuous diffusion coefficients, Malliavin differentiability can fail while sharp Besov regularity survives.","keywords":["stochastic differential equations","coupling method","Malliavin Sobolev space","Malliavin Besov spaces","real interpolation","Hölder continuous diffusion","backward stochastic differential equations","Wiener space"],"falsifier":"Take the explicit diffusion coefficient $\\sigma_\\theta(s,x)=1+\\sum_{\\ell\\ge1}\\mathbf{1}_{I_\\ell}(s)S^\\theta_{n_\\ell}(x)$ from Theorem 6.1, with the parameters $(t_\\ell)$ and $(n_\\ell)$ chosen in its proof, and compute $\\|X_1\\|_{D_{1,2}}$ directly. If a finite Malliavin derivative exists, Theorem 6.1 is refuted; the proof's contradiction argument predicts the $D_{1,2}$ norm must be infinite.","tokens_in":2297,"feed_emoji":"🎲","tokens_out":2690,"duration_ms":117027,"temperature":0.7,"pith_summary":"The paper asks how smooth the solution of an SDE is as a function of the driving Brownian path when the diffusion coefficient is only Hölder continuous rather than Lipschitz. It proves two complementary statements with one coupling method: in the Lipschitz and path-dependent case, solutions are Malliavin differentiable and lie in real-interpolation Besov spaces; in the scalar Hölder case, Malliavin differentiability can fail even for bounded non-degenerate diffusions, yet a sharp Besov-type estimate survives. The surviving estimate, Corollary 7.11(3), controls the $L_p$ distance between the original solution and its cut-off coupled version by $(c-a)^{1/(2p)}$ plus explicit terms, and Proposition 7.12 shows the exponent $1/(2p)$ is optimal. This matters because the same Besov regularity of the forward diffusion drives the $L_p$-variation of backward SDEs, so the result extends BSDE regularity theory to non-smooth diffusions such as the CIR process.","feed_headline":"Hölder diffusion can break Malliavin differentiability","feed_subtitle":"A coupling proof shows the break is real, then pins the surviving Lp rate to its sharpest possible value.","key_machinery":"The machinery is a coupling of two Wiener spaces: fix a cut-off coupling function $\\varphi=\\mathbf{1}_{(a,c]}$ and replace the Brownian motion on the interval $(a,c]$ by an independent copy, with $W^{\\varphi}_s=W_s$ for $s\\le a$, $W^{\\varphi}_s=W_a+W'_s-W'_a$ for $a<s\\le c$, and $W^{\\varphi}_s-W^{\\varphi}_c=W_s-W_c$ afterwards. Transference operators $C_0$ and $C_T$ from [14] move random variables and predictable processes from the original Wiener space to the coupled one, preserving finite-dimensional distributions (Lemma 3.7). For the Hölder SDE, a classical regularization argument (Lemma 7.5) shows the difference $D_s=X^{t,\\xi}_s-X^{t,\\xi,\\varphi}_s$ satisfies $|D_s|=|A|+\\int_c^s \\mathrm{sign}_0(D_u)\\,dD_u$, leading to the domination Lemma 7.7 that bounds $\\sup_s|D_s|$ by $|A|$ plus a martingale term, with the Hölder exponent $\\theta\\in[1/2,1)$ entering through $|\\sigma|\\le L_\\sigma|D|^{\\theta}$. The law identities (7.12)–(7.13) then convert the pathwise estimates into the $L_p$ estimates of Theorem 7.3; Proposition 7.12, based on the SDE $dX_s=|X_s|^{\\theta}\\,dW_s$, shows the resulting exponent $1/(2p)$ is optimal.","core_discovery":"The central claim is that regularity of SDE solutions on Wiener space is governed by which coupling is used. For Lipschitz, path-dependent, random coefficients with a BMO drift, the uniform coupling $\\varphi_r\\equiv r$ shows that solutions belong to the Malliavin Sobolev space $D_{1,2}$ and to the real-interpolation Besov spaces $B^{\\eta}_{p,q}$, with quantitative bounds. In dimension one, for a bounded diffusion coefficient $\\sigma$ that is $\\theta$-Hölder with $\\theta\\in[1/2,1)$, the same Malliavin differentiability fails: Theorem 6.1 constructs examples $\\sigma_\\theta(s,x)=1+\\sum_\\ell \\mathbf{1}_{I_\\ell}(s)S^{\\theta}_{n_\\ell}(x)$ for which $X_1\\notin D_{1,2}$ for every starting point, even with zero drift. Nevertheless, using only cut-off couplings $\\varphi=\\mathbf{1}_{(a,c]}$, Corollary 7.11(3) gives a sharp $L_p$ estimate for the sup-norm effect of the coupling, and Proposition 7.12 shows the resulting exponent $1/(2p)$ is optimal for $p\\in[3-2\\theta,\\infty)$. The paper's message is that Malliavin differentiability is too strong a notion for Hölder diffusions, but Besov regularity survives with the best possible rate.","pith_inferences":["Editorial inference: the counterexample's dyadic Schauder blocks suggest the failure of Malliavin differentiability is generic for Hölder moduli that are saturated on infinitely many scales, not an artifact of a special construction.","Editorial inference: a direct testable extension is to multidimensional systems with diagonal Hölder diffusion coefficients; the dimension should enter the cut-off coupling estimates, and the sharp exponent may become dimension-dependent.","Editorial inference: the paper leaves the boundary case $\\theta=1/2$ open; determining whether the Besov estimates persist there would clarify whether the regularity threshold coincides with the pathwise-uniqueness threshold.","Editorial inference: because the Besov rate is derived from cut-off couplings, the same rate should appear as a strong-approximation error for Euler-type schemes for these diffusions; checking this numerically would be a direct test of the paper's implications."],"forward_implications":["In the Lipschitz, path-dependent, random-coefficient setting, solutions inherit Malliavin differentiability from an $\\mathcal{F}_t$-measurable starting point, with explicit bounds in terms of a fractional potential $(U,V)$ of the coefficients.","For scalar Hölder diffusions with $\\theta\\in[1/2,1)$, $X_T$ need not be in $D_{1,2}$; the counterexample is built from Schauder blocks and has a bounded, non-degenerate diffusion coefficient.","Corollary 7.11(3) gives a quantitative control of the cut-off coupling error in $L_p$, and for $p\\in[3-2\\theta,\\infty)$ the leading rate is $(c-a)^{1/(2p)}$ as $c\\downarrow a$.","Proposition 7.12 shows this exponent is optimal: a lower-bound example with $\\sigma(x)=|x|^{\\theta}$ satisfies $\\|X_T-X_T^{(a,c]}\\|_{L_p} \\ge c_{\\theta,p}(c-a)^{1/(2p)}$.","Applied to BSDEs (Corollary 9.4), the forward Besov regularity transfers to the $Y$ and $Z$ components, giving $L_p$-variation rates of order $(c-a)^{1/(2p)}$ for CIR-type forward processes."],"supporting_citations":[{"why":"supplies the coupling/decoupling construction, the transference operators, and the Besov-space characterizations that the paper extends.","marker":"[14]"},{"why":"provides the Euler-approximation and Hölder-coefficient estimates used in Lemmas 7.4–7.7.","marker":"[15]"},{"why":"supplies the Malliavin calculus background, chain rule, and absolute continuity of law used in Theorem 6.1.","marker":"[30]"},{"why":"provides strong-solution existence, pathwise uniqueness, and the comparison theorem used for the Hölder SDEs and the lower bound.","marker":"[20]"},{"why":"supplies the Schauder-function characterization of Hölder functions used to build the counterexample diffusion coefficient.","marker":"[6]"},{"why":"earlier $L_p$-variation and fractional-smoothness results for BSDEs that motivate the Besov regularity target and the BSDE application.","marker":"[12]"}],"fun_headline_variants":["Malliavin differentiability fails for Hölder SDEs","Coupling yields sharp Lp rates for Hölder SDEs","Hölder diffusion loses Malliavin smoothness, gains Besov","Sharp Besov regularity survives when Malliavin fails","Cut-off coupling gives optimal Lp rates for SDEs"],"cache_read_input_tokens":49408,"weakest_assumption_plain":"The load-bearing premise is that the transference machinery of [14] extends to the cut-off coupling $\\varphi=\\mathbf{1}_{(a,c]}$ for SDEs whose diffusion coefficient is only Hölder continuous; the distributional identities (7.12)–(7.13), asserting that the transferred coefficient processes have the same law as the original ones, are what turn the pathwise estimates of Lemma 7.7 into the $L_p$ estimates of Theorem 7.3.","fun_headline_variants_meta":{"raw":{"variants":["Malliavin differentiability fails for Hölder SDEs","Coupling yields sharp Lp rates for Hölder SDEs","Hölder diffusion loses Malliavin smoothness, gains Besov","Sharp Besov regularity survives when Malliavin fails","Cut-off coupling gives optimal Lp rates for SDEs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001595,"raw_usage":{"total_tokens":6352,"prompt_tokens":937,"completion_tokens":5415,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":553,"completion_tokens_details":{"reasoning_tokens":5341}},"tokens_in":553,"tokens_out":5415,"duration_ms":33960,"temperature":1.0,"reasoning_tokens":5341,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T15:34:19.888522+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the explicit diffusion coefficient $\\sigma_\\theta(s,x)=1+\\sum_{\\ell\\ge1}\\mathbf{1}_{I_\\ell}(s)S^\\theta_{n_\\ell}(x)$ from Theorem 6.1, with the parameters $(t_\\ell)$ and $(n_\\ell)$ chosen in its proof, and compute $\\|X_1\\|_{D_{1,2}}$ directly. If a finite Malliavin derivative exists, Theorem 6.1 is refuted; the proof's contradiction argument predicts the $D_{1,2}$ norm must be infinite.","supporting_citations":[{"cited_title":"Geiss and J","cited_arxiv_id":null,"evidence_quote":"supplies the coupling/decoupling construction, the transference operators, and the Besov-space characterizations that the paper extends."},{"cited_title":"Gy¨ ongy and M","cited_arxiv_id":null,"evidence_quote":"provides the Euler-approximation and Hölder-coefficient estimates used in Lemmas 7.4–7.7."},{"cited_title":"Nualart, The Malliavin Calculus and Related Topics, S pringer, 2006","cited_arxiv_id":null,"evidence_quote":"supplies the Malliavin calculus background, chain rule, and absolute continuity of law used in Theorem 6.1."},{"cited_title":"Karatzas and S.E","cited_arxiv_id":null,"evidence_quote":"provides strong-solution existence, pathwise uniqueness, and the comparison theorem used for the Hölder SDEs and the lower bound."},{"cited_title":"Ciesielski, On the isomorphisms of the space Hα and m, Bull","cited_arxiv_id":null,"evidence_quote":"supplies the Schauder-function characterization of Hölder functions used to build the counterexample diffusion coefficient."},{"cited_title":"Geiss, S","cited_arxiv_id":null,"evidence_quote":"earlier $L_p$-variation and fractional-smoothness results for BSDEs that motivate the Besov regularity target and the BSDE application."}],"review_version":1}