{"id":"8ddbaa9d-7f8d-4eaf-8629-0ae9474e0e9c","arxiv_id":"2504.21268","paper_version":1,"verdict":"REJECT","confidence":"HIGH","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The averaging principle (Theorem 4.7) fails for the equation dX^ε_t = E[X^ε_t | F^0_t] dt with random initial condition, contradicting the paper's central claim.","lead":"This paper claims to prove an averaging principle for stochastic differential equations that depend on conditional distributions and are driven by both Brownian and fractional Brownian noise. A simple linear example shows the main theorem is false under the paper's own assumptions.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 4.4 misidentifies the averaged law: X̄_t depends on Ω0 through ξ, so Theorem 4.7 fails (linear-mean counterexample).","rationale":"The central claim is the averaging principle, Theorem 4.7. The reader's critique identifies a false assertion in Lemma 4.4. I checked it in good faith and found the counterexample is even starker: with time-independent linear mean-field drift and no diffusion, the conditional and unconditional equations have different solutions for every ε, so the limit cannot be zero. This is not a disagreement with a known consensus; it is an internal inconsistency between the statement of Lemma 4.4 and the definition of the averaged equation. I do not see a way to repair the estimate while keeping the theorem as stated; changing the averaged equation to use L(X̄_t | F^0_t) might salvage a different statement, but that would be a different theorem. The wellposedness section is not the basis for rejection. Thus the reader's REJECT verdict stands unchanged.","tokens_in":16744,"tokens_out":8279,"duration_ms":88037,"concrete_test":"Run the explicit counterexample: take d = 1, σ_W = σ_H = 0, b(t, x, μ) = mean(μ), and ξ a non-degenerate Bernoulli random variable. Solve (4.1) as X^ε_t = ξ e^t and (4.2) as X̄_t = ξ + Eξ(e^t - 1); then compute E sup_{0≤t≤T}|X^ε_t - X̄_t|^2 = Var(ξ)(e^T - 1)^2. If this quantity is nonzero, Theorem 4.7 is false as stated.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 4.4 is the load-bearing step and it is false. The text claims \"the process X̄_t does not depend on Ω0\", but X̄_0 = ξ and ξ is F^0-measurable; for any t > 0 the averaged solution is a functional of ξ, so its P1-fiber laws vary with ω0. Consequently W_2^2(L(X^ε_t|F^0_t), L(X̄_t)) is not controlled by E_1|X^ε_t - X̄_t|^2: the right side couples the conditional law with the fiber law L(X̄_t(ω0, ·)), while the left side compares it with the P0-mixture L(X̄_t). A concrete instance satisfying all stated assumptions: d = 1, σ_W = σ_H = 0, b(t, x, μ) = ∫ y μ(dy). Then X^ε_t is F^0_t-measurable and (4.1) gives X^ε_t = ξ e^t. The averaged equation (4.2) has \\bar b(x, μ) = ∫ y μ(dy), so X̄_t = ξ + ∫_0^t E X̄_s ds and E X̄_t = Eξ e^t, hence X̄_t = ξ + Eξ(e^t - 1). Therefore E sup_{0≤t≤T}|X^ε_t - X̄_t|^2 = Var(ξ)(e^T - 1)^2 > 0, contradicting Theorem 4.7. The paper's own assumptions allow zero diffusion, so this counterexample is within scope. The error is structural, not a mere estimate: the averaged equation uses the unconditional law while the original equation uses the conditional law, and the common randomness in ξ prevents the conditional information from disappearing in the limit.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies conditional McKean-Vlasov stochastic differential equations driven simultaneously by a standard Brownian motion and a fractional Brownian motion with Hurst parameter H > 1/2. The first contribution is a wellposedness theorem (Theorem 3.1) obtained by a fixed-point iteration on the space of square-integrable processes. The second, and central, contribution is an averaging principle (Theorem 4.7): the solution of the conditional distribution dependent equation with fast time scales is claimed to converge in mean-square sup norm, as the fast parameter tends to zero, to the solution of an associated averaged distribution dependent equation. The proof of the averaging principle relies on Lemma 4.4, which estimates Wasserstein distances between conditional laws and the law of the averaged process.","tokens_in":17158,"tokens_out":6747,"duration_ms":66172,"significance":"If the averaging principle were correct, it would provide a useful extension of the classical Khasminskii averaging framework to conditional McKean-Vlasov equations with common noise, a setting relevant to particle systems in random environments. The paper contains a genuinely detailed fixed-point argument for wellposedness and explicit estimates for the fractional Brownian component, and it engages with the relevant recent literature. However, the central advertised claim is false: Lemma 4.4 contains a load-bearing error, and Theorem 4.7 is contradicted by a simple counterexample that satisfies all of the paper's assumptions. The significance of the paper is therefore negative with respect to its main result.","major_comments":[{"comment":"The assertion \"It is easy to see the process X̄_t does not depend on Ω0\" is false. The averaged equation (4.2) has initial condition X̄_0 = ξ, and ξ is F^0-measurable; for every t > 0 the averaged solution X̄_t remains a functional of ξ, so its P1-fiber laws vary with ω0. Consequently the displayed inequality W_2^2(L(X^ε_t|F^0_t), L(X̄_t)) ≤ E1|X^ε_t - X̄_t|^2 compares a conditional law with the unconditional law and is not justified. This inequality is used in the estimates K12, K22, and K32 inside the proof of Theorem 4.7, so the Gronwall argument cannot close once the error is removed.","section":"Section 4, Lemma 4.4"},{"comment":"Theorem 4.7 is false as stated. Take d = 1, σ_W = σ_H = 0, b(t, x, μ) = ∫ y μ(dy), and a non-deterministic initial condition ξ ∈ L^2. All of Assumptions 2.1 and 4.1 are satisfied. Equation (4.1) reduces to dX^ε_t = X^ε_t dt because X^ε_t is F^0_t-measurable and hence L(X^ε_t|F^0_t) = δ_{X^ε_t}; the solution is X^ε_t = ξ e^t. The averaged equation (4.2) has b̄(x, μ) = ∫ y μ(dy), so X̄_t = ξ + ∫_0^t E X̄_s ds, which gives X̄_t = ξ + E[ξ](e^t - 1). Then E sup_{0≤t≤T}|X^ε_t - X̄_t|^2 = Var(ξ)(e^T - 1)^2 > 0, contradicting the claimed limit.","section":"Theorem 4.7"},{"comment":"The structural source of the error is that the unconditional averaged law L(X̄_t) is used as a substitute for the conditional law L(X̄_t|F^0_t). In a conditional McKean-Vlasov equation with a common random initial condition, averaging over P1 does not erase the common randomness contributed by ξ, and the P0-mixture law of X̄_t can be very different from the fiberwise conditional law. A correct averaging statement would need a conditional averaged equation, which is not the equation studied in the paper.","section":"Section 4, proof of Theorem 4.7"}],"minor_comments":[{"comment":"There are numerous typographical errors, including \"Browinan motion\" in Section 2 and \"McKean-Vlasov stochastic differential equations\" spelled inconsistently.","section":"Throughout"},{"comment":"The notation in the second line of (4.17) drops the conditional law: it writes b(X^ε_r, L(X^ε_r)) where the context requires b(X^ε_r, L(X^ε_r|F^0_r)). This is confusing because the conditional law is essential in the equation being analyzed.","section":"Equation (4.17)"},{"comment":"Several references are incomplete: [12] lacks a journal or volume information, and [13] appears without a journal volume or page range.","section":"References"}],"recommendation":"reject","confidential_remarks":"The counterexample in the major comments is decisive and requires no exotic construction; it fits exactly within the stated assumptions. The error is structural: the averaged equation uses the unconditional law, while the original equation is driven by the conditional law, and the common initial randomness prevents the two from agreeing in the limit. I see no local repair short of reformulating the averaged equation as a conditional McKean-Vlasov equation, which would change the paper's stated scope. The wellposedness part may be salvageable, but the advertised averaging principle cannot stand."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem of this paper is false. A concrete counterexample lives entirely inside the paper's assumptions: take d=1, σ_W=σ_H=0, and b(t,x,μ)=mean(μ). Then (4.1) becomes dX^ε_t = mean(L(X^ε_t|F^0_t))dt, and since there is no noise, X^ε_t is F^0-measurable; hence its conditional law is a point mass and the equation reduces to dX^ε_t = X^ε_t dt, so X^ε_t=ξe^t. The averaged equation (4.2) is dX̄_t = E[X̄_t]dt with X̄_0=ξ, giving X̄_t = ξ + Eξ(e^t−1). Then E sup_{0≤t≤T}|X^ε_t−X̄_t|² = Var(ξ)(e^T−1)² > 0 if Var(ξ)>0, contradicting Theorem 4.7. This is not a pathology; the drift is Lipschitz in the Wasserstein sense and all assumptions in Sections 2 and 4 are satisfied.\n\nThe root cause is Lemma 4.4. The proof asserts that X̄_t does not depend on Ω0, but X̄_0=ξ and ξ is F0-measurable. For every t>0 the averaged solution is a functional of ξ, so its P1-fiber laws vary with ω0. The inequality W₂²(L(X^ε_t|F^0_t), L(X̄_t)) ≤ E₁|X^ε_t−X̄_t|² is therefore invalid: the left side compares a fiber law with the unconditional mixture law, while the right side couples that mixture law with a random fiber. Lemma 4.5 and Lemma 4.6 rely on the same flawed step, and the Gronwall argument in Theorem 4.7 cannot close.\n\nWhat the paper does well: it targets a real gap, combining conditional McKean–Vlasov structure with mixed Brownian/fractional Brownian drivers. The wellposedness theorem (Theorem 3.1) is plausible; the fixed-point iteration is standard, and the fBm estimate (3.5) follows known techniques. The exposition is honest and the reference list is appropriate. But the averaging principle is not a minor glitch—it is the central advertised result, and it fails structurally because the common randomness in the initial condition prevents the conditional information from disappearing in the limit.\n\nThe paper is not publishable in its current form. A serious referee could still help: the wellposedness part deserves scrutiny, and the error, while real, is subtle enough that a detailed report would be useful to the authors. I would not cite Theorem 4.7. If the authors restrict the paper to the existence/uniqueness result and either remove the averaging claim or formulate a genuinely conditional averaged equation, there may be a publishable core. As is, the averaging principle should be rejected.","headline":"The averaging principle is false: a simple linear-mean counterexample with zero noise satisfies all stated assumptions, so Theorem 4.7 does not hold; the wellposedness part may still be salvageable.","tokens_in":17650,"tokens_out":3918,"would_cite":false,"duration_ms":39697,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","60G22"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves unique strong solutions and an averaging limit for conditional distribution-dependent SDEs driven by Brownian motion and fractional Brownian motion.","keywords":["conditional McKean-Vlasov SDE","fractional Brownian motion","averaging principle","wellposedness","Wasserstein distance","mean-field SDE","multiscale stochastic differential equation","fixed point theorem"],"falsifier":"Set all coefficients to zero and take $\\xi$ to be any non-degenerate $\\mathcal{F}^0$-measurable random variable. Then $X^{\\varepsilon}_t = \\bar X_t = \\xi$, so $E(\\sup_{0\\le t\\le T}|X^{\\varepsilon}_t-\\bar X_t|^2)=0$, but for each $\\omega_0$, $W_2^2(L(X^{\\varepsilon}_t|\\mathcal{F}^0_t)(\\omega_0), L(\\bar X_t)) = E|\\xi-\\xi(\\omega_0)|^2$, whose expectation is $2\\operatorname{Var}(\\xi)>0$. This directly contradicts the first inequality of Lemma 4.4 and shows the proof of Theorem 4.7 cannot cover random initial data as stated.","tokens_in":16541,"feed_emoji":"📉","tokens_out":11427,"duration_ms":107378,"temperature":0.7,"pith_summary":"This paper studies stochastic differential equations whose coefficients depend on the conditional law of the solution given a background filtration, with independent standard Brownian and fractional Brownian drivers of Hurst parameter $H>1/2$. It establishes that, under Lipschitz and time-averaging assumptions, such an equation has a unique strong solution, and that the solution of the fast-oscillating equation converges in mean square, uniformly in time, to the solution of the averaged equation as the time-scale parameter $\\varepsilon$ tends to zero. The contribution is a wellposedness-and-averaging result for conditional mean-field systems in a common random environment.","feed_headline":"Fast noise averages out in conditional distribution-dependent SDEs","feed_subtitle":"Unique strong solutions exist; fast fluctuations vanish to an averaged fractional-noise limit as $\\varepsilon\\to 0$.","key_machinery":"The load-bearing object is the fixed-point map $\\Psi: Y \\mapsto X^{\\mu}$, where $\\mu_t = L(Y_t|\\mathcal{F}^0_t)$ and $X^{\\mu}$ solves the auxiliary SDE with coefficients frozen at the conditional law $\\mu_t$; iterating $\\Psi$ and showing that some power is a strict contraction yields the unique strong solution of the original equation. The second mechanism is the fractional Brownian integral estimate (3.5), which bounds the second moment of the fBm integral by a constant times $t^{2H-1}$ times the $L^2$-norm of the integrand, using the kernel representation with $\\kappa\\in(1-H,1/2)$. For the averaging principle, the main tool is a partition of $[0,T]$ into intervals of length $\\sqrt{\\varepsilon}$, on which the fast coefficients are compared with their time averages; Assumption 4.1 forces the averaged discrepancy to vanish as $\\varepsilon\\to 0$.","core_discovery":"The central claim is Theorem 4.7: for the conditional McKean-Vlasov equation\n$$$dX^{{\\varepsilon}}$_t = b(t/\\varepsilon, $X^{{\\varepsilon}}$_t, L($X^{{\\varepsilon}}$_t|\\mathcal{F}^0_t))\\,dt + \\sigma_W(t/\\varepsilon, $X^{{\\varepsilon}}$_t, L($X^{{\\varepsilon}}$_t|\\mathcal{F}^0_t))\\,dW_t + \\sigma_H(t/\\varepsilon, L($X^{{\\varepsilon}}$_t|\\mathcal{F}^0_t))\\,dB^H_t$$\nwith initial value $\\xi$, the mean-square pathwise distance to the averaged solution $\\bar X$, defined by\n$$d\\bar X_t = \\bar b(\\bar X_t, L(\\bar X_t))\\,dt + \\bar\\sigma_W(\\bar X_t, L(\\bar X_t))\\,dW_t + \\bar\\sigma_H(L(\\bar X_t))\\,dB^H_t,$$\nvanishes: $\\lim_{\\varepsilon\\to 0} E(\\sup_{0\\le t\\le T}|X^{\\varepsilon}_t - \\bar X_t|^2)=0$. The route is to prove existence and uniqueness by a fixed-point contraction on the space of square-integrable processes, using an auxiliary unconditioned equation for each fixed conditional-law input, then to control the difference between the fast system and its average by partitioning the time axis into intervals of length $\\sqrt{\\varepsilon}$ and applying the averaging assumption to the leftover drift and diffusion discrepancies. The uniqueness and averaging results cover all three coefficients, including the fractional-noise coefficient.","pith_inferences":["The paper's argument appears to require the initial value $\\xi$ to be deterministic: if $\\xi$ is a non-degenerate $\\mathcal{F}^0$-measurable random variable, the averaged process $\\bar X$ depends on the background randomness through its starting point, contradicting the statement in Lemma 4.4 that $\\bar X$ does not depend on $\\Omega_0$; in the pure-noise case $b=\\sigma_W=\\sigma_H=0$, the key inequ","A natural reformulation, not stated in the paper, is to compare $X^{\\varepsilon}_t$ with the conditional averaged law $L(\\bar X_t|\\mathcal{F}^0_t)$ rather than the unconditional law, or to declare $\\xi$ deterministic; either change would let the Gronwall step in Theorem 4.7 close.","The proof's explicit error terms suggest a quantitative convergence rate once $K_2(T)$ is given a rate, with main contributions of order $\\sqrt{\\varepsilon}\\,K_2(1/\\sqrt{\\varepsilon})$ plus the Holder-in-time terms $\\sqrt{\\varepsilon}+\\varepsilon+\\varepsilon^H$.","Because the fractional estimate (3.5) uses $H>1/2$ through the kernel exponent, extending the averaging principle to $H\\le 1/2$ would require a different integral bound, such as a Young or rough-path estimate; this is not addressed in the paper."],"forward_implications":["Unique strong solutions exist for conditional distribution-dependent SDEs driven jointly by a standard Brownian motion and a fractional Brownian motion with $H>1/2$, under Lipschitz and growth conditions on all coefficients.","The multiscale system $X^{\\varepsilon}$ is well approximated by the single-scale averaged system $\\bar X$: the mean-square uniform error over any finite time horizon tends to zero as $\\varepsilon\\to 0$.","The averaging limit equation is itself well posed, since Assumption 4.1 together with Assumption 2.1 imply Lipschitz continuity of the averaged coefficients in the Wasserstein metric.","Averaging applies not only to the drift but also to the Brownian and fractional diffusion coefficients, so the effective noise retains both the standard and fractional components."],"supporting_citations":[{"why":"Supplies the existence theorem for the auxiliary distribution-independent SDE (3.1), the base case of the fixed-point construction.","marker":"[16]"},{"why":"Supplies the fractional Brownian integral estimate (3.5) with the kernel bound used to control the $\\sigma_H$ term.","marker":"[17]"},{"why":"Supplies the iteration argument that makes a power of the fixed-point map a strict contraction and the Wasserstein comparison used in Lemma 4.4.","marker":"[14]"},{"why":"Supplies the Wasserstein-moment bound $W_\\theta^2(L(X_t),\\delta_0) \\le E|X_t|^2$ used to close the Gronwall estimates.","marker":"[18]"}],"fun_headline_variants":["Mixed Brownian-fractional noise averages out in conditional SDEs","Rough and Brownian noise average out in McKean-Vlasov SDEs","Fast fluctuations vanish in conditional distribution-dependent SDEs","Averaging principle for conditional SDEs with mixed noise"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof assumes the averaged process is independent of the background randomness, but the initial value is allowed to be random on that background, so the key Wasserstein comparison between conditional and unconditional laws can fail.","fun_headline_variants_meta":{"raw":{"variants":["Mixed Brownian-fractional noise averages out in conditional SDEs","Rough and Brownian noise average out in McKean-Vlasov SDEs","Fast fluctuations vanish in conditional distribution-dependent SDEs","Averaging principle for conditional SDEs with mixed noise"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001438,"raw_usage":{"total_tokens":5789,"prompt_tokens":933,"completion_tokens":4856,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":4788}},"tokens_in":549,"tokens_out":4856,"duration_ms":36026,"temperature":1.0,"reasoning_tokens":4788,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:09:21.539650+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Set all coefficients to zero and take $\\xi$ to be any non-degenerate $\\mathcal{F}^0$-measurable random variable. Then $X^{\\varepsilon}_t = \\bar X_t = \\xi$, so $E(\\sup_{0\\le t\\le T}|X^{\\varepsilon}_t-\\bar X_t|^2)=0$, but for each $\\omega_0$, $W_2^2(L(X^{\\varepsilon}_t|\\mathcal{F}^0_t)(\\omega_0), L(\\bar X_t)) = E|\\xi-\\xi(\\omega_0)|^2$, whose expectation is $2\\operatorname{Var}(\\xi)>0$. This directly contradicts the first inequality of Lemma 4.4 and shows the proof of Theorem 4.7 cannot cover random initial data as stated.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the existence theorem for the auxiliary distribution-independent SDE (3.1), the base case of the fixed-point construction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the fractional Brownian integral estimate (3.5) with the kernel bound used to control the $\\sigma_H$ term."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the iteration argument that makes a power of the fixed-point map a strict contraction and the Wasserstein comparison used in Lemma 4.4."},{"cited_title":", Salkeld W., Tugaut J., Freidlin-Wentzell LDP in path spa ce for McKean- Vlasov equations and the functional iterated logarithm law, Ann","cited_arxiv_id":null,"evidence_quote":"Supplies the Wasserstein-moment bound $W_\\theta^2(L(X_t),\\delta_0) \\le E|X_t|^2$ used to close the Gronwall estimates."}],"review_version":1}