{"id":"23c3650b-22ad-413c-8efc-48ba6f17c0e6","arxiv_id":"2412.02948","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The claimed complete description of bicausal Monge transports between laws of SDEs is invalid because the sufficiency direction drops the requirement that the transported path generates the same information.","lead":"This paper tries to fully describe 'bicausal' couplings between random paths, which move the law of one stochastic differential equation to another without using future information. The central characterization is too broad: it includes simple rotations of Brownian motion that are not reversible in the adapted sense, so the main theorem fails.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lévy transform is not a counterexample; genuine gap is Lemma 3.10's unjustified identification of the H^eta-drift with b(T).","rationale":"The reader's core counterexample, the Lévy transform, is actually bicausal under the paper's Definition 2.1: it is a stochastic integral against an O_d-valued integrand, hence an F^X-Brownian motion, and any F^X-martingale is a martingale with respect to the product filtration generated by (X,T(X)) because σ(X_s,T_s)⊆F^X_s. The reader's claim that Eq. (12) only tests A×W^d overlooks that C=A∩T^{-1}(B)∈H^η_s when T is causal, so the martingale test extends to all rectangles. The paper's Eq. (12) is notationally sloppy but the underlying argument is repairable. The genuinely load-bearing weakness is different: Lemma 3.10's converse asserts that the H^η-semimartingale decomposition of T has drift b(T) merely from T#η=ν. That assertion is not justified, since semimartingale drift is filtration-dependent and can differ by a process adapted to the larger filtration but not to T's own filtration, particularly in degenerate directions. This gap affects the proof of Lemma 3.10 and its uses in Corollary 3.7 and Proposition 4.3, but not necessarily the main characterization Theorem 3.4, whose sufficiency condition (9) supplies the missing martingale property. Therefore the appropriate verdict is CONDITIONAL: the central claim may be correct, but the stated Lemma 3.10 needs either a proof of the drift identification or a strengthened hypothesis.","tokens_in":24102,"tokens_out":50955,"duration_ms":536840,"concrete_test":"Construct or rule out an adapted H^η-semimartingale T on the degenerate example d=2, σ=(1,0), b=0, η=ν=law of (B,0), with T#η=ν and nonzero H^η-drift in Ker σ(T). Existence of such T invalidates Lemma 3.10; non-existence shows the missing step is automatic and the central theorem stands.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Reader's attack misidentifies the failure. The Lévy transform T(X)=∫ sign(X)dX is exactly of the form in Corollary 3.6 with Q=sign(X)∈O_d; it is an F^X-Brownian motion and hence an F^X-martingale. For any A,B∈H_s, the set C=A∩T^{-1}(B) lies in H^η_s, so the martingale property of N=T against H^η_s yields the required product-filtration martingale property. Equivalently, any pair of Brownian motions on a common filtration is bicausal, so the strict inclusion of image filtrations does not imply non-bicausality. Thus the Lévy objection does not land. The real issue is in Lemma 3.10's converse: it defines N_t=T_t-˜z-∫b(s,T)ds and uses E^η(N_t 1_A)=E^η(N_s 1_A), which requires N to be an H^η-martingale. But T being an H^η-semimartingale with T#η=ν does not by itself make N the martingale part: the H^η-finite-variation part of T can differ from ∫b(T)dt by an adapted drift that is not visible to T's own filtration, especially when σ is degenerate and the difference lies in Ker σ(T). The proof gives no argument excluding this, so Lemma 3.10 as stated is unsupported. Theorem 3.4's sufficiency may be salvageable because equation (9) explicitly sets the H^η-drift to b, but the proof as written appeals to Lemma 3.10.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies causal and bicausal couplings between probability measures on the space of continuous paths, with the measures assumed to be (possibly weak) solutions of SDEs. The main results are a characterization of bicausal couplings as joint laws of two weak solutions driven by a pair of Brownian motions on a common filtration (Theorem 3.2), and a claimed complete description of bicausal Monge transports in terms of stochastic integrals with orthogonal-matrix-valued integrands (Theorem 3.4). Corollaries specialize to the Wiener measure, give existence and non-existence criteria, and lead to density results and to equality of the bicausal Monge and Kantorovich costs.","tokens_in":24334,"tokens_out":33462,"duration_ms":365003,"significance":"If correct, the structural description of bicausal Monge maps between SDE laws would be a substantial contribution to adapted transport theory, with a clean statement for Wiener measure and connections to the H-hypothesis and Émery's almost Brownian filtrations. The paper also gives a useful characterization of all bicausal couplings as common-filtration weak solutions. The central claims are ambitious and potentially very useful, but the current version contains a serious proof gap in the key lemma and a false existence corollary, so the manuscript is not yet in a publishable state.","major_comments":[{"comment":"The converse direction of Lemma 3.10 is not proved. The equality of the first and last terms in (12) is asserted without justification: it is valid only if N_t = T_t - z - ∫_0^t b(s,T) ds is already known to be an (η,(H^η_t))-martingale. Semimartingality of T under (H^η_t) gives only a decomposition T_t = A_t + M_t with an arbitrary (H^η_t)-predictable finite-variation part A_t; nothing in the assumptions forces A_t = ∫_0^t b(s,T) ds. The proof silently identifies the H^η-drift of T with the drift of the target SDE, which is exactly the point that needs to be established. Since the sufficiency proof of Theorem 3.4 invokes this lemma, the main characterization is not established as written. I do not regard the Lévy transform as a counterexample to the lemma: it is of the form in Corollary 3.6 and is in fact bicausal; the genuine issue is the drift identification, not the size of the image filtration.","section":"Section 3.4.2, Lemma 3.10 (Eq. (12))"},{"comment":"Corollary 3.5 is false as stated. Take η = W^d (σ = I, b = 0, z = 0) and ν = δ_0 (σ = 0, b = 0, z = 0). Then max_ω dim Ker σ(s,ω) = 0 < d = min_ω dim Ker σ(s,ω), so the corollary asserts that no bicausal Monge map exists. But T ≡ 0 is a bicausal Monge map: the induced plan (Id,0)#η is causal from η to ν, and the reverse plan is causal because the disintegration of η given Y = 0 is the constant kernel η, which is measurable with respect to the trivial filtration H^ν_t. The inequality therefore has the wrong direction; the correct no-existence condition should be max dim Ker σ > min dim Ker σ (source more degenerate than target). This error affects the paper's existence criteria.","section":"Corollary 3.5"},{"comment":"The proof of Proposition 4.3 invokes Lemma 3.10 to conclude that T^n is a bicausal Monge map. In that application the conclusion is likely correct because X^n and Y^n generate the same filtration and Y^n is a Brownian motion in that filtration, so the H^η-drift condition holds. However, the proof as written relies on the unproved converse of Lemma 3.10 rather than on the explicit martingale structure, and this should be repaired if the lemma is weakened or corrected.","section":"Section 4.2, Proposition 4.3"}],"minor_comments":[{"comment":"The proof of the converse should explicitly state the required identification of the H^η-finite-variation part of T with ∫_0^· b(s,T) ds, and give an argument for it; simply asserting that the first and last terms of (12) are equal is circular.","section":"Section 3.4.2, Lemma 3.10"},{"comment":"The statement of Corollary 3.5 contains no proof; given the counterexample above, the authors should either supply a corrected statement with the reversed inequality or delete the corollary and explain the correct conditions.","section":"Corollary 3.5"},{"comment":"The notation in Corollary 3.8 is confusing: the text appears to write 'from μ^{σ,0}_z to μ^{σ,0}_z' with the same symbol σ for both marginals, while the proof clearly distinguishes σ and σ. The statement should be reformulated with separate symbols for the two SDE coefficients.","section":"Corollary 3.8"},{"comment":"The proof of Corollary 4.7 is omitted with a reference to [32, Lemma 5.14]; since this is an application of the density result, the omission may be acceptable, but the dependence on the still-unproved Lemma 3.10 should be made explicit.","section":"Section 4.3, Corollary 4.7"}],"recommendation":"major_revision","confidential_remarks":"The reader's report identifies a gap in Lemma 3.10, and I agree that this is a genuine problem. However, the reader's Lévy-transform counterexample does not land: the Lévy transform is exactly a stochastic integral with an O_d-valued integrand and is bicausal. The more serious problems are (i) the missing drift identification in the converse of Lemma 3.10 and (ii) the false statement of Corollary 3.5, which has a simple counterexample. These are load-bearing but appear to be locally fixable, so I recommend major revision rather than outright rejection. The authors should also carefully re-check all dimension/inequality conditions in the existence corollaries."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's my read. The paper is not the clean reject the reader's report suggests, but it is also not ready as is. The headline result — a complete description of bicausal Monge maps between laws of SDEs as semimartingales of the form (9) — is genuinely new and, I suspect, correct in spirit. Theorem 3.2 (bicausal couplings are exactly joint laws of weak solutions driven by a common Brownian pair) is a clean contribution that generalizes the scalar results in [8] and is likely to be useful. Corollary 3.6, the Wiener-to-Wiener description in terms of stochastic integrals of O_d-valued integrands, is elegant and would be a natural tool for adapted Wasserstein problems. The density arguments using Émery's almost Brownian filtrations are a nice touch.\n\nThe reader's verdict leans on the Lévy transform as a counterexample. That doesn't work: under the Lévy coupling, X and T are both Brownian motions on the common filtration generated by X, so by the paper's own Theorem 3.1 the coupling is bicausal. Strictly smaller image filtration by itself does not break reverse causality. So that part of the reader's rationale is wrong.\n\nThe real hole is in the converse direction of Lemma 3.10. The proof only checks the martingale property of the target process on sets A×W^d, which is not enough to certify reverse causality against all rectangles. More substantively, the proof assumes the H^η-semimartingale decomposition of T has drift b(s,T). That is not automatic from T#η=ν: the drift under the enlarged filtration H^η can differ from the drift under T's own filtration, especially when σ is degenerate and the difference lives in the kernel. The argument needs to exclude that possibility, and it doesn't. As stated, Lemma 3.10 is unsupported. Since Theorem 3.4's sufficiency invokes Lemma 3.10, the proof needs repair even if the theorem itself is true — and I think it may be, because (9) explicitly pins the drift to b(T), which is exactly the missing condition.\n\nSo the paper has a significant, localized proof gap rather than a refuted central claim. It deserves a serious referee. I'd send it out, and ask the referee to focus on whether the H^η-drift of a semimartingale transport is forced to be b(T) under the assumptions of Lemma 3.10, or whether the lemma needs an extra hypothesis. For readers working on adapted Wasserstein causality, the paper is worth reading for Theorem 3.2 and the density results, but they should be cautious about Lemma 3.10 until it's fixed.","headline":"Novel and likely correct in spirit, but Lemma 3.10 has a real unpatched proof gap; the reader's Lévy-transform counterexample does not actually land.","tokens_in":24964,"tokens_out":18159,"would_cite":true,"duration_ms":192710,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60G44","60H10","49Q22"],"pacs":[],"model":"deepseek-v4-flash","headline":"All bicausal Monge transports between SDE laws are stochastic integrals driven by rotation-valued integrands.","keywords":["causal transport","bicausal couplings","Monge maps","path space","stochastic differential equations","Wiener measure","semimartingales","adapted Wasserstein distance"],"falsifier":"The one-dimensional map $T(X)_t=\\int_0^t \\operatorname{sign}(X_s)\\,dX_s$ pushes Wiener measure to itself and is a semimartingale under it. If the converse direction of Lemma 3.10 were valid, the induced coupling would be bicausal; the original Wiener path, however, is not adapted to the filtration generated by $T(X)$, so the reverse-causality condition fails. Verifying this non-adaptation directly would settle the converse direction.","tokens_in":23778,"feed_emoji":"🌀","tokens_out":12105,"duration_ms":112270,"temperature":0.7,"pith_summary":"This paper sets out to characterize bicausal Monge transports between laws of stochastic processes on path space, in the case where the two laws are weak solutions of stochastic differential equations. It claims that a measurable map is a bicausal Monge transport exactly when its coordinate process is a semimartingale under the source law and satisfies a transport SDE with a rotation-valued matrix process $Q$; for Wiener measures this reduces to stochastic integrals $\\int_0^\\cdot Q_s\\,dX_s$ with adapted $O_d$-valued integrands. If correct, the result turns the measure-theoretic question of causality into a differential question about SDEs. It also yields explicit existence obstructions, density of Monge transports among all bicausal couplings, and equality of Monge and Kantorovich costs for regular SDEs.","feed_headline":"All bicausal Monge transports are rotation-valued stochastic integrals","feed_subtitle":"For SDE laws the maps solve a transport equation with rotation-matrix integrands; Wiener measures give the cleanest form.","key_machinery":"The central object is a bicausal coupling: a transport plan between two path-space measures whose conditional distributions are non-anticipative in both time directions. The proof machinery has three load-bearing components: the H-hypothesis reformulation of causality (Theorem 2.1), a martingale representation theorem for degenerate diffusions (Proposition 3.9), and the transport SDE (9)--(10) with its rotation-valued coefficient $Q$. The SDE is what converts measure preservation into a differential condition: the pushed-forward diffusion must align with the target diffusion through $Q$, and condition (10) enforces the alignment on kernels of degenerate diffusion coefficients.","core_discovery":"On its own terms, the paper's central result is Theorem 3.4: for $\\eta=\\mu_z^{\\sigma,b}$ and $\\nu=\\mu_{\\tilde z}^{\\bar\\sigma,\\bar b}$, a map $T:W^d\\to W^d$ is a bicausal Monge map from $\\eta$ to $\\nu$ if and only if $T$ is an $(\\eta,H_t^\\eta)$-semimartingale and satisfies $T_t=\\tilde z+\\int_0^t b(s,T)\\,ds+\\int_0^t \\sigma(s,T)Q_s\\sigma^\\dagger(s,X)\\,dM_s^\\eta$ together with the alignment condition $\\sigma(s,T)Q_s=\\sigma(s,T)Q_s\\sigma^\\dagger(s,X)\\sigma(s,X)$, where $Q$ is adapted and orthogonal-matrix-valued. Corollary 3.6 draws the Wiener-space conclusion: every bicausal Monge transport of Wiener measure is a stochastic integral of an adapted $O_d$-valued integrand. The paper also claims that when the diffusion coefficients are regular and invertible, all bicausal Monge transports between strong SDE solutions factor through the Itô maps and rotation-valued stochastic integrals, and that such Monge transports are dense among all bicausal couplings.","pith_inferences":["Strengthening the converse to require that the original path be adapted to the transported path would preserve the classification while excluding stochastic-integral maps whose image filtration is strictly smaller.","The kernel-dimension obstruction in Corollary 3.5 gives a design principle: pairs of SDEs with incompatible diffusion-kernel dimensions can only admit Kantorovich, not Monge, bicausal transports.","Because bicausal Monge maps are dense among all bicausal couplings for regular SDEs, numerical solvers for adapted transport could parameterize near-optimal plans by rotation-valued integrands rather than by general couplings."],"forward_implications":["Every bicausal Monge transport between $d$-dimensional Wiener measures has the form $T(X)=\\int_0^\\cdot Q_s\\,dX_s$ for an adapted $O_d$-valued process $Q$.","Between strong solutions of SDEs with invertible diffusion coefficients, every bicausal Monge transport factors as an Itô map composed with a rotation-valued stochastic integral and a second Itô map (Corollary 3.7).","If the kernel dimensions of the two diffusion coefficients are incompatible, no bicausal Monge map exists (Corollary 3.5).","Bicausal Monge transports are dense in the set of all bicausal couplings for SDEs with regular, invertible coefficients (Propositions 4.5 and 4.6).","For continuous costs with integrable separable growth, the bicausal Monge problem and the bicausal Kantorovich problem have the same optimal value (Corollary 4.7)."],"supporting_citations":[{"why":"Defines causal transport plans and their Monge–Kantorovich problems on path space, supplying the causal-coupling notion the paper extends.","marker":"[29]"},{"why":"Provides the martingale representation theorem for degenerate diffusions used to derive the transport SDE in Theorem 3.4.","marker":"[37]"},{"why":"Establishes the H-hypothesis equivalence for filtrations, which Theorem 2.1 uses to reformulate causality as a martingale preservation property.","marker":"[12]"},{"why":"Supplies the almost Brownian filtrations and density of mutually adapted Brownian motions used in the density results of Section 4.2.","marker":"[18]"},{"why":"Gives the weak-uniqueness/pathwise-uniqueness criteria used in Proposition 4.1 to decide when a bicausal coupling is induced by a Monge map.","marker":"[13]"},{"why":"Studies adapted Wasserstein distances between laws of SDEs, providing the one-dimensional bicausal transport results that this work generalizes.","marker":"[8]"},{"why":"Supplies the adapted Wasserstein cost structure and financial context used in the explicit examples of Section 4.4.","marker":"[5]"},{"why":"Shows that polar decomposition can be done measurably, which is used to obtain the progressively measurable rotation process $Q$.","marker":"[3]"}],"fun_headline_variants":["Wiener Monge maps are rotation-stochastic integrals","Bicausal Monge transport equals stochastic integral with rotations","Monge transports for SDE laws: rotation-valued integrals","Every Wiener bicausal Monge map is a rotated stochastic integral"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The classification rests on the claim that any measurable map pushing one SDE law to another is automatically bicausal once its coordinate process is a semimartingale under the starting law.","fun_headline_variants_meta":{"raw":{"variants":["Wiener Monge maps are rotation-stochastic integrals","Bicausal Monge transport equals stochastic integral with rotations","Monge transports for SDE laws: rotation-valued integrals","Every Wiener bicausal Monge map is a rotated stochastic integral"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000275,"raw_usage":{"total_tokens":1621,"prompt_tokens":898,"completion_tokens":723,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":656}},"tokens_in":514,"tokens_out":723,"duration_ms":6783,"temperature":1.0,"reasoning_tokens":656,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T22:57:07.328745+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"The one-dimensional map $T(X)_t=\\int_0^t \\operatorname{sign}(X_s)\\,dX_s$ pushes Wiener measure to itself and is a semimartingale under it. If the converse direction of Lemma 3.10 were valid, the induced coupling would be bicausal; the original Wiener path, however, is not adapted to the filtration generated by $T(X)$, so the reverse-causality condition fails. Verifying this non-adaptation directly would settle the converse direction.","supporting_citations":[{"cited_title":"Lassalle , Causal transport plans and their Monge–Kantorovich problem s, Stochastic Analysis and Applications, 36 (2018), pp","cited_arxiv_id":null,"evidence_quote":"Defines causal transport plans and their Monge–Kantorovich problems on path space, supplying the causal-coupling notion the paper extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the martingale representation theorem for degenerate diffusions used to derive the transport SDE in Theorem 3.4."},{"cited_title":"Br ´ emaud and M","cited_arxiv_id":null,"evidence_quote":"Establishes the H-hypothesis equivalence for filtrations, which Theorem 2.1 uses to reformulate causality as a martingale preservation property."},{"cited_title":"´Emery, On certain almost brownian filtrations , in Annales de l’Institut Henri Poincare (B) Probability and Statistics, vol","cited_arxiv_id":null,"evidence_quote":"Supplies the almost Brownian filtrations and density of mutually adapted Brownian motions used in the density results of Section 4.2."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the weak-uniqueness/pathwise-uniqueness criteria used in Proposition 4.1 to decide when a bicausal coupling is induced by a Monge map."},{"cited_title":"Backhoff-Veraguas, S","cited_arxiv_id":null,"evidence_quote":"Studies adapted Wasserstein distances between laws of SDEs, providing the one-dimensional bicausal transport results that this work generalizes."},{"cited_title":"Backhoff, D","cited_arxiv_id":null,"evidence_quote":"Supplies the adapted Wasserstein cost structure and financial context used in the explicit examples of Section 4.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows that polar decomposition can be done measurably, which is used to obtain the progressively measurable rotation process $Q$."}],"review_version":1}