Quantitative homogenisation of stochastic PDEs with multiscale structure: Hilbert-Schmidt noise, convergence rates, and the integer-cell assumption

Hairer and Kelly (2012) initiated the quantitative homogenisation theory for one-dimensional linear parabolic stochastic PDEs whose differential operator and Gaussian forcing both oscillate at a common microscopic scale. Under algebraic-decay assumptions on the Fourier coefficients of the cell-periodic noise, they identified the homogenised limit, classified when the limit is surprising (for space-time white noise the effective noise is amplified by the L2 norm of the invariant cell measure), proved convergence with polynomial rates for the first-order limit, and obtained rescaled second-order limits, the latter only in law and without rates. Several structural restrictions were left open: the Hilbert-Schmidt noise class is excluded from the stated theorems, no rates are available for the rescaled limits, the domain must contain an integer number of periodic cells, and the theory covers only additive noise. Later work (2023-2025) resolved the additive-noise restriction for periodic coefficients by establishing homogenisation for nonlinear singular SPDEs, discovering that homogenisation and renormalisation interact through additional counterterms. The problem asks for these remaining components of the quantitative theory.

Background

Homogenisation theory describes the effective large-scale behaviour of equations whose coefficients vary rapidly on a small spatial scale $\varepsilon$. Its one-dimensional prototype is a diffusion on $[0,2\pi]$ with generator $L=b(x)\partial_x+\frac12\sigma^2(x)\partial_x^2$, where $b$ and $\sigma$ are smooth $2\pi$-periodic functions, $\sigma$ is bounded away from zero, and the centering condition $\int_0^{2\pi}b(v)/\sigma^2(v)\,dv=0$ holds. The diffusively rescaled generator $L^\varepsilon=\frac1\varepsilon b(x/\varepsilon)\partial_x+\frac12\sigma^2(x/\varepsilon)\partial_x^2$ then converges, in the sense that the associated diffusions converge to a Brownian motion with variance given by an effective diffusivity $\mu>0$ determined by $b$ and $\sigma$; equivalently, solutions of parabolic equations $\partial_t u^\varepsilon=L^\varepsilon u^\varepsilon+f$ converge to solutions of a constant-coefficient limit equation.

Hairer and Kelly asked what survives of this picture when the forcing $f$ is replaced by a genuinely random, distribution-valued term. Their model is the stochastic PDE (SPDE)

$du^\varepsilon(x,t)=L^\varepsilon u^\varepsilon(x,t)\,dt+\sum_{k\in\mathbb{Z}}q_k(x/\varepsilon)\,e^{ikx}\,dW_k(t)$

on $[0,2\pi]$ with periodic boundary conditions, zero initial condition and $\varepsilon^{-1}\in\mathbb{N}$ (the domain contains an integer number of cells of length $2\pi\varepsilon$). The $W_k$ are independent complex-valued Brownian motions with $W_{-k}=\overline{W_k}$, and the $q_k$ are $2\pi$-periodic functions; this structure encodes cell-translation invariance of the noise, meaning that the material has the same microscopic structure in every cell. Solutions are understood in the mild sense $u^\varepsilon(t)=\int_0^t S^\varepsilon(t-s)Q^\varepsilon\,dW(s)$, where $S^\varepsilon$ is the semigroup generated by $L^\varepsilon$ and $Q^\varepsilon e_k=q_k(x/\varepsilon)e_k$. Space-time white noise, the centred Gaussian field $\xi$ with covariance $\mathbb{E}\xi(s,x)\xi(t,y)=\delta(s-t)\delta(x-y)$, is the special case $q_k\equiv 1$.

The headline phenomenon is that for space-time white noise the naive guess is wrong: instead of $\partial_t u=\mu\partial_x^2u+\xi$, the true limit is $\partial_t u=\mu\partial_x^2u+\|\rho\|\,\xi$, where $\rho$ is the normalised invariant measure of the cell diffusion with generator $L$ and $\|\cdot\|$ is the normalised $L^2$ norm. By Jensen's inequality $\|\rho\|\ge 1$ with equality if and only if $\rho$ is constant (equivalently, $L$ is in divergence form), so non-trivial microstructure always amplifies the noise. The mechanism is a resonance between the Fourier modes of the fast diffusion, which concentrate near wave numbers that are integer multiples of $1/\varepsilon$, and the corresponding modes of the rough forcing; such resonances are negligible for sufficiently regular forcing, which is why earlier SPDE homogenisation results (Ichihara 2004; Wang, Cao and Duan 2007), which assumed Hilbert-Schmidt noise, saw only the classical limit.

Hairer and Kelly then classified the possible limits by the decay of the norms $\|q_k\|$. Theorem 1.3 assumes algebraic decay, $\|q_k\|\lesssim 1\wedge|k|^{-\alpha}$ for some $\alpha\in(0,1)$ (plus an $H^1$ bound on the normalised profiles $q_k/\|q_k\|$ when $\alpha\le 1/2$; $H^s$ denotes the usual $L^2$-Sobolev space on $[0,2\pi]$), and proves convergence in mean square against test functions $\varphi\in H^s$ for large enough $s$, with a polynomial rate: $\mathbb{E}\sup_{t\le T}|\langle u^\varepsilon(t)-u(t),\varphi\rangle|^2\le C_T\varepsilon^\theta$. The limit is the unsurprising homogenised SPDE $du=\mu\partial_x^2u\,dt+\sum_k\langle q_k,\rho\rangle e^{ikx}\,dW_k(t)$. Theorem 1.6 assumes a convergent profile, $\| |k|^\alpha q_k-\bar q\|\to 0$ in $L^2$, together with the centering $\langle q_k,\rho\rangle=0$; it shows that the rescaled solutions $\varepsilon^{-\alpha}\hat u^\varepsilon$, for processes $\hat u^\varepsilon$ equal in law to $u^\varepsilon$ but reconstructed on a different probability space, converge to a limit driven by space-time white noise arising from constructive interference in the very high modes; this second-order term bridges the surprising and unsurprising regimes. Theorem 1.7 treats the $\alpha=0$ class (including space-time white noise) under the assumption that $q_k\to\bar q$ in $H^1$ with $\sum_k(1\wedge|k|^{-\eta})\|q_k-\bar q\|_{H^1}^2<\infty$; the limiting noise combines the first-order coefficient $\langle q_k,\rho\rangle$ with the resonant contribution of the profile $\bar q$. The convergences in Theorems 1.6 and 1.7 hold only in law, with no rates, and the paper states that in general nothing stronger is possible.

The noise classes involved are subtle. The Hilbert-Schmidt condition $\sum_k\|q_k\|^2<\infty$, the assumption of the earlier literature, which makes the forcing a genuine $L^2$-valued process, neither implies nor is implied by the algebraic-decay conditions: a sufficiently sparse sequence with only logarithmic decay is square-summable yet violates the decay assumption. Remark 1.4 of the paper records that with only the Hilbert-Schmidt assumption one should still be able to prove via a tightness argument that the weak limit exists and solves the unsurprising homogenised SPDE, but that the authors "will not treat this case as it is somewhat incongruous with the existing framework"; Remark 2.8 further notes that square-summability to a fixed profile, $\sum_k\|q_k-\bar q\|^2<\infty$ with $\bar q\in H^1$, suffices for Theorem 1.7, which for $\bar q=0$ is precisely the Hilbert-Schmidt condition, though only as an asserted sufficient condition without a stated theorem, without rates, and without second-order asymptotics.

After 2012 the centre of gravity of this field moved to nonlinear equations. Singular SPDEs, such as the dynamical $\Phi^4$ models (stochastic heat equations with polynomial nonlinearities) and the parabolic Anderson model (a stochastic heat equation whose random potential multiplies the solution), are equations whose forcing is so rough that solutions exist only after subtracting divergent counterterms, a procedure called renormalisation, carried out through the theories of regularity structures and para-controlled distributions, the two standard solution frameworks for such equations. Between 2023 and 2025 a series of works established periodic homogenisation for such singular SPDEs: Hairer and Singh for $\Phi^4_2$, Chen and Xu for $P(\phi)_2$, Chen, Fehrman and Xu for the two-dimensional generalised parabolic Anderson model, Chen and Xu (via operator-dependent para-controlled calculus) for $\Phi^4_3$ and KPZ, and Hairer and Singh in a general regularity-structures framework for oscillatory g-PAM and $\Phi^4_3$. These works discovered that homogenisation and renormalisation interact: additional explicit constants depending on the coefficient field must be subtracted, the nonlinear analogue of the noise enhancement factor $\|\rho\|$. All of them assume periodic coefficients and an integer number of cells. What they do not touch are the quantitative questions of the original linear program (rates for Hilbert-Schmidt noise, rates for the rescaled second-order limits, second-order asymptotics without profile structure). Chen, Fehrman and Xu separately remark that combining singular SPDEs with stochastic homogenisation, meaning homogenisation of equations with stationary random coefficients, "will be a much greater challenge, and will be worth investigation in the long term"; that remark concerns nonlinear singular SPDEs requiring renormalisation rather than the linear equation treated here, and non-periodic microstructure lies outside the scope of the present problem.

Problem Statement

The problem is to complete the quantitative homogenisation theory of Hairer and Kelly (2012) for the linear parabolic SPDE

$du^\varepsilon(x,t)=L^\varepsilon u^\varepsilon(x,t)\,dt+\sum_{k\in\mathbb{Z}}q_k(x/\varepsilon)\,e^{ikx}\,dW_k(t)$

on $[0,2\pi]$ (periodic boundary conditions, zero initial condition, smooth $2\pi$-periodic coefficients $b$ and $\sigma$ with $\sigma$ bounded away from zero and the centering condition $\int_0^{2\pi}b(v)/\sigma^2(v)\,dv=0$), by removing its structural restrictions while retaining its two quantitative objectives: identification of the effective limiting SPDE and a convergence rate. It has two primary components, (A) and (B), together with a secondary technical component (D); component (A) comprises the sub-questions (A1)-(A3). The sub-questions (A1), (A2), (A3), (B) and (D) are independent of one another, and each admits a rigorous affirmative or negative resolution.

(A) Hilbert-Schmidt noise. Assume only $\sum_{k\in\mathbb{Z}}\|q_k\|^2<\infty$, with no algebraic decay and no convergent profile for the $q_k$ (for example, sparse sequences with merely logarithmic decay, which are square-summable but fail the decay assumption of Theorem 1.3). (A1) Prove a stated theorem identifying the limit: does $u^\varepsilon$ converge to the unsurprising homogenised SPDE $du=\mu\partial_x^2u\,dt+\sum_k\langle q_k,\rho\rangle e^{ikx}\,dW_k(t)$, as the tightness route sketched in Remark 1.4 and the sufficient condition of Remark 2.8 (with $\bar q=0$) suggest, or does the limit differ for some Hilbert-Schmidt noise? (A2) Determine whether a quantitative rate holds, in the sense of an analogue of Theorem 1.3's bound $\mathbb{E}\sup_{t\le T}|\langle u^\varepsilon(t)-u(t),\varphi\rangle|^2\le C_T\varepsilon^\theta$ for Sobolev test functions $\varphi$, and how any such rate depends on the decay profile of the $\|q_k\|$; or prove that no algebraic rate exists in general within this class. (A3) Determine whether an analogue of the second-order asymptotics of Theorems 1.6 and 1.7 exists for Hilbert-Schmidt noise without profile structure, that is, whether some rescaling of $u^\varepsilon$ by a power of $\varepsilon$ isolates a leading vanishing term with a non-trivial limit driven by resonant high modes, and identify that limit; or prove that no such second-order term exists beyond the first-order limit.

(B) Rates for the rescaled limits. Under the source's profile assumption for Theorem 1.6 (there exist $\alpha\in(0,1)$ and a profile $\bar q$ with $\bigl\||k|^\alpha q_k-\bar q\bigr\|_{L^2}\to0$ as $|k|\to\infty$, together with the centering $\langle q_k,\rho\rangle=0$) or its $H^1$-summability assumption for Theorem 1.7 (there exist $\bar q\in H^1$ and $\eta\in[0,1)$ with $\sum_{k\in\mathbb{Z}}(1\wedge|k|^{-\eta})\|q_k-\bar q\|_{H^1}^2<\infty$), or explicitly stated strengthenings of either, determine whether the convergences in law of Theorem 1.6 ($\varepsilon^{-\alpha}\hat u^\varepsilon\to v$) and Theorem 1.7 ($\hat u^\varepsilon\to\hat u$) admit quantitative rates; this requires first making the notion of a rate precise for convergence in law of the reconstructed processes. Either establish such rates, or prove that no algebraic rate is possible in general, which would make precise the source's assertion that "in general, nothing stronger is possible".

(D) Integer-cell assumption. A secondary technical extension from the source, it asks to remove the assumption that the domain contains an integer number of cells ($\varepsilon^{-1}\in\mathbb{N}$), retaining the same pair of conclusions, an identified effective limit and a quantitative convergence statement.

The components share one scientific objective: for each structural restriction that the Hairer-Kelly framework imposes, deliver the same pair of conclusions, an identified effective limit and a quantitative convergence statement, outside that restriction. Components (A) and (B) are two faces of a single quantitative question inside the same one-dimensional linear framework, (A) for an excluded noise class and (B) for an excluded convergence mode. Rigorous negative results, such as an explicit Hilbert-Schmidt sequence whose limit differs from the expected one, or a proof that algebraic rates are impossible, resolve the corresponding sub-question.

References

1. Martin Hairer and David Kelly. Stochastic PDEs with multiscale structure. Electronic Journal of Probability 17 (2012), 1-38. DOI: 10.1214/ejp.v17-1807, https://doi.org/10.1214/ejp.v17-1807 (preprint arXiv:1202.1775, https://arxiv.org/abs/1202.1775).
2. Martin Hairer and Harprit Singh. Periodic space-time homogenisation of the ϕ^4_2 equation. Journal of Functional Analysis 288 (5) (2025), article 110762. DOI: 10.1016/j.jfa.2024.110762, https://doi.org/10.1016/j.jfa.2024.110762 (preprint arXiv:2311.15788, https://arxiv.org/abs/2311.15788).
3. Yilin Chen and Weijun Xu. Periodic homogenisation for P(phi)_2. Preprint (2023), arXiv:2311.16398, https://arxiv.org/abs/2311.16398.
4. Yilin Chen, Benjamin Fehrman, and Weijun Xu. Periodic homogenisation for two dimensional generalised parabolic Anderson model. Probability Theory and Related Fields 194 (2026), 1235-1286. DOI: 10.1007/s00440-026-01465-1, https://doi.org/10.1007/s00440-026-01465-1 (preprint arXiv:2401.05718, https://arxiv.org/abs/2401.05718).
5. Yilin Chen and Weijun Xu. Operator dependent para-controlled calculus, periodic homogenisation and singular PDEs. Preprint (2024), arXiv:2404.16320, https://arxiv.org/abs/2404.16320.
6. Martin Hairer and Harprit Singh. Homogenisation of singular SPDEs. Preprint (2025), arXiv:2510.19339, https://arxiv.org/abs/2510.19339.
7. Annika Lang, Per Ljung, and Axel Målqvist. Localized Orthogonal Decomposition for a Multiscale Parabolic Stochastic Partial Differential Equation. Multiscale Modeling and Simulation 22 (1) (2024), 204-229. DOI: 10.1137/23m1569216, https://doi.org/10.1137/23m1569216 (preprint arXiv:2304.14049, https://arxiv.org/abs/2304.14049).
8. Wei Wang, Daomin Cao, and Jinqiao Duan. Effective Macroscopic Dynamics of Stochastic Partial Differential Equations in Perforated Domains. SIAM Journal on Mathematical Analysis 38 (5) (2007), 1508-1527. DOI: 10.1137/050648766, https://doi.org/10.1137/050648766.
9. Naoyuki Ichihara. Homogenization problem for stochastic partial differential equations of Zakai type. Stochastics and Stochastic Reports 76 (3) (2004), 243-266. DOI: 10.1080/10451120410001714107, https://doi.org/10.1080/10451120410001714107.