{"id":"f4a79225-0c29-457b-bc29-f1d12cf96fef","arxiv_id":"2412.08483","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Two new Carleman estimates yield Lipschitz and Hölder stability and an inverse-source uniqueness theorem for coupled stochastic mean field game equations with common noise.","lead":"This paper proves stability and uniqueness results for inverse problems in mean field games with common noise, where all players share a random shock. The proofs rely on two new Carleman inequalities for the coupled stochastic Hamilton-Jacobi-Bellman and Fokker-Planck system.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The solution class for U is only L^2, but the Carleman estimates used require the noise coefficient to lie in H^1; unless U is proved to have H^1 regularity, Theorem 1.1 and 1.3 do not follow as written.","rationale":"The reader's weakest assumption identifies exactly the gap that I find most load-bearing: the proofs require H^1 regularity for the stochastic integrand U, but the stated solution class only gives L^2. This is not an external disagreement with a consensus assumption; it is an internal mismatch between the hypotheses of Theorems 2.1 and 2.3 and the solution classes stated in Theorems 1.1 and 1.3. The gap is concretely fixable by adding an H^1 assumption on U (or proving one from the well-posedness of the backward stochastic parabolic equation), so the appropriate verdict remains CONDITIONAL rather than REJECT. I also note that the boundary Carleman estimates in Theorems 2.3 and 2.4 are stated without proof and, with a time-only weight, the usual boundary integrals require careful handling; this is a secondary concern. The main computations otherwise follow the standard linearization and interpolation pattern, and the two whole-space Carleman estimates are proven in detail. My recommendation is to keep the conditional acceptance, with the explicit condition that the regularity of U be clarified or supplied.","tokens_in":17916,"tokens_out":3865,"duration_ms":39257,"concrete_test":"Derive or supply an energy estimate for ∇U from the backward component of (1.1) under Assumptions 1.1-1.3, showing U belongs to L^2_F(0,T;H^1). Specifically, test whether E∫_0^T∫ |∇U|^2 dxdt is bounded by the data norms assumed in the theorems. If the estimate holds, the gap closes; if it requires extra terminal regularity or smallness not assumed, then Theorems 1.1-1.3 must be restated with the stronger hypothesis. A complementary test: attempt to construct a weak solution of (1.1) with U in L^2_F(0,T;L^2) but not in L^2_F(0,T;H^1), checking whether the current assumptions admit such a case and whether the Carleman estimates would fail for it.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Theorems 2.1 and 2.3 are stated for equations (2.2) and (2.20) whose stochastic integrand f2 is required to be in L^2_F(0,T;H^1). In the proof of Theorem 1.1, the estimate (2.3) is applied to the u-equation (3.9) with f2 = U, yet the theorem assumes only U in L^2_F(0,T;L^2). In Section 4, the proof of Theorem 1.3 defines V = U/R and then sets w = χ v_{x1}, W = χ V_{x1}; applying Theorem 2.3 to (4.6) requires W in L^2_F(0,T;H^1(G)). This demands two x1-derivatives of U (one to form V_{x1}, one for ∇W), which is far beyond L^2. No regularity theorem is cited or proved to upgrade U. If the natural BSDE regularity of U does not provide H^1, the Carleman estimates cannot be applied and the displayed inequalities (3.12) and (4.14) are unjustified. The gap is in the hypotheses, not necessarily in the conclusions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two inverse problems for mean field games with common noise, described by the coupled forward-backward stochastic parabolic system (1.1). The first problem (IP1) asks whether the terminal density observation, together with initial and terminal data, determines the solution pair (ρ,u) with Lipschitz or Hölder stability; the second problem (IP2) asks whether an x1-independent source term r in a factorized running cost can be uniquely recovered from terminal density and lateral Cauchy data. The main results are Theorem 1.1 (Lipschitz stability), Theorem 1.2 (Hölder stability), and Theorem 1.3 (unique determination of the source term). The proofs rely on two whole-space Carleman estimates (Theorems 2.1 and 2.2), proved in detail via Itô's formula, and two boundary Carleman estimates (Theorems 2.3 and 2.4), stated without proof. The paper linearizes the Hamiltonian under Assumptions 1.1–1.3 and reduces the inverse problems to weighted energy estimates.","tokens_in":18214,"tokens_out":15631,"duration_ms":152584,"significance":"If the results are valid, the paper provides the first stability and uniqueness results for inverse problems in MFGs with common noise, extending a substantial line of work on deterministic MFG inverse problems (e.g., [14,15]) to the stochastic setting. The whole-space Carleman estimates in Section 2 are proved carefully, and the treatment of the Hamiltonian nonlinearity through the coefficients B1–B5 and F1,F2 is systematic. The authors correctly identify the main stochastic difficulties, namely the gradient noise in the Fokker-Planck equation and the nonhomogeneous correction term U in the HJB equation. The use of the relation between β and β̂ to control the gradient drift is a notable technical contribution. However, the manuscript as written has a serious regularity gap: the Carleman estimates require the stochastic integrand and the solution to lie in higher Sobolev spaces than the stated solution classes provide. Since this gap affects the central applications of the estimates, the results cannot be considered fully established without repairs.","major_comments":[{"comment":"Theorem 2.1 requires the noise integrand f2 to lie in L^2_F(0,T;H^1(R^n)) and the solution w to lie in L^2_F(0,T;H^2(R^n)), while Theorem 1.1 only assumes U∈L^2_F(0,T;L^2(R^n)) and (ρ_i,u_i)∈L^2_F(0,T;H^1(R^n)). The proof applies (2.3) to the u-equation (3.9) with f2=U, which is not justified for U of class L^2 only. Moreover, the term -β div U in (3.9) is not well-defined as an L^2-valued dt-integrand when U is merely L^2. The same problem occurs for ρ: Theorem 2.2 is applied to (3.8), but that theorem requires p∈L^2_F(0,T;H^2(R^n)), whereas the stated solution class for ρ in Theorem 1.1 is only H^1(R^n). The authors should either strengthen the solution classes (e.g., U∈L^2_F(0,T;H^1(R^n)) and u,ρ∈L^2_F(0,T;H^2(R^n))) or prove and cite a regularity result showing that Assumption 1.3 upgrades U and ρ to the required classes.","section":"§3, application of Theorem 2.1 to (3.9) and Theorem 2.2 to (3.8)"},{"comment":"The proof of Theorem 1.3 applies the boundary Carleman estimate (2.21) to w=χ v_{x1} with noise integrand W=χ V_{x1}, and (2.23) to p=χ ρ_{x1}. For (2.21) to apply, one needs W∈L^2_F(0,T;H^1(G)) and w∈L^2_F(0,T;H^2(G)); for (2.23), one needs p∈L^2_F(0,T;H^2(G)). The hypotheses of Theorem 1.3 only assume U∈L^2_F(0,T;L^2(G)) and u∈L^2_F(0,T;H^1(G)). Even with Assumption 1.3 (u∈W^{2,∞}, ρ∈W^{1,∞}), the required H^2 regularity on the unbounded slab G=(0,1)×R^{n-1} does not follow, since bounded second derivatives do not imply square-integrability on unbounded domains. Consequently, the estimates (4.14)-(4.16) are not justified as written. The solution classes in Theorem 1.3 need to be strengthened, or a separate approximation/regularization argument must be supplied.","section":"§4, equations (4.6)-(4.16)"},{"comment":"Theorems 2.3 and 2.4 are essential for the proof of Theorem 1.3, but they are stated without proof, with the remark that their proofs are 'very similar' to those of Theorems 2.1 and 2.2. This is not sufficient as it stands: (2.20) contains the additional drift coupling f3·f2, which is absent in the whole-space case and forces the parameters λ0(f3), µ0(f3) to be chosen in a way that is not quantified; the Dirichlet boundary condition w=0 on (0,T)×Γ also generates boundary terms in the weighted integration-by-parts argument that do not appear in the whole-space proof. The authors should provide complete proofs of these two estimates, or a precise reduction to Theorems 2.1 and 2.2 with full details of the boundary and f3 terms.","section":"§2, Theorems 2.3 and 2.4"}],"minor_comments":[{"comment":"The line 'Put v = u/R and V = V/R' contains a typo; it should read 'Put v = u/R and V = U/R'.","section":"§4, equation before (4.4)"},{"comment":"In the left-hand side of (2.3) and (2.21), the |f2|^2 term is written with weight θ, whereas the proof of Theorem 2.1 and the intermediate estimate (2.7) contain θ^2; the displayed inequalities appear to have missing squares.","section":"§2, equations (2.3) and (2.21)"},{"comment":"The last integral in (2.23) is over R^n, while all other integrals are over G; this is presumably a typo and should be ∫_G.","section":"§2, Theorem 2.4, equation (2.23)"},{"comment":"In the statement of Theorem 1.2, the norm ‖u1-u2‖ is written as L^2(ε,T;H^1(R^n)) without the subscript F, whereas the ρ term uses L^2_F; the u term should also be L^2_F for consistency.","section":"§1, Theorem 1.2"},{"comment":"In (4.12) and (4.13), the final bound is written as M^2∫_G p(T,y)^2 dy, but p=χρ_{x1} depends on (t,y); this should be ∫_G |χ(t)ρ(t,y)|^2 dy or a similar expression involving p(t,y), with the time variable made explicit.","section":"§4, equations (4.12)-(4.13)"}],"recommendation":"major_revision","confidential_remarks":"The regularity gap is the main obstacle: it is not merely cosmetic, because the model itself contains div U, so the stated L^2 class for U is too weak even for the equation to hold in the strong sense used by the Carleman estimates. The authors can likely repair this by enlarging the solution classes and adding a short regularity argument, or by citing standard BSDE regularity results. The omitted proofs of Theorems 2.3 and 2.4 are also essential; if they are genuinely routine variants, including them in the revision should be feasible. I see no evidence of circular reasoning or improper citation practice; the manuscript is a serious contribution that needs technical completion rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real step forward — the first Carleman treatment of inverse problems for MFGs with common noise — and the two global Carleman estimates are actually proved. But read the regularity hypotheses carefully: the theorems as stated do not give the U-regularity that the proofs require. That is the thing to focus on if you referee it.\n\nThe genuinely new content is in Section 2: Theorems 2.1 and 2.2 for forward-backward stochastic parabolic operators with the specific β, β̂ structure. The proofs are detailed and I did not find a false step. The linearization of the Hamiltonian in Section 3 is handled cleanly, with all coefficients bounded under Assumptions 1.1–1.3, and the interpolation argument in Theorem 1.2 is standard. The paper is honest about the hard parts, and the self-citations to [21, 23, 25, 28] are legitimate background rather than padding. No circularity detected.\n\nThe soft spots are real. The Carleman estimates require the noise coefficient f2 to lie in L²_F(0,T;H¹). In Theorem 1.1 the u-equation is (3.9) with f2 = U, yet the theorem only assumes U ∈ L²_F(0,T;L²(Rⁿ)). So (2.3) cannot be applied as written. The same problem appears in Theorem 1.3, where the proof defines W = χ(U/R)_{x1} and applies Theorem 2.3 to (4.6): this needs W ∈ L²_F(0,T;H¹(G)), i.e. roughly two x1-derivatives of U, far beyond the stated L²(G). No regularity lemma is supplied. This is a genuine gap in the hypotheses, not a contradiction in the equations — but it is load-bearing for all three theorems.\n\nAlso, the boundary Carleman estimates Theorems 2.3 and 2.4, which carry Theorem 1.3, are stated without proof. I believe they are similar to the global versions, but saying “we omit them” is thin for results that are essential. And Theorem 1.1 assumes terminal data h ∈ L²_{F_T}(L²) in the system, but the statement and proof need H¹; that should be made consistent.\n\nBottom line: I think the claims are probably right, and the gaps are fixable by adding the missing regularity assumptions or proving that the BSDE component U inherits H¹. As it stands, the paper is conditional. I would accept it for peer review, not desk-reject: the Carleman estimates are novel, the topic is timely, and a careful referee can tell the authors exactly what to add. I would bring it to reading group to check whether the U-regularity rescue works.","headline":"A serious and mostly detailed extension of Carleman inverse problems to MFGs with common noise, but the theorems as stated overreach the regularity of U.","tokens_in":18680,"tokens_out":4438,"would_cite":true,"duration_ms":45218,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35R30","60H15","49N80"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves Lipschitz and Hölder stability for the stochastic mean-field-game system with common noise and a uniqueness theorem for an x1-independent source, all via two new Carleman estimates.","keywords":["mean field games","common noise","inverse problems","Carleman estimates","Lipschitz stability","Hölder stability","inverse source problem","stochastic Fokker-Planck-HJB system"],"falsifier":"On the one-dimensional slab G=(0,1), take β=0, R≡1, and a source difference r1-r2 supported in time; solve the linearized forward-backward system with U chosen in L2 but not in H1 and check whether the lateral Cauchy data plus terminal density still force r1=r2. If such a U makes the Carleman estimate (2.3) inapplicable, the missing H1 hypothesis is exposed; if the identity still holds, the H1 condition is unnecessary.","tokens_in":17686,"feed_emoji":"🔍","tokens_out":10464,"duration_ms":102636,"temperature":0.7,"pith_summary":"This paper studies two inverse problems for mean-field games with common noise, described by a coupled forward-backward stochastic system of Fokker-Planck and Hamilton-Jacobi-Bellman equations. It proves that the solution pair is Lipschitz stable in the terminal and initial observations (Theorem 1.1), that it is Hölder stable when only the two terminal observations are used (Theorem 1.2), and that an unknown source term depending only on the transverse variables is uniquely determined by the terminal density together with lateral Cauchy data on two parallel boundaries (Theorem 1.3). The engine is a pair of new Carleman estimates for the forward and backward stochastic parabolic operators of the system. If correct, these results extend the deterministic mean-field-game inverse program to the common-noise setting that arises in financial markets, traffic flow, and other systems with a shared random factor.","feed_headline":"Sources in common-noise mean-field games recovered from boundary data","feed_subtitle":"Two new Carleman estimates yield Lipschitz and Hölder stability and source uniqueness.","key_machinery":"The load-bearing objects are the two Carleman estimates of Section 2 (Theorems 2.1 and 2.2, with the slab versions 2.3 and 2.4). A Carleman estimate is a weighted integral inequality for solutions of a parabolic equation, built from an exponential weight θ=$e^{{λ(t+2)^μ}}$; the proofs multiply the equation by a judiciously chosen test expression, apply the stochastic chain rule (Itô's formula), and integrate so that weighted norms of the unknown, its gradient, and its Laplacian are bounded by the source terms and by boundary, terminal, and initial data. The structural relation βhat=(1+$β^{2}$)/2 between the common-noise coefficient β and the diffusion coefficient βhat is what lets the undesirable stochastic cross terms be absorbed. For the inverse source problem, the additional devices are dividing the HJB equation by R, differentiating in x1 so the unknown r—independent of x1—drops out, inserting a time cut-off χ, and using the lateral Cauchy data to bring the estimate to a close.","core_discovery":"The central claim is Theorem 1.3: two solutions of the coupled stochastic FP-HJB system with the same terminal cost, the same terminal density, and equal lateral Cauchy data for ρ and u together with their first and second x1-derivatives on the two sides of the slab G=(0,1)×$R^{{n-1}}$ must have the same x1-independent source factors r1 and r2, almost surely, whenever the multiplier R is bounded away from zero. Theorems 1.1 and 1.2 are the companion stability statements: for two solutions satisfying the a priori bounds of Assumption 1.3, the H1 difference of the pair is controlled by the measurements—Lipschitz when terminal value, terminal density, and initial density are all observed, Hölder when only the two terminal measurements are available and the estimate is taken on [ε,T] with ε∈(0,T).","pith_inferences":["The paper proves uniqueness but not stability for the inverse source problem; writing a quantitative version of the x1-differentiation argument would likely yield a conditional Hölder estimate for r, but that step is absent.","The constants in the Carleman estimates grow like e^{2λ(T+2)^μ}, so a numerical reconstruction from these measurements would be severely ill-conditioned without regularization; the paper does not address algorithms.","One could test whether Theorem 1.3 survives with fewer boundary observations, for instance only ρ and u on the lateral boundary rather than their first and second x1-derivatives; the proof as written uses all six conditions in (1.7)."],"forward_implications":["Adding terminal density observation restores uniqueness for the coupled forward-backward system, which the paper notes is otherwise rare without a monotonicity assumption.","With the full set of three measurements, the solution map is Lipschitz; with only the two terminal measurements, the map is Hölder on any subinterval [ε,T].","The source uniqueness theorem implies that an x1-independent cost or interaction source can be recovered from terminal density plus boundary polls in a small neighbourhood of two parallel planes, because the Cauchy data in (1.7) can be approximated from such neighbourhood data.","Because the estimates hold for every β∈[0,1], the deterministic MFG inverse results appear as the limiting case β=0 of the common-noise framework."],"supporting_citations":[{"why":"Supplies the two-Carleman-estimate template for the deterministic FP-HJB system that the paper adapts to common noise.","marker":"[14]"},{"why":"Provides the Hölder-stability-with-fewer-measurements result that Theorem 1.2 mirrors in the stochastic setting.","marker":"[15]"},{"why":"Establishes the stochastic parabolic Carleman estimates and flags the non-homogeneous U·dW terms as the main difficulty the new estimates must control.","marker":"[28]"},{"why":"Gives the Carleman estimate for stochastic parabolic equations that underlies the inverse-problem methodology extended here.","marker":"[23]"},{"why":"Shows stability for an inverse stochastic parabolic problem, supporting the use of Carleman estimates for stochastic equations.","marker":"[21]"},{"why":"Applies Carleman estimates directly to the mean-field-game system for Lipschitz stability and uniqueness in the deterministic case.","marker":"[13]"},{"why":"Develops the lateral-Cauchy-data Carleman approach for deterministic MFGs that the source uniqueness proof in Theorem 1.3 follows.","marker":"[16]"}],"fun_headline_variants":["Carleman estimate settles inverse MFG source recovery","Common-noise MFG: source uniqueness from boundary data","Lipschitz and Hölder stability for inverse MFG with common noise","Carleman-based proof: unique source in common-noise MFGs","Boundary data recovers sources in common-noise MFGs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the correction process U is spatially differentiable in a square-integrable sense, even though the stated solution class only assumes it is square-integrable without derivatives.","fun_headline_variants_meta":{"raw":{"variants":["Carleman estimate settles inverse MFG source recovery","Common-noise MFG: source uniqueness from boundary data","Lipschitz and Hölder stability for inverse MFG with common noise","Carleman-based proof: unique source in common-noise MFGs","Boundary data recovers sources in common-noise MFGs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000912,"raw_usage":{"total_tokens":3860,"prompt_tokens":829,"completion_tokens":3031,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":445,"completion_tokens_details":{"reasoning_tokens":2946}},"tokens_in":445,"tokens_out":3031,"duration_ms":24453,"temperature":1.0,"reasoning_tokens":2946,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T17:46:36.016323+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the one-dimensional slab G=(0,1), take β=0, R≡1, and a source difference r1-r2 supported in time; solve the linearized forward-backward system with U chosen in L2 but not in H1 and check whether the lateral Cauchy data plus terminal density still force r1=r2. If such a U makes the Carleman estimate (2.3) inapplicable, the missing H1 hypothesis is exposed; if the identity still holds, the H1 condition is unnecessary.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the two-Carleman-estimate template for the deterministic FP-HJB system that the paper adapts to common noise."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Hölder-stability-with-fewer-measurements result that Theorem 1.2 mirrors in the stochastic setting."},{"cited_title":"Tang and X","cited_arxiv_id":null,"evidence_quote":"Establishes the stochastic parabolic Carleman estimates and flags the non-homogeneous U·dW terms as the main difficulty the new estimates must control."},{"cited_title":"Lü, Carleman estimate for stochastic parabolic equations and inverse stochastic parabolic problems","cited_arxiv_id":null,"evidence_quote":"Gives the Carleman estimate for stochastic parabolic equations that underlies the inverse-problem methodology extended here."},{"cited_title":"Liao and Q","cited_arxiv_id":null,"evidence_quote":"Shows stability for an inverse stochastic parabolic problem, supporting the use of Carleman estimates for stochastic equations."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Applies Carleman estimates directly to the mean-field-game system for Lipschitz stability and uniqueness in the deterministic case."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Develops the lateral-Cauchy-data Carleman approach for deterministic MFGs that the source uniqueness proof in Theorem 1.3 follows."}],"review_version":1}