Higher-order expansion for fluctuations around the homogenised semilinear random PDE of Hairer–Pardoux

Hairer and Pardoux (Arch. Ration. Mech. Anal. 239, 2021) proved a law of large numbers and a first-order central limit theorem for a semilinear parabolic PDE on $[0,1]^d$, $d\le 3$, driven by a critically scaled (amplitude $\varepsilon^{-1}$ under parabolic rescaling), highly oscillating, stationary and possibly non-Gaussian random potential, with homogeneous Dirichlet or Neumann boundary conditions: after subtracting the deterministic homogenised limit $u^{(0)}$ and, in $d\in\{2,3\}$, a first-order corrector $u^{(1)}$ whose Neumann boundary condition is modified by a boundary layer, the rescaled fluctuations converge to a Gaussian process solving a linear SPDE driven by space-time white noise. The paper itself conjectures the full higher-order expansion $\varepsilon^{-d/2}(u_\varepsilon-\sum_{k=0}^{\lfloor d/2\rfloor}\varepsilon^k u^{(k)})\to v$ with recursively defined deterministic correctors, and records the obstructions: its techniques yield no moment bounds on $u_\varepsilon$ (the source notes that even in $d\le 3$ the expansion of $\mathbb{E}u_\varepsilon$ would require additional work on this account); the boundary power-counting breaks at $d=4$, where correction terms charging codimension-2 faces are expected; the noise assumptions (tree-based cumulant summability plus an $L^p$ moment condition) are probably non-optimal; and the counterterm constants are identified only in principle via the BPHZ tree algebra.

Background

Homogenisation theory describes the large-scale behaviour of PDEs whose coefficients or potentials oscillate rapidly on a small spatial scale: when the oscillations mix sufficiently, the solution converges to that of an effective deterministic equation — a law of large numbers (LLN) — and the refined question is a central limit theorem (CLT) describing the rescaled fluctuations around that effective solution. Hairer and Pardoux (2021) studied the critical regime of this problem for semilinear parabolic equations: on $[0,T]\times[0,1]^d$ with $d\le 3$ and either homogeneous Dirichlet or Neumann boundary conditions, consider

$$\partial_t u_\varepsilon=\Delta u_\varepsilon+H(u_\varepsilon)+G(u_\varepsilon)\,\eta_\varepsilon,\qquad u_\varepsilon(0,x)=u_0(x),$$

where $H,G:\mathbb{R}\to\mathbb{R}$ are smooth nonlinearities and the random potential is a stationary, centred, possibly non-Gaussian field $\eta$ under parabolic rescaling with critical amplitude,

$$\eta_\varepsilon(t,x)=\varepsilon^{-1}\,\eta(\varepsilon^{-2}t,\,\varepsilon^{-1}x).$$

The field $\eta$ is assumed to have jointly controlled cumulants (the connected parts of moments, used to encode mixing: the cumulant of order $p$ decays as a power of the parabolic distance between its arguments at both small and large scales, with a tree-based summability condition) and, in $d\in\{2,3\}$, an $L^p$ moment condition with $p=(d+2)/c$ on the decay exponent $c$ — Assumption 2.1 of the paper, satisfied for example by a mollified Poisson field. For the linear case ($G$ linear, $H=0$) the LLN and CLT were obtained by Bal (2010) (Gaussian, time-independent potential) and Yu Gu and Guillaume Bal (2015) (non-Gaussian case); Bal and Jing (2016) treated semilinear equations with random potentials in a non-critical amplitude regime. The critical amplitude $\varepsilon^{-1}$ is the same at which the related Wong–Zakai problem for stochastic PDEs was analysed by Hairer and Pardoux (2015).

The source paper proves: (i) an LLN — $u_\varepsilon\to u^{(0)}$ in probability, uniformly on $[0,T]\times[0,1]^d$, where $u^{(0)}$ solves a deterministic parabolic PDE with renormalised drift $H_\eta(u)=H(u)+c_\eta\,G'(u)G(u)$, with $c_\eta$ an explicit constant obtained by integrating the heat kernel against the second cumulant of $\eta$ (Eq. (1.4) of the source); and (ii) a CLT — in $d\in\{2,3\}$, after subtracting $u^{(0)}$ and a first-order deterministic corrector $u^{(1)}$ (solving a linear parabolic equation whose inhomogeneity is built from $\nabla u^{(0)}$ and constants obtained by integrating the heat kernel against the second and third cumulants of $\eta$), one has convergence in law, in a Hölder space of low regularity,

$$\lim_{\varepsilon\to 0}\varepsilon^{-d/2}\big(u_\varepsilon-u^{(0)}-\varepsilon u^{(1)}\big)=v,$$

where $v$ solves a linear SPDE driven by additive space-time white noise (the generalised Gaussian field with covariance $\delta(t-t')\delta(x-x')$). In $d=1$ the limit is a centred Gaussian (Ornstein–Uhlenbeck-type) process; in $d=2$ it is non-centred unless $u^{(1)}$ is subtracted. A striking feature is that under Neumann boundary conditions the corrector $u^{(1)}$ obeys a non-homogeneous Neumann condition even though $u_\varepsilon$, $u^{(0)}$ and $v$ all satisfy homogeneous ones — a boundary-layer effect analogous to the boundary renormalisation arising for singular stochastic PDEs (SPDEs whose noise is too irregular for classical solution theories) on domains with boundaries, as developed by Gerencsér and Hairer (2019). The proofs rely on Hairer's theory of regularity structures (a framework that lifts a singular equation to a graded algebraic structure of models and modelled distributions on which analytic operations and renormalisation can be performed) together with BPHZ-type counterterms (the systematic renormalisation scheme of Bruned, Chandra, Chevyrev and Hairer (2021) assigning to each decoration tree a renormalisation constant) adapted to the non-Gaussian cumulant structure.

The paper stops at first order. Remark 1.6 records the expectation that in any dimension a full expansion should hold with recursively defined correctors, and identifies the obstruction: the techniques give no moment bounds on the solution, so even in $d\le 3$ the passage to the expansion of the expectation $\mathbb{E}u_\varepsilon$ requires additional work. In the model-convergence analysis the authors moreover show that a boundary power-counting sum diverges exactly at $d=4$, and predict that correction terms charging faces of codimension 2 (the $(d-2)$-dimensional edges of the cube) must be added there.

Problem Statement

Prove, or disprove, the higher-order fluctuation expansion conjectured in Remark 1.6 (Eq. (1.11)) of Hairer–Pardoux (2021). For the semilinear random PDE

$$\partial_t u_\varepsilon=\Delta u_\varepsilon+H(u_\varepsilon)+G(u_\varepsilon)\,\eta_\varepsilon,\qquad \eta_\varepsilon(t,x)=\varepsilon^{-1}\eta(\varepsilon^{-2}t,\varepsilon^{-1}x),$$

on $[0,T]\times[0,1]^d$ with homogeneous Dirichlet or Neumann boundary conditions, with $G$ of class $C^5$ and $H$ of class $C^4$, with an initial condition $u_0$ satisfying the regularity assumptions of the source's Theorem 1.2, and with $\eta$ a stationary centred random field satisfying Assumption 2.1 of the source, resolve with complete proofs: prove or disprove the conjectured expansion of Remark 1.6 (Proposition 1), in full or on an identified range of dimensions that includes at least one dimension $d\ge 4$, and prove or refute the expectation expansion (Proposition 2) — Proposition 2 implies Proposition 1, so a proof of Proposition 2 also establishes Proposition 1 and a disproof of Proposition 1 also refutes Proposition 2:

1. Full expansion in general dimension: in every $d\ge 1$ — in particular in $d\ge 4$, where the source's boundary power-counting breaks down — the convergence in law

$$\lim_{\varepsilon\to 0}\varepsilon^{-d/2}\Big(u_\varepsilon-\sum_{k=0}^{\lfloor d/2\rfloor}\varepsilon^k u^{(k)}\Big)=v$$

with convergence in $C^\alpha([0,T]\times[0,1]^d)$ for every $\alpha<1-d/2$ and every $d\ge 1$ (for $d\le 3$, as in the source's Theorem 1.2), where the deterministic correctors $u^{(k)}$ are defined recursively by linear parabolic equations $\partial_t u^{(k)}=\Delta u^{(k)}+H_\eta'(u^{(0)})u^{(k)}+\Psi_k$ with explicitly identified inhomogeneities $\Psi_k$ (functions of the lower correctors and their derivatives) and with the correct boundary conditions, including whatever modification of the Neumann condition the boundary layers induce at each order.

2. Expectation expansion: $\varepsilon^{-d/2}(u_\varepsilon-\mathbb{E}u_\varepsilon)\to v$ in law and $\mathbb{E}u_\varepsilon=\sum_{k=0}^{\lfloor d/2\rfloor}\varepsilon^k u^{(k)}+o(\varepsilon^{d/2})$ as $\varepsilon\to 0$.

Consistency requirement: for $d\le 3$ the correctors and the limit must reproduce Theorems 1.1–1.2 of the source (the $k=0,1$ correctors and the limiting SPDE for $v$). Remark 1.7 of the source indicates that the form of the higher corrector equations and their constants can in principle be derived from the BPHZ counterterm algebra of Bruned–Chandra–Chevyrev–Hairer, with numerical constants differing from those of that algebra because the source computes a 'centering' rather than a 'recentering' of the original equation.

Two auxiliary directions recorded in the source may be addressed within the same objective: weakening the noise Assumption 2.1 (the tree-based cumulant homogeneity and the $L^p$ condition, which the authors deem probably non-optimal) while preserving the above results; and giving a systematic algebraic/combinatorial computation of all counterterm coefficients for arbitrary trees, including their boundary analogues.

References

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