Wellposedness and averaging principle for conditional distribution dependent SDEs driven by standard Brownian motions and fractional Brownian motions

Li Tan; Shengrong Wang

REVIEW 3 major objections 3 minor 18 references

Wellposedness and averaging principle for conditional distribution dependent SDEs driven by standard Brownian motions and fractional Brownian motions

T0 review · 3 major / 3 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper proves unique strong solutions and an averaging limit for conditional distribution-dependent SDEs driven by Brownian motion and fractional Brownian motion.

desk verdict 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. read the letter →

arxiv 2504.21268 v1 pith:ZX7SR33M submitted 2025-04-30 math.PR

Li Tan , Shengrong Wang This is my paper · ORCID

classification math.PR MSC 60H1060G22

keywords conditionalMcKean-VlasovSDEfractionalBrownianmotionaveragingprinciplewellposednessWassersteindistancemean-fieldmultiscalestochasticdifferentialequationfixedpointtheorem

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $\varepsilon\to 0$.

What would settle it

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.

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Extended reading notes

Core claim

The central claim is Theorem 4.7: for the conditional McKean-Vlasov equation $$$dX^{{\varepsilon}}$_t = b(t/\varepsilon, $X^{{\varepsilon}}$_t, L($X^{{\varepsilon}}$_t|\mathcal{F}^0_t))\,dt + \sigma_W(t/\varepsilon, $X^{{\varepsilon}}$_t, L($X^{{\varepsilon}}$_t|\mathcal{F}^0_t))\,dW_t + \sigma_H(t/\varepsilon, L($X^{{\varepsilon}}$_t|\mathcal{F}^0_t))\,dB^H_t$$ with initial value $\xi$, the mean-square pathwise distance to the averaged solution $\bar X$, defined by $$d\bar X_t = \bar b(\bar X_t, L(\bar X_t))\,dt + \bar\sigma_W(\bar X_t, L(\bar X_t))\,dW_t + \bar\sigma_H(L(\bar X_t))\,dB^H_t,$$ vanishes: $\lim_{\varepsilon\to 0} E(\sup_{0\le t\le T}|X^{\varepsilon}_t - \bar X_t|^2)=0$. 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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

The 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.

What 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.

The 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.

Referee Report

3 major / 3 minor

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.

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 (3)

[Section 4, Lemma 4.4] 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.
[Theorem 4.7] 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 4, proof of Theorem 4.7] 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.

minor comments (3)

[Throughout] There are numerous typographical errors, including "Browinan motion" in Section 2 and "McKean-Vlasov stochastic differential equations" spelled inconsistently.
[Equation (4.17)] 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.
[References] Several references are incomplete: [12] lacks a journal or volume information, and [13] appears without a journal volume or page range.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the averaging argument is structurally non-circular, though Lemma 4.4 contains a genuine mathematical error about dependence on the background space.

full rationale

The derivation is not circular in the sense prohibited by the rubric. The existence proof is a standard fixed-point contraction on L^2, and the averaging proof compares X^ε with X̄ through a triangular decomposition of the drift, diffusion and fractional-noise terms, then closes with Gronwall; the target convergence is not an input to any of the estimates. No parameter is fitted to the final quantity, and no 'prediction' is defined in terms of the quantity it predicts. The only self-citations, [14] for the iteration method and for Lemma 4.4's method, are methodological; the relevant inequality is restated in the paper rather than merely imported, so the citation is not load-bearing as an unverified external theorem. The serious defect is mathematical, not circular: Lemma 4.4 asserts an inequality because 'the process X̄_t does not depend on Ω0', but X̄_0 = ξ is F^0-measurable, so X̄_t does depend on ω0 through the initial condition. That makes the key estimate false, but a false estimate is not the same as deriving the conclusion from its own assumption. Accordingly, the circularity score is minimal.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper relies on two explicit domain assumptions (2.1 and 4.1) plus an unstated and false premise in Lemma 4.4 that the averaged process is independent of the background filtration. No free parameters are fitted to data, and no new entities are invented.

assumptions (3)

domain assumption Assumption 2.1: coefficients b, σ_W, σ_H are Lipschitz in state and Wasserstein distance, with square-growth bounded by a non-decreasing bounded function K1(t).
Used throughout the paper to prove wellposedness and moment bounds. It is a standard regularity condition for McKean-Vlasov SDEs.
domain assumption Assumption 4.1: time-averages of b, σ_W, σ_H converge to averaged coefficients at rate K2(T) → 0.
This is the ergodicity condition that underpins the averaging principle. It is stated explicitly and commonly used in averaging literature.
ad hoc to paper The averaged process X̄_t is independent of the background space Ω0.
This premise is used in Lemma 4.4 to justify the inequality W_2^2(L(X^ε_t|F^0_t), L(X̄_t)) ≤ E1|X^ε_t - X̄_t|^2. It is not stated as an assumption and is false because X̄_0 = ξ, a random variable on Ω0.

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Pith. "Pith review of Wellposedness and averaging principle for conditional distribution dependent SDEs driven by standard Brownian motions and fractional Brownian motions." pith.science (2026). https://pith.science/paper/ZX7SR33M

@misc{pith2026250421268,
  author       = {Pith},
  title        = {Pith review of: Wellposedness and averaging principle for conditional distribution dependent SDEs driven by standard Brownian motions and fractional Brownian motions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZX7SR33M}},
  note         = {Machine review of arXiv:2504.21268}
}

abstract

In this paper, we study a conditional distribution dependent stochastic differential equations driven by standard Brownian motion and fractional Brownian motion with Hurst exponent $H>\frac{1}{2}$ simultaneously. First, the existence and uniqueness of the equation is established by the fixed point theorem. Then, we show that the solutions of conditional distribution dependent stochastic differential equations can be approximated by the solutions of the associated averaged distribution dependent stochastic differential equations.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

18 extracted references · 17 canonical work pages

[1]

McKean H.P., A class of Markov processes associated with nonlinea r parabolic equa- tions, Proc. Natl. Acad. Sci. USA 56: 1907-1911, 1996

work page 1907
[2]

Sznitman A.S., Topics in propagation of chaos, Notes in Math., 1464 : 165-251, 1991

work page 1991
[3]

Mishura Y., Veretennikov A.Y., Existence and uniqueness theorem s for solutions of McKean-Vlasov stochastic equations, Theor. Probab. Math. Sta t., 46867923, 2020

work page 2020
[4]

Dynam., 19(06): 1950042, 2019

Mehri S., Stannat W., Weak solutions to Vlasov-McKean equations under Lyapunov- type conditions, Stoch. Dynam., 19(06): 1950042, 2019

work page 2019
[5]

De Raynal P., Strong well posedness of McKean-Vlasov stochas tic diﬀerential equations with H¨ older drift, Stoc. Proc. Appl., 130(1): 79-107, 2020

work page 2020
[6]

Pascucci A., Rondelli A., Veretennikov A.Y., Existence and uniquen ess results for strongly degenerate McKean-Vlasov equations with rough coe ﬃcients, arXiv: 2409.14451, 2024

arXiv 2024
[7]

Huang X., Wang F.Y., Distribution dependent SDEs with singular coeﬃ cients, Stoch. Proc. Appl., 129(11): 4747-4770, 2019

work page 2019
[8]

Dyn., 22: 00093R2, 2022

Dieye M., Diop A., Mckibben M.A., Existence of solutions for mean-ﬁeld integrodiﬀer- ential equations with delay, Stoch. Dyn., 22: 00093R2, 2022

work page 2022

Show all 18 references

[9]

Fan X., Huang X., Suo Y., et al., Distribution dependent SDEs driven b y fractional Brownian motions, Stoc. Proc. Appl., 151: 23-67, 2022

2022
[10]

Shen G., Xiang J., Wu J.L., Averaging principle for distribution depen dent stochastic diﬀerential equations driven by fractional Brownian motion and sta ndard Brownian motion, J. Diﬀer. Equations, 321: 381-414, 2022. 17

2022
[11]

Galeati L., Harang F.A., Mayorcas A., Distribution dependent SDE s driven by additive fractional Brownian motion, Probab. Theor. Rel., 185: 251-309, 2 023
[12]

Erny X., Locherbach E., Loukianova D., White-noise driven condit ional McKean-Vlasov limits for systems of particles with simultaneous and random jumps, a rXiv: 2103.04100

arXiv
[13]

Shao J., Wei D., Propagation of chaos and conditional McKean-V lasov SDEs with regime-switching, Front. Math. China, DOI: s11464-021-0960-3 , 2021

2021
[14]

Shao J., Tian T., Wang S., Conditional Mckean-Vlasov SDEs with jum ps and Markovian regime-switching: Wellposedness, propagation of chaos, averagin g priciple, J. Math. Anal. Appl., 534: 128080, 2024

2024
[15]

Khasminskii R., On the principle of averaging the Itˆ o stochastic diﬀerential equations, Kibernetika, 4: 260-279, 1968

1968
[16]

Guerra J., Nualart D., Stochastic diﬀerential equations driven b y fractional Brownian motion and standard Brownian motion, Stoch. Anal. Appl., 26: 1053- 1075, 2008

2008
[17]

Fan X., Huang X., Suo Y., Yuan C., Distribution dependent SDEs driv en by fractional Brownian motions, Stoch. Proc. Appl., 151: 23-67, 2022

2022
[18]

, Salkeld W., Tugaut J., Freidlin-Wentzell LDP in path spa ce for McKean- Vlasov equations and the functional iterated logarithm law, Ann

Dos Reis G. , Salkeld W., Tugaut J., Freidlin-Wentzell LDP in path spa ce for McKean- Vlasov equations and the functional iterated logarithm law, Ann. Ap pl. Probab. 29: 1487-1540, 2019. 18

2019

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

McKean H.P., A class of Markov processes associated with nonlinea r parabolic equa- tions, Proc. Natl. Acad. Sci. USA 56: 1907-1911, 1996

work page 1907

[2] [2]

Sznitman A.S., Topics in propagation of chaos, Notes in Math., 1464 : 165-251, 1991

work page 1991

[3] [3]

Mishura Y., Veretennikov A.Y., Existence and uniqueness theorem s for solutions of McKean-Vlasov stochastic equations, Theor. Probab. Math. Sta t., 46867923, 2020

work page 2020

[4] [4]

Dynam., 19(06): 1950042, 2019

Mehri S., Stannat W., Weak solutions to Vlasov-McKean equations under Lyapunov- type conditions, Stoch. Dynam., 19(06): 1950042, 2019

work page 2019

[5] [5]

De Raynal P., Strong well posedness of McKean-Vlasov stochas tic diﬀerential equations with H¨ older drift, Stoc. Proc. Appl., 130(1): 79-107, 2020

work page 2020

[6] [6]

Pascucci A., Rondelli A., Veretennikov A.Y., Existence and uniquen ess results for strongly degenerate McKean-Vlasov equations with rough coe ﬃcients, arXiv: 2409.14451, 2024

arXiv 2024

[7] [7]

Huang X., Wang F.Y., Distribution dependent SDEs with singular coeﬃ cients, Stoch. Proc. Appl., 129(11): 4747-4770, 2019

work page 2019

[8] [8]

Dyn., 22: 00093R2, 2022

Dieye M., Diop A., Mckibben M.A., Existence of solutions for mean-ﬁeld integrodiﬀer- ential equations with delay, Stoch. Dyn., 22: 00093R2, 2022

work page 2022