{"id":"8c1308a4-b078-4700-b9c5-7d9eb0e68a06","arxiv_id":"2605.24232","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Lipschitz L² stability estimates for OT maps in terms of 2-MK distance (and C^{1,α} under Hölder) plus explicit second variation of quadratic MK distance via Monge-Ampère linearization.","lead":"This paper derives quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains under the quadratic cost, via linearization of the Monge-Ampère equation, plus an explicit second-variation formula for the 2-Monge-Kantorovich distance. A smart generalist might read it to see how small density perturbations translate into controlled changes in transport maps and costs.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's assessment that the verdict must remain UNVERDICTED because proofs and technical details cannot be examined is correct. No load-bearing concern can be formulated or refuted until the full text is available.","tokens_in":1743,"tokens_out":217,"duration_ms":19097,"concrete_test":"Retrieve the full manuscript and verify that the linearization step (claimed to yield the Lipschitz L² stability) holds under the stated Hölder assumptions on the densities; if the derivation in the relevant section contains an unstated regularity requirement, the map-stability claim would need adjustment.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The provided context supplies only the abstract; the full manuscript (including the linearization of the Monge-Ampère equation, the precise characterization of the linear response, and the derivation of the second variation) is not available for inspection. Without those sections, no internal inconsistency, unjustified assumption, or gap in the argument can be located or tested.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper establishes quantitative stability estimates for optimal transport maps between non-degenerate densities on uniformly convex domains for the quadratic cost. Under Hölder regularity assumptions on the densities, it proves Lipschitz L² stability estimates for the maps in terms of the 2-Monge-Kantorovich distance between source and target densities, and C^{1,α} stability in terms of L^p distances. Removing the continuity assumption yields a Lipschitz L² stability estimate for the Brenier potentials in terms of the L² distance between the densities. The proofs rely on a precise characterization of the linear response of the Brenier potential obtained by linearizing the Monge-Ampère equation in divergence form along smooth interpolations; as an application, an explicit formula for the second variation of the quadratic Monge-Kantorovich distance is derived.","tokens_in":1802,"tokens_out":582,"duration_ms":9518,"significance":"If the central claims hold, the results supply useful quantitative stability controls for optimal transport maps and potentials under standard structural assumptions (uniform convexity of domains and non-degeneracy of densities). The linearization approach to the Monge-Ampère equation and the explicit second-variation formula constitute concrete contributions that can be applied in stability analysis and related PDE problems. The distinction between map stability under Hölder regularity and potential stability without continuity is clearly delineated.","major_comments":[{"comment":"§3 (linearization step): the derivation of the linearized Monge-Ampère equation in divergence form assumes sufficient smoothness of the interpolation to justify differentiation under the integral; the manuscript should explicitly verify that the error terms arising from this differentiation remain controlled by the 2-MK distance under the stated Hölder assumptions.","section":"§3"},{"comment":"Theorem 1.2 (map stability): the Lipschitz constant in the L² estimate appears to depend on the uniform convexity constants and the non-degeneracy bounds; the proof should confirm that this dependence is explicit and does not deteriorate when the Hölder exponent α approaches zero.","section":"Theorem 1.2"}],"minor_comments":[{"comment":"The abstract states the results for 'pairs of source and target densities' but the precise interpolation path between densities is only defined later; a brief sentence in the abstract or introduction clarifying the interpolation would improve readability.","section":"Abstract"},{"comment":"Notation for the 2-Monge-Kantorovich distance is introduced without an explicit formula in the opening paragraph; adding the standard definition W_2^2(μ,ν) = inf ∫ |x-y|^2 dγ would help readers unfamiliar with the notation.","section":"Introduction"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for the careful reading and constructive comments. Below we address the two major comments point by point, indicating the changes we will make to the manuscript.","responses":[{"response":"We agree that the passage from the smooth interpolation to the linearized equation requires explicit control of the error terms arising from differentiation under the integral. In the revised manuscript we will add a short lemma in §3 that bounds these remainder terms in terms of the 2-MK distance, using only the Hölder regularity of the densities and the uniform convexity of the domains. The argument proceeds by approximating the given Hölder densities by smooth ones whose 2-MK distance is controlled, applying the smooth linearization, and passing to the limit with the error estimate.","revision_made":"yes","referee_comment":"[§3] §3 (linearization step): the derivation of the linearized Monge-Ampère equation in divergence form assumes sufficient smoothness of the interpolation to justify differentiation under the integral; the manuscript should explicitly verify that the error terms arising from this differentiation remain controlled by the 2-MK distance under the stated Hölder assumptions."},{"response":"The Lipschitz constant in Theorem 1.2 is explicit and depends on the uniform convexity constants, the non-degeneracy bounds, and the Hölder exponent α. As α → 0 the constant necessarily deteriorates; this is consistent with the paper’s separation of results, since map stability in L² requires positive Hölder regularity while the continuous case yields only potential stability. We will insert a remark immediately after the statement of Theorem 1.2 that records the explicit dependence on all parameters and notes the blow-up as α → 0. No change to the theorem statement itself is required.","revision_made":"partial","referee_comment":"[Theorem 1.2] Theorem 1.2 (map stability): the Lipschitz constant in the L² estimate appears to depend on the uniform convexity constants and the non-degeneracy bounds; the proof should confirm that this dependence is explicit and does not deteriorate when the Hölder exponent α approaches zero."}],"tokens_in":1409,"tokens_out":460,"duration_ms":38282,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The paper's core contribution is a linear-response analysis that produces Lipschitz L² stability for the transport maps when the densities are Hölder, plus C^{1,α} stability under L^p perturbations, and a clean formula for the second variation of the quadratic Monge-Kantorovich distance. The same linearization also gives L² stability for the Brenier potentials once continuity of the densities is dropped. These are presented as direct consequences of the differentiated equation rather than indirect arguments.\n\nThe approach is straightforward and stays inside the usual setting of non-degenerate densities on uniformly convex domains. The distinction between map stability (which needs Hölder) and potential stability (which does not) is handled without extra smallness assumptions, which keeps the statements usable.\n\nThe main limitation is that the quantitative constants and error terms depend on the usual ellipticity and convexity quantities; the paper does not claim uniformity beyond that. No circularity appears in the derivation, and the cited background is standard. The second-variation formula is explicit enough to be checked independently.\n\nThis is for people working on quantitative stability in optimal transport or on second-variation questions for transport costs. It is narrow but technically sharp, so a serious referee should see it.","headline":"Linearization of the divergence-form Monge-Ampère equation yields both the stability estimates and an explicit second-variation formula for the quadratic MK distance.","tokens_in":2306,"tokens_out":325,"would_cite":false,"duration_ms":18873,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Under Hölder regularity, optimal transport maps between non-degenerate densities on uniformly convex domains are Lipschitz stable in L² with respect to the 2-Monge-Kantorovich distance.","keywords":["optimal transport","Monge-Ampère equation","stability estimates","Brenier potentials","Monge-Kantorovich distance","linearization","second variation","convex domains"],"falsifier":"Finding a counterexample consisting of non-degenerate Hölder densities on a uniformly convex domain where the L² difference of the optimal maps is not bounded by a constant times the 2-Monge-Kantorovich distance between the densities.","tokens_in":2641,"feed_emoji":"","tokens_out":727,"duration_ms":50723,"temperature":0.7,"pith_summary":"The paper derives quantitative stability estimates for optimal transport maps with quadratic cost. When source and target densities are Hölder continuous and non-degenerate on uniformly convex domains, the maps satisfy a Lipschitz bound in L² controlled by the 2-Monge-Kantorovich distance between the densities. They also obtain C^{1,α} stability controlled by L^p distances. Without the Hölder assumption the Brenier potentials are still Lipschitz stable in L² with respect to the L² distance of the densities. The key step is a characterization of the linear response of the Brenier potential obtained by linearizing the Monge-Ampère equation, which additionally yields an explicit second variation formula for the Monge-Kantorovich distance.","feed_headline":"Optimal transport maps stable in L2 to 2-MK perturbations of densities","feed_subtitle":"Linearization of the Monge-Ampère equation yields the stability and a second variation formula.","key_machinery":"The linearized Monge-Ampère equation in divergence form, used to characterize the first variation of the Brenier potential along interpolations of the densities.","core_discovery":"Under the stated assumptions the linear response of the Brenier potential to perturbations of the source and target densities is given by the solution of the linearized Monge-Ampère equation in divergence form. This response implies the Lipschitz stability estimates for the maps and potentials, and it produces the explicit formula for the second variation of the quadratic Monge-Kantorovich distance.","pith_inferences":["This linearization technique could be applied to derive stability results for other cost functions in optimal transport.","The stability estimates suggest quantitative convergence rates for approximations of transport maps when densities are perturbed by discretization error.","Connections to the geometry of the Wasserstein space may follow from the second variation formula.","Testing the formula on explicit examples such as uniform densities on balls would verify the second variation explicitly."],"forward_implications":["The optimal transport map depends Lipschitz continuously on the pair of densities when measured in the 2-Monge-Kantorovich metric and L² norm.","C^{1,α} regularity of the map is controlled by L^p distances of the densities under Hölder assumptions.","Brenier potentials remain stable in L² even if the densities lose continuity.","The second variation of the 2-Monge-Kantorovich distance admits an explicit integral expression derived from the linearization."],"fun_headline_variants":["L2 stability of OT maps via Monge-Ampere linearization","Second variation formula for quadratic 2-MK distance","Brenier potentials L2 stable to density perturbations","Linearized MA equation yields OT map stability estimates"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"Densities are non-degenerate on uniformly convex domains, with Hölder continuity needed for the map stability claims.","fun_headline_variants_meta":{"raw":{"variants":["L2 stability of OT maps via Monge-Ampere linearization","Second variation formula for quadratic 2-MK distance","Brenier potentials L2 stable to density perturbations","Linearized MA equation yields OT map stability estimates"]},"model":"grok-4.3","cost_usd":0.006471,"raw_usage":{"total_tokens":3006,"prompt_tokens":620,"num_sources_used":0,"completion_tokens":63,"cost_in_usd_ticks":64712000,"prompt_tokens_details":{"text_tokens":620,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2323,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":620,"tokens_out":63,"duration_ms":27543,"temperature":1.0,"reasoning_tokens":2323,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-30T14:45:49.331771+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"Finding a counterexample consisting of non-degenerate Hölder densities on a uniformly convex domain where the L² difference of the optimal maps is not bounded by a constant times the 2-Monge-Kantorovich distance between the densities.","supporting_citations":[],"review_version":1}