Variance renormalisation in regularity structures -- the case of $2d$ gPAM

Variance renormalisation in regularity structures – the case of 2d gPAM February 20, 2026 Máté Gerencsér1 and Yueh-Sheng Hsu2 1 TU Wien, Austria. Email: mate.gerencser@tuwien.ac.at 2 TU Wien, Austria. Email: yueh-sheng.hsu@tuwien.ac.at Abstract We consider the variance renormalisation of a singular SPDE for which a Da Prato-Debussche trick is not applicable. The example taken is the 2-dimensional generalised parabolic Anderson model (gPAM), driven by a much rougher than white noise, necessitating both a multiplicative and an additive renormalisation. To handle the discrepancy between the regularity structures of the approximate and the limiting equations, we consider models that lift 0 noises to nontrivial models, in analogy with “pure area” from rough paths. The convergence to such a model is shown for the BPHZ model over the vanishing noise via graphical computations. Contents 1 Introduction 1 2 The deterministic step 4 2.1 Regularity structure setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Pure area noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Proof of Theorem 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 The probabilistic step 11 3.1 The graphical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.2 The new white noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 Completely vanishing trees . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.4 Generalising the graphical argument . . . . . . . . . . . . . . . . . . . . . . 25 3.5 The critical Feynman graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.6 Concluding the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 1 Introduction The theory of regularity structures, introduced in [Hai14] and developed to large generality in [CH16, BHZ19, BCCH20], provides a renormalised local solution the- ory to a large class of singular stochastic PDEs: equations driven by very rough random fields, where due to the low regularity, products on the right-hand side of the equation have no classically well-defined meaning. The roughness of the noise arXiv:2602.17369v1 [math.PR] 19 Feb 2026 Introduction 2 has to satisfy two essential constraints, the first and more well-known is scaling sub- criticality. For example, consider the well-known case of the KPZ equation, formally given as ∂th = ∆h + |∇h|2 −∞+ ξ. (1.1) Given a noise ξ with regularity1 α, if one (formally) rescales the equation in a way that leaves the linear part invariant, hλ(t, x) = λ−α−2h(λ2t, λx), then hλ solves (1.1) with another noise ˜ξ following the same law as ξ but the nonlinearity multiplied by λ2+α. This suggests that the scaling critical exponent is αc = −2, and for α > αc the solution on small scales should look like the Gaussian process solving the linear equation. For critical (α = αc) or supercritical noise (α < αc) the solution theory of regularity structures (or of paracontrolled distributions [GIP15]) do not apply. We remark that while αc does not depend on the dimension, the criticality of the choice of space-time white noise does: as it has regularity (−d −2)/2, for (1.1) it is subcritical in spatial dimension d = 1, critical for d = 2, and supercritical for d ≥3. There is another exponent αv (that does depend on the dimension) that ensures that the variance of all of the stochastic objects required for the theory of regularity structures and constructed in [CH16] remains finite. The condition α > αv is another crucial assumption, as an infinite variance can not be removed by an additive counterterm. For example, for (1.1) in dimension d = 1, one has αv = −7/4, which is precisely the barrier of the theory instead of α > αc = −2, c.f. [Hos16]. Recently [Hai25, GT25] considered the regime α < αv in the case of (1.1) in dimension 1. The discussion in [Hai25, Sec. 1.2] hints that if αc < αv, then the subcritical toolbox applies for all α, but in the regime α ≤αv one can expect a different type of result: when replacing the noise by its smooth approximation in the equation, then to tame the exploding variance, a multiplicative renormalisation is required and a nontrivial limit in law is obtained. This is formulated in a general prediction in [Hai25]. The results of [Hai25, GT25] are about (1.1) and the proofs depend on the Da Prato-Debussche trick of subtracting the linear solution and treating the (more regular) remainder equation. It should be mentioned that [Hai25] also treats a similar problem for stochastic ordinary differential equation driven by fractional Brownian motions with Hurst parameter H ≤1/4 where the noise is not additive. The goal of the present paper is to confirm the prediction of [Hai25] for a “more nonlinear” SPDE, where the Da Prato-Debussche trick is not applicable, and there- fore several of the issues that are treated in an ad hoc manner in [Hai25, GT25] have to be handled within the theory of regularity s

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