{"id":"69357a1c-c8c6-4fbc-b2af-e3c10c14879a","arxiv_id":"2506.14939","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The projection method for coarse-graining stochastic differential equations does not generally match stochastic averaging under time-scale separation, and this paper gives conditions and counterexamples showing when they agree and when they do not.","lead":"Many physical systems are modeled by equations that mix slow variables we care about with fast variables that are hard to simulate. This paper compares two standard ways to remove the fast variables and shows that they can give different answers even when the fast variables move on a much shorter time scale.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.3 is conditional on the obtuse angle condition [C3] for the implicitly defined Gyöngy coefficients; the paper provides no class of systems where [C3] is verified and Note 5.4 defers this check to future work.","rationale":"The reader's weakest assumption identifies the same load-bearing concern: the proof of Theorem 5.3 depends on the non-autonomous theory in Theorem 5.1, and the condition [C3] is the part of Assumption A1 that is both essential for uniqueness of the evolution system of measures and impossible to verify from the original system without knowing the full conditional law of the process. The paper is honest about this in Note 5.4, but the consequence is that the central theorem is a conditional statement whose hypothesis is left uninstantiated. The reader's verdict CONDITIONAL is appropriate: the argument is structurally plausible and the conclusion may well be true for many systems, but the main theorem currently rests on an assumption that is not connected to checkable conditions on f,g,alpha,beta and is known to be critical (Example 5.3 shows the long-time identification fails when related structural conditions fail). I do not see a reason to change the verdict: the concern does not disprove the theorem, but it does mean the theorem's practical scope is unresolved. The proposed numerical test would provide a first concrete data point on whether [C3] is ever satisfied in a nonlinear system, which would either mitigate or sharpen the concern.","tokens_in":31237,"tokens_out":23296,"duration_ms":237805,"concrete_test":"In the scalar additive-noise case, [C3] reduces to d_x f_G(t,x) <= -lambda0 for all t,x. For a uniformly elliptic two-dimensional system with known invariant measure, e.g. dX = (-X^3 + Y)dt + dU, dY = -Y dt + dW, numerically solve the Fokker-Planck equation for rho_t(x,y) on a finite time horizon, compute f_G(t,x) = E[Y | X_t=x] plus the -X^3 term, and estimate its x-derivative; if sup_{t,x} d_x f_G(t,x) > 0, then [C3] fails for a smooth system satisfying assumptions i)-ii) of Theorem 5.3, showing that the sufficient condition is not automatically met and requires case-by-case certification. If the sup is negative, evaluate the full commutator expression for [C3] using the estimated diffusion coefficient alpha_G(t,x); this settles whether Assumption iii) can hold in a nontrivial nonlinear example.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim of the paper is Theorem 5.3, and its proof is a two-step chaining: Proposition 3.2 identifies the law of the Gyöngy SDE (8) with the x-marginal of the full system, and Theorem 5.1 transfers the long-time limit of the non-autonomous coefficients (8) to their autonomous limit, which is asserted to be the projection dynamics (11). The decisive hypothesis is Assumption iii), in particular [C3], the obtuse angle condition on the commutator of the Stratonovich drift and diffusion vector fields of (8). These fields are defined through the time-dependent conditional density rho_t(y|x), which is not explicit outside Gaussian/linear settings. Note 5.4 concedes that checking Assumption iii) is difficult and defers it to future work. If [C3] fails, the uniqueness of the evolution system of measures in Theorem 5.1 is not available, and the identification of the long-time behaviour of XG with that of XP breaks down; Section 5.3 gives a concrete illustration of such a breakdown, albeit outside uniform ellipticity. Thus the main theorem's applicability is gated by an unverified, intrinsically nonlocal condition on the full conditional law of the process, and the paper does not exhibit any nontrivial system in which [C3] is certified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies coarse-graining of stochastic differential equations by comparing the classical averaging principle with the projection method (PM), which constructs an effective dynamics by conditional expectation with respect to the equilibrium conditional distribution of the unresolved variables. The authors link the PM to Gyöngy's mimicking-marginals method, prove that the PM generator leaves the marginal of the original invariant measure infinitesimally invariant (Proposition 3.1), give necessary and sufficient conditions for the equilibrium conditional distribution to coincide with the frozen-process invariant measure (Proposition 4.1), and state a theorem (Theorem 5.3) under which the Gyöngy dynamics and the PM dynamics have the same long-time behavior, so that the PM samples the correct marginal. The comparison between PM and averaging is illustrated by examples and counterexamples, including a formal asymptotic expansion of the conditional distribution in the slow-fast limit. The main theorem relies on a non-autonomous SDE result (Theorem 5.1) adapted from the authors' prior work [CCDO21] and [ALL13].","tokens_in":31518,"tokens_out":19672,"duration_ms":168350,"significance":"If the main results hold, the paper provides a reversibility-independent justification of the projection method and clarifies its relationship to Gyöngy's exact mimicking construction. Proposition 3.1 and Proposition 4.1 are clean and useful, and the counterexample in Section 5.3 is instructive. The formal expansion in Section 4.2 is also a helpful heuristic. However, the central Theorem 5.3 is conditional on the obtuse angle condition [C3] imposed on the implicitly defined Gyöngy coefficients, and the paper does not exhibit any non-trivial system for which this condition is verified; Note 5.4 explicitly defers this check to future work. This substantially limits the practical applicability of the main theorem. In addition, several displayed formulas in the examples are incorrect. With corrections and a more honest framing of the theorem's scope, the paper could be a valuable contribution to the coarse-graining literature.","major_comments":[{"comment":"The central theorem is gated by the obtuse angle condition [C3], which is imposed on the Stratonovich vector fields of the Gyöngy SDE (8). Since those fields are defined through the time-dependent conditional density ρ_t(y|x), which is explicit only in Gaussian or linear settings, [C3] is not checkable for typical systems. Note 5.4 concedes that checking Assumption iii) 'might be difficult' and defers it to future work. The paper provides no non-trivial class of systems for which [C3] is certified; the only sufficient condition mentioned (the two-dimensional case with α=σ, where [C3] reduces to ∂_x f_G ≤ −λ_0) is not applied to any example. The introduction's claim that the PM works for 'a large class of SDEs' is therefore not substantiated by the results. The authors should either add a non-trivial class of systems where [C3] is verified, or explicitly state in the abstract and introduction that Theorem 5.3 is conditional and that no instance of its hypotheses is provided.","section":"§5.2, Theorem 5.3 and Assumption A1 [C3], with Note 5.4"},{"comment":"The paper claims to provide sufficient conditions under which the PM and averaging coincide in the limit of time-scale separation, but the derivation of the limit (29) is formal. The text states: 'To turn the above formal expansion into an actual proof one needs to combine these results on Poisson equations together with a more careful study of the difference ρε − ρ(x)(y) − ερ1(x,y), which needs to be shown to converge to zero as ε → 0. While not too difficult, this is not within the scope of this paper.' Thus the rigorous comparison of PM and averaging in the slow-fast regime is not established. Since this comparison is one of the paper's two advertised contributions, the formal status of this part should be clearly flagged in the abstract and introduction; as written, the reader may overestimate the strength of the results.","section":"§4.2, formal asymptotic expansion and the limit (29)"}],"minor_comments":[{"comment":"The conditional variance in equation (26) is incorrect. For the joint Gaussian invariant measure with covariance matrix (25), the conditional variance of Y given X=x is 1 − (ε/(1+ε))^2 / (ε/(1+ε)) = 1/(1+ε), not (1−ε)/(1+ε). For ε=1, the stated formula gives variance zero, contradicting the direct computation in Section 5.3 where the correct value is 1/2.","section":"Example 2.1, equation (26)"},{"comment":"The conditional variance formula is incorrect. Given the stated covariance matrix Σε, the conditional variance is 1 − ε^2/((1+ε)(1+2ε)) = (1+3ε+ε^2)/(1+3ε+2ε^2), not (1−ε^2)/(1+3ε+4ε^2). The qualitative conclusion that ρε(·|x) converges to N(0,1) as ε→0 remains correct with the corrected formula, but the displayed expression should be fixed.","section":"Example 4.6"},{"comment":"The 'for instance' construction of the vector field A is incorrect: the condition is ∇_x V = ∇_y A, so the correct formula is A(x,y) = ∫_0^y ∂_x V(x,w) dw, not ∫_0^y ∂_y V(x,w) dw as written.","section":"Example 4.4"},{"comment":"The bullet list in Note 4.2 is malformed: bullets appear mid-sentence ('• As a direct consequence of Proposition 4.1, • if α12(x,y)=0 ... • states that L′ρ=0'). Please reformat so that the text reads as complete, coherent sentences.","section":"Note 4.2"},{"comment":"The abstract displays 'A VERAGING' instead of 'AVERAGING' in the title; this typo should be corrected.","section":"Title/Abstract"},{"comment":"The claim that 'it follows by Itô's formula and a standard Gronwall estimate that ... the projected dynamics converges pathwise' is not proved or even sketched. A brief indication of the estimate would strengthen the example.","section":"Example 4.6, pathwise convergence claim"}],"recommendation":"major_revision","confidential_remarks":"The paper's central theorem is a relatively direct consequence of the authors' own previous work [CCDO21] combined with Gyöngy's construction; the novelty lies in the framework and the caveats. The authors are honest about the difficulty of verifying [C3], but the absence of any positive example where the main theorem applies is a serious weakness for a journal publication. The explicit formula errors in Examples 2.1, 4.6, and 4.4 should be corrected. I see no circularity problem, since Theorem 5.1 is an external published result and Proposition 3.2 is proved in the paper. The paper would be acceptable after the authors provide at least one non-trivial instance where [C3] is certified, or substantially temper the claims in the abstract and introduction."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things you should know. First, the paper gives a genuinely useful clarification: for slow-fast SDEs, averaging and the projection method (PM) do not generally coincide, and it provides a simple Gaussian counterexample (Example 2.1) that makes the point cleanly. Second, the main theorem (5.3) is a conditional result: it guarantees that the PM produces the correct invariant marginal only when the coefficients of the Gyöngy SDE satisfy an obtuse angle condition ([C3]) that is, by the authors' own admission in Note 5.4, difficult to check in practice. That is not a hidden flaw—they say it plainly—but it does limit the theorem's immediate applicability.\n\nWhat is actually new: the rigorous link between the Gyöngy method and the PM (Proposition 3.2), the characterization of when the frozen-process invariant measure coincides with the equilibrium conditional distribution (Proposition 4.1), and the formal expansion in Section 4.2 showing conditions under which the PM and averaging agree in the ε→0 limit. The paper also does a service by showing, via two non-reversible examples, that reversibility is not the deciding factor. That corrects an easy misreading of the literature.\n\nThe soft spots are real but not fatal. The explicit conditional variance formulas in Examples 4.5 and 4.6 are wrong: in 4.5 the variance should be 1/(1+ε), not (1−ε)/(1+ε), and the variance formula in 4.6 does not match the stated covariance matrix when you compute it directly. In both cases the qualitative conclusion survives because the drift uses the conditional mean (which is correct) and the diffusion coefficient in those examples is constant. Still, displayed errors in a counterexample section are a poor look, and a referee should demand the formulas be corrected.\n\nThe bigger question is whether Theorem 5.3, gated as it is by [C3], is a theorem users can actually deploy. My reading: it is honest sufficient-condition work. The authors show a one-dimensional check (if ∂ₓfG ≤ −λ0 then [C3] holds) and they explicitly delegate the general check to future work. That is a limitation, but not an invalidation. The theorem's logic is sound given the imported results from [ALL13, CCDO21]; the citation pattern is fine, and the reliance on their own prior work is legitimate because those results are external and published.\n\nWho is this for? Researchers working on effective dynamics, multiscale SDEs, or model reduction. They will find the exact statement of when PM and averaging disagree useful, and the counterexample is a good teaching tool. I would bring it to the reading group and would cite it for the counterexample (after checking the corrected formulas). It deserves a serious referee—not a desk reject—with a request for corrected examples and a more explicit discussion of how restrictive [C3] actually is.","headline":"A useful, honest clarification of when projection-type coarse graining and averaging disagree, with sufficient-condition results that are checkable in principle but hard to verify in practice; the explicit errors in the counterexample section are sloppy but not load-bearing.","tokens_in":32055,"tokens_out":3640,"would_cite":true,"duration_ms":33565,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["60H10","34F05","60J60","34K33","35B40","82C31"],"pacs":[],"model":"deepseek-v4-flash","headline":"Under the assumptions of Theorem 5.3, the projection-method reduced dynamics has the same long-time statistics as the slow variables of the full SDE, and its equality with averaging holds only under explicit conditions.","keywords":["coarse graining","stochastic differential equations","projection method","averaging","Gyöngy mimicking marginals","equilibrium conditional density","slow-fast systems","two-parameter Markov semigroups"],"falsifier":"Take the uniform-elliptic modification of Example 2.1, i.e. add an independent noise term $\\sqrt{2}\\,dU_t$ to the slow equation (66), so the joint law is Gaussian with an explicit invariant measure. Compute the projection-method coefficients and the Gyöngy coefficients from the Gaussian conditional densities, check the obtuse-angle inequality of condition [C3] directly, and compare the invariant measure of the projected dynamics with the marginal $\\bar\\rho=N(0,1/2)$; a mismatch while [C3] holds would refute Theorem 5.3, and a match in a regime where [C3] fails would show the condition is too strong.","tokens_in":31046,"feed_emoji":"🧩","tokens_out":10774,"duration_ms":103896,"temperature":0.7,"pith_summary":"Two standard ways to reduce a system of stochastic differential equations to the variables that matter are classical averaging, which exploits a separation of time scales, and the projection method, which replaces the eliminated variables by their equilibrium conditional distribution given the remaining ones. This paper's first claim is that, under explicit conditions, the projection-method reduced dynamics has a unique invariant measure, equal to the marginal of the full system's invariant measure on the slow variables; long-time averages of the reduced dynamics therefore reproduce the slow-variable statistics of the original system. The proof works by connecting the projection method to the Gyöngy mimicking-marginals construction, an exact but hard-to-simulate rewriting of the slow dynamics, and then showing the two have the same long-time behaviour. The paper's second claim is that, contrary to a common belief, the projection method and averaging do not in general coincide under time-scale separation; they coincide exactly under a kernel condition on the generators, and a degenerate-noise example shows the projection method can even produce a constant dynamics where averaging gives a nontrivial Ornstein-Uhlenbeck process.","feed_headline":"Projection coarse graining matches the original SDE's slow statistics","feed_subtitle":"Only under a contraction condition the reduced dynamics inherits the full system's slow-variable statistics.","key_machinery":"The load-bearing object is the Gyöngy SDE (8), $$dX^G(t)=f_G(t,X^G(t))\\,dt+\\$alpha_G^{{1/2}}$(t,X^G(t))\\,dU_t,$$ whose drift and diffusion are obtained by integrating the original coefficients against the conditional density $\\rho_t(y|x)$ of the full system. Because $\\rho_t(y|x)$ evolves in time, (8) is a non-autonomous SDE; the projection method $X^P$ is exactly the autonomous limit obtained by replacing $\\rho_t(y|x)$ with the equilibrium conditional density $\\rho(y|x)$. The argument runs through the theory of evolution systems of measures: condition [C3]---the obtuse-angle condition on the commutator $[\\sigma^{[i]}\\cdot\\nabla, B\\cdot\\nabla]$ of the drift and diffusion vector fields---ensures uniqueness of the evolution system of measures for the non-autonomous dynamics, so its long-time limit is the invariant measure of the limit autonomous equation, which is the projection dynamics. A second structural identity, Proposition 4.1, characterizes when the invariant measure of the frozen fast dynamics equals the equilibrium conditional density in terms of the kernels of the dual generators $L'_x$ and $L'_y$.","core_discovery":"The central discovery is that the autonomous SDE produced by the projection method, $$dX^P(t)=f_P(X^P(t))\\,dt+\\$alpha_P^{{1/2}}$(X^P(t))\\,dU_t, \\quad f_P(x)=\\int f(x,y)\\rho(y|x)\\,dy,\\quad \\alpha_P(x)=\\int \\$\\alpha$\\$\\alpha$^T(x,y)\\rho(y|x)\\,dy,$$ inherits its long-time behaviour from the Gyöngy SDE whose coefficients use the time-dependent conditional density $\\rho_t(y|x)$ instead of the equilibrium conditional density $\\rho(y|x)$. Theorem 5.3 states that if the original system has smooth, uniformly elliptic, ergodic coefficients, the Gyöngy SDE is well posed, and its non-autonomous coefficients satisfy Assumption A1---uniform ellipticity, a Lyapunov condition, the obtuse-angle commutator condition [C3], and convergence to the autonomous coefficients---then $X^P$ admits a unique invariant measure, equal to the marginal $\\bar\\rho$ of the invariant measure $\\rho$ of the full system, so $\\lim_{t\\to\\infty}\\mathbb{E}h(X^P(t))=\\int h(x)\\bar\\rho(x)\\,dx$ for bounded continuous $h$. The proof does not rely on reversibility or an explicit formula for $\\rho$, and it makes rigorous the relation between the projection method and Gyöngy's exact marginal-mimicking method.","pith_inferences":["Editorial inference: the commutator condition [C3] reduces in one dimension to the monotonicity test $\\partial_x f_G(t,x)\\le -\\lambda_0$, suggesting a cheap practical diagnostic---compute the conditional-drift derivative and check for contraction before trusting the projected reduced model.","Editorial inference: the formal expansion in Section 4.2 ties the validity of the $\\varepsilon\\to0$ limit to the strict positivity of the limiting marginal; when the slow marginal degenerates (as in Example 4.5), the equilibrium conditional density cannot converge to the frozen measure, so a practical rule emerges: expect projection and averaging to disagree whenever the slow-variable invariant me","Editorial inference: the same two-parameter-semigroup framework could be applied to other conditional-expectation reductions, such as nonlinear reaction coordinates, by substituting the reaction-coordinate conditional law for $\\rho_t(y|x)$.","Editorial inference: the degenerate-noise failure of the projection method indicates that the method should be applied only when the resolved variables carry their own noise; adding a small ellipticity to the slow equation and letting it vanish may provide a selection criterion among the many invariant measures, a possibility the paper does not explore."],"forward_implications":["If the conditions of Theorem 5.3 hold, practitioners can use the projection-method SDE to compute long-time averages of the slow variables without simulating the fast variables, with the guarantee that the limiting statistics coincide with the marginal $\\bar\\rho$ of the full invariant measure.","In the scale-separation limit, the equality of projection and averaging holds precisely when $\\lim_{\\varepsilon\\to0}\\rho_\\varepsilon(y|x)\\to\\rho^{(x)}(y)$, which the paper shows is equivalent to the marginal of the invariant measure remaining strictly positive in the limit and to the kernel condition of Proposition 4.1 being satisfied.","When the slow equation has degenerate noise, as in the Ornstein-Uhlenbeck example (66)-(67), the projection method can produce a zero-drift, zero-noise dynamics with a continuum of invariant measures, whereas averaging produces a nontrivial OU process; the two methods are then not only different, the projection method fails to capture the correct long-time behaviour.","For reversible (gradient) systems the two methods coincide because the frozen invariant measure equals the equilibrium conditional law; the paper's examples show that non-reversibility alone does not determine whether they coincide."],"supporting_citations":[{"why":"supplies the exact marginal-mimicking construction (8) whose law equals the slow marginal of the full system","marker":"[Gy¨ o86]"},{"why":"introduced the projection method via conditional expectations for SDEs and proved the reversible gradient case that this paper generalizes","marker":"[LL10]"},{"why":"supplies the long-time behaviour theorem for non-autonomous Kolmogorov equations used to identify the limiting invariant measure","marker":"[ALL13]"},{"why":"extends the non-autonomous theory to time- and space-dependent diffusions, relaxing the boundedness assumptions needed for the Gyöngy coefficients","marker":"[CCDO21]"},{"why":"provides the known invariance result for the projected marginal that Proposition 3.1 recaps and Theorem 5.3 strengthens","marker":"[ZHS16]"},{"why":"is the standard reference for the averaging principle and defines the frozen process and averaged coefficients used in the comparison","marker":"[PS08]"}],"fun_headline_variants":["Projection method for SDEs: when does it truly match?","Coarse graining SDEs: projection vs averaging, not always equal","Projection coarse graining: exact only under contraction","SDE reduction: projection matches slow stats only with conditions","New theorem clarifies when projection coarse graining works"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The time-dependent Gyöngy coefficients must satisfy the obtuse-angle commutator condition [C3], and because these coefficients are defined through the conditional density $\\rho_t(y|x)$, which is usually not known explicitly, the condition is hard to check; the paper states this difficulty explicitly in Note 5.4 and leaves it to future work.","fun_headline_variants_meta":{"raw":{"variants":["Projection method for SDEs: when does it truly match?","Coarse graining SDEs: projection vs averaging, not always equal","Projection coarse graining: exact only under contraction","SDE reduction: projection matches slow stats only with conditions","New theorem clarifies when projection coarse graining works"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000615,"raw_usage":{"total_tokens":2877,"prompt_tokens":988,"completion_tokens":1889,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":604,"completion_tokens_details":{"reasoning_tokens":1804}},"tokens_in":604,"tokens_out":1889,"duration_ms":14009,"temperature":1.0,"reasoning_tokens":1804,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T19:46:33.236363+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the uniform-elliptic modification of Example 2.1, i.e. add an independent noise term $\\sqrt{2}\\,dU_t$ to the slow equation (66), so the joint law is Gaussian with an explicit invariant measure. Compute the projection-method coefficients and the Gyöngy coefficients from the Gaussian conditional densities, check the obtuse-angle inequality of condition [C3] directly, and compare the invariant measure of the projected dynamics with the marginal $\\bar\\rho=N(0,1/2)$; a mismatch while [C3] holds would refute Theorem 5.3, and a match in a regime where [C3] fails would show the condition is too strong.","supporting_citations":[],"review_version":2}