Critical and supercritical singular stochastic PDEs and the short-range critical Glauber conjecture: the surviving open core of the theory of regularity structures

Determine whether a canonical renormalised solution theory exists for singular stochastic PDEs at or beyond criticality - or prove that no canonical theory exists - and prove or disprove that the constructed $\Phi^4_3$ Markov process describes the critical Glauber dynamics of genuinely short-range three-dimensional ferromagnets. Of the five issues identified in Hairer's 'A theory of regularity structures' (2014), three are resolved: the probabilistic model-construction step is now a black box (the BPHZ theorem for regularity structures, which also covers non-Gaussian driving noise); the dynamical $\Phi^4_3$ model is globally well-posed with the $\Phi^4_3$ Gibbs measure invariant under its dynamics; and the dynamical $\Phi^4_3$ model arises as the scaling limit of the Glauber dynamics of Ising-Kac ferromagnets in two and three dimensions. The renormalisation-selection issue has been reframed by canonical BPHZ renormalisation, the residual freedom parametrising genuinely different equations. The surviving open core is the critical regime: (A) a canonical renormalised solution theory - or a definitive impossibility, triviality, or non-uniqueness result - for singular SPDEs at or beyond criticality, where local subcriticality fails and renormalisation is no longer finite-dimensional (benchmarks: dynamical $\Phi^4$ in four spatial dimensions, with the critical $\Phi^4_4$ measure already known to be Gaussian, and KPZ in two spatial dimensions); and (B) the conjecture that the constructed $\Phi^4_3$ Markov process describes the critical Glauber dynamics of short-range three-dimensional ferromagnets, proven so far only for long-range Kac interactions.

Background

Many physically important evolution equations are formally semilinear stochastic partial differential equations (SPDEs) driven by extremely irregular random forcing - typically space-time white noise $\xi$, a distribution-valued random field rougher than any function. Examples include the parabolic Allen-Cahn / $\Phi^4$ dynamics

$$\partial_t \Phi = \Delta \Phi - \Phi^3 + \xi,$$

(the stochastic quantisation of Euclidean quantum field theory, i.e. the Langevin dynamics whose formal equilibrium is the $\Phi^4$ Gibbs measure), the Kardar-Parisi-Zhang (KPZ) equation for randomly growing interfaces, whose nonlinearity is the square of the interface gradient, and the parabolic Anderson model for a particle in a random potential. Since the solution $\Phi$ is typically distribution-valued, products like $\Phi^3$ have no classical meaning and these equations are ill-posed as written.

Hairer's theory of regularity structures (2014) gives a black-box solution theory for a large class of such equations. A regularity structure is a graded vector space of abstract symbols, carrying a structure group, that generalises the space of Taylor polynomials: it records the local 'jet' of a candidate solution around each point, including symbols representing iterated convolutions of the noise against singular kernels. A model assigns to every symbol a concrete distribution on space-time, generalising the polynomial model that describes smooth functions, and a reconstruction theorem maps abstract fixed points back to genuine distributions. Applying this to a random rough input proceeds in three steps: an algebraic step (build the structure), an analytic step (solve an abstract fixed-point problem in spaces of modelled distributions), and a probabilistic step: regularise the noise, lift it to a model, renormalise (subtract diverging constants - counterterms - so that finitely many ill-defined products acquire limits), and prove convergence of the renormalised models.

The scope is the class of locally subcritical equations - in quantum-field-theory language, super-renormalisable - meaning that under the parabolic scaling that zooms in on small scales the nonlinearity vanishes and only finitely many symbols of negative homogeneity arise, so the renormalisation is a finite-dimensional (finitely many counterterms) procedure. For the $\Phi^4$ dynamics the dividing line is the spatial dimension: subcritical for $d<4$, critical at $d=4$, supercritical for $d\ge 5$; for KPZ-type equations the boundary is two spatial dimensions. The 2014 paper treats two flagship applications, including the dynamical $\Phi^4_3$ model - the stochastic quantisation of the $\Phi^4_3$ Euclidean field theory on the three-dimensional torus, whose renormalisation involves one linearly and one logarithmically divergent counterterm (the logarithmic subdivergence is what makes three dimensions the hardest subcritical case). It constructs, for the first time, a Markov process symmetric (i.e. reversible, with that measure as its invariant equilibrium) with respect to the finite-volume $\Phi^4_3$ Gibbs measure whose rigorous construction in the 1970s was a landmark of constructive quantum field theory.

The paper is explicit about what it does not achieve: (i) the probabilistic step is carried out with Gaussian tools (Wiener chaos, the orthogonal decomposition of functionals of a Gaussian field into iterated stochastic integrals) and 'we do not have a completely general theory at the moment'; (ii) solutions are local in time, and it is conjectured that they are global for almost every initial condition drawn from the Gibbs measure, which is correspondingly expected to be invariant; (iii) it is 'natural to conjecture' that the constructed Markov process describes the Glauber dynamics (the reversible single-spin-flip relaxation dynamics) of three-dimensional ferromagnets near their critical temperature; (iv) critical and supercritical equations - where infinitely many counterterms would be needed - lie outside the theory, and for $\Phi^4$ in $d\ge 4$ the paper states one does not expect any non-trivial meaning; (v) the finite-dimensional renormalisation freedom (a two-parameter group for $\Phi^4_3$) parametrises different modified equations, and a principled canonical choice beyond symmetry considerations is not settled.

Between 2014 and 2026 the literature resolved most of this list: the probabilistic step became a general black box (the BPHZ theorem for regularity structures, named after the Bogoliubov-Parasiuk-Hepp-Zimmermann renormalisation prescription of perturbative quantum field theory, and valid even for non-Gaussian driving noise); the dynamical $\Phi^4_3$ model was shown to be globally well-posed with the $\Phi^4_3$ Gibbs measure invariant; and the Glauber-dynamics conjecture was proved for ferromagnets with long-range Kac interactions (mean-field-type interactions in which each spin couples weakly to all spins within a large radius, sent to infinity relative to the lattice spacing in the scaling limit) in two and three dimensions. What survives, and constitutes the problem posed here, is the critical regime itself: a renormalised solution theory (or a rigorous impossibility statement) for singular SPDEs at or beyond criticality, and the identification of the $\Phi^4_3$ dynamics with the critical Glauber dynamics of genuinely short-range magnets.

Problem Statement

The unresolved core of the open issues left by 'A theory of regularity structures' (2014) is the critical regime, in two linked strands:

(A) Critical and supercritical singular SPDEs. Determine, for semilinear stochastic PDEs driven by rough (e.g. white-noise-type) input for which local subcriticality fails, whether a canonical renormalised solution theory exists, and if so construct it; or prove that no canonical theory exists (that the renormalised limit is trivial (Gaussian), non-unique, or dependent on the chosen regularisation and renormalisation scheme). The canonical benchmarks - the concrete test cases of the class problem, each an open instance in its own right - are the dynamical $\Phi^4$ equation in four spatial dimensions ($\partial_t \Phi = \Delta \Phi - \Phi^3 + \xi$ on the four-dimensional torus, at the critical dimension) and the KPZ equation in two spatial dimensions; a result for either benchmark resolves that instance, while strand (A) asks for the general-class statement. Here 'canonical' means the limiting solution is independent of the regularisation of the noise, as in the subcritical theory; a 'solution theory' comprises a notion of solution, well-posedness (or a precisely delimited weak well-posedness), identification of the counterterms, and universality. Any answer must be consistent with the known marginal triviality of the critical four-dimensional $\Phi^4_4$ measure - i.e. the theorem that its scaling limit is Gaussian, so that no non-trivial critical $\Phi^4$ measure exists in four dimensions (a four-dimensional dynamical theory must explain how the dynamics degenerates consistently with this) - and with the known non-uniqueness phenomena for certain supercritical equations.

(B) Critical Glauber identification. Prove or disprove that the Markov process constructed for the dynamical $\Phi^4_3$ model describes the critical Glauber dynamics of three-dimensional ferromagnets with genuinely short-range (e.g. nearest-neighbour or finite-range) interactions, excluding the Kac scaling regime in which the interaction range diverges relative to the lattice spacing: that is, that the suitably rescaled coarse-grained spin field of such a system near its critical temperature converges in distribution to the solution of the renormalised dynamical $\Phi^4_3$ model, with the renormalisation (length-scale) prescription adapted to the distance from criticality. This is known for long-range Kac interactions in the Kac scaling regime and open for genuinely short-range interactions.

The two strands form one objective - completing the theory of singular stochastic dynamics at criticality and its identification with physical critical systems - the continuum theory (A) and its principal physical benchmark (B); each strand is an open problem in its own right and neither is resolved by resolving the other.

References

1. Martin Hairer (2014). A theory of regularity structures. Inventiones mathematicae 198, 269-504. DOI: 10.1007/s00222-014-0505-4. https://doi.org/10.1007/s00222-014-0505-4 (arXiv:1303.5113, https://arxiv.org/abs/1303.5113)
2. Ajay Chandra; Martin Hairer (2016). An analytic BPHZ theorem for regularity structures. arXiv:1612.08138. https://arxiv.org/abs/1612.08138 (preprint)
3. Yvain Bruned; Martin Hairer; Lorenzo Zambotti (2019). Algebraic renormalisation of regularity structures. Inventiones mathematicae 215(3), 1039-1156. DOI: 10.1007/s00222-018-0841-x. https://doi.org/10.1007/s00222-018-0841-x (arXiv:1610.08468, https://arxiv.org/abs/1610.08468)
4. Yvain Bruned; Ajay Chandra; Ilya Chevyrev; Martin Hairer (2021). Renormalising SPDEs in regularity structures. Journal of the European Mathematical Society 23(3), 869-947. DOI: 10.4171/jems/1025. https://doi.org/10.4171/jems/1025 (arXiv:1711.10239, https://arxiv.org/abs/1711.10239)
5. Martin Hairer; Rhys Steele (2024). The BPHZ Theorem for Regularity Structures via the Spectral Gap Inequality. Archive for Rational Mechanics and Analysis 248, Article 9. DOI: 10.1007/s00205-023-01946-w. https://doi.org/10.1007/s00205-023-01946-w (arXiv:2301.10081, https://arxiv.org/abs/2301.10081)
6. Yvain Bruned; Franck Gabriel; Martin Hairer; Lorenzo Zambotti (2022). Geometric stochastic heat equations. Journal of the American Mathematical Society 35(1), 1-80. DOI: 10.1090/jams/977. https://doi.org/10.1090/jams/977 (arXiv:1902.02884, https://arxiv.org/abs/1902.02884)
7. Jean-Christophe Mourrat; Hendrik Weber (2017). The dynamic $\Phi^4_3$ model comes down from infinity. Communications in Mathematical Physics 356(3), 673-753. DOI: 10.1007/s00220-017-2997-4. https://doi.org/10.1007/s00220-017-2997-4 (arXiv:1601.01234, https://arxiv.org/abs/1601.01234)
8. Martin Hairer; Konstantin Matetski (2018). Discretisations of rough stochastic PDEs. The Annals of Probability 46(3). DOI: 10.1214/17-aop1212. https://doi.org/10.1214/17-aop1212 (arXiv:1511.06937, https://arxiv.org/abs/1511.06937)
9. Martin Hairer; Jonathan Mattingly (2018). The strong Feller property for singular stochastic PDEs. Annales de l'Institut Henri Poincaré, Probabilités et Statistiques 54(3). DOI: 10.1214/17-aihp840. https://doi.org/10.1214/17-aihp840 (arXiv:1610.03415, https://arxiv.org/abs/1610.03415)
10. Rongchan Zhu; Xiangchan Zhu (2018). Dirichlet form associated with the $\Phi^4_3$ model. Electronic Journal of Probability 23. DOI: 10.1214/18-ejp207. https://doi.org/10.1214/18-ejp207 (arXiv:1703.09987, https://arxiv.org/abs/1703.09987)
11. Sergio Albeverio; Seiichiro Kusuoka (2020). The invariant measure and the flow associated to the $\Phi^4_3$-quantum field model. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 20(4), 1359-1427. DOI: 10.2422/2036-2145.201809_008. https://doi.org/10.2422/2036-2145.201809_008 (arXiv:1711.07108, https://arxiv.org/abs/1711.07108)
12. Paweł Duch; Martin Hairer; Jaeyun Yi; Wenhao Zhao (2025). Ergodicity of infinite volume $\Phi^4_3$ at high temperature. arXiv:2508.07776. https://arxiv.org/abs/2508.07776 (preprint)
13. Michael Aizenman; Hugo Duminil-Copin (2021). Marginal triviality of the scaling limits of critical 4D Ising and $\lambda\phi_4^4$ models. Annals of Mathematics 194(1). DOI: 10.4007/annals.2021.194.1.3. https://doi.org/10.4007/annals.2021.194.1.3 (arXiv:1912.07973, https://arxiv.org/abs/1912.07973)
14. Martina Hofmanová; Rongchan Zhu; Xiangchan Zhu (2023). A class of supercritical/critical singular stochastic PDEs: existence, non-uniqueness, non-Gaussianity, non-unique ergodicity. Journal of Functional Analysis 285(5), 110011. DOI: 10.1016/j.jfa.2023.110011. https://doi.org/10.1016/j.jfa.2023.110011 (arXiv:2205.13378, https://arxiv.org/abs/2205.13378)
15. Jean-Christophe Mourrat; Hendrik Weber (2017). Convergence of the two-dimensional dynamic Ising-Kac model to $\Phi^4_2$. Communications on Pure and Applied Mathematics 70(4), 717-812. DOI: 10.1002/cpa.21655. https://doi.org/10.1002/cpa.21655 (arXiv:1410.1179, https://arxiv.org/abs/1410.1179)
16. Paolo Grazieschi; Konstantin Matetski; Hendrik Weber (2025). The dynamical Ising-Kac model in 3D converges to $\Phi^4_3$. Probability Theory and Related Fields 191, 671-778. DOI: 10.1007/s00440-024-01316-x. https://doi.org/10.1007/s00440-024-01316-x (arXiv:2303.10242, https://arxiv.org/abs/2303.10242)
17. Tom Klose; Avi Mayorcas (2024). Large deviations of the $\Phi^4_3$ measure via stochastic quantisation. arXiv:2402.00975. https://arxiv.org/abs/2402.00975 (preprint)
18. Ajay Chandra; Hendrik Weber (2015). Stochastic PDEs, regularity structures, and interacting particle systems. arXiv:1508.03616. https://arxiv.org/abs/1508.03616 (lecture notes)
19. Francesco Caravenna; Rongfeng Sun; Nikos Zygouras (2023). The critical 2d stochastic heat flow. Inventiones mathematicae 233(1), 325-460. DOI: 10.1007/s00222-023-01184-7. https://doi.org/10.1007/s00222-023-01184-7 (arXiv:2109.03766, https://arxiv.org/abs/2109.03766)
20. Giuseppe Cannizzaro; Quentin Moulard; Fabio Toninelli (2025). Superdiffusive central limit theorem for the stochastic Burgers equation at the critical dimension. arXiv:2501.00344. https://arxiv.org/abs/2501.00344 (preprint)
21. Giuseppe Cannizzaro; Tom Klose; Quentin Moulard (2026). Superdiffusive central limit theorem for a class of driven diffusive systems at the critical dimension. arXiv:2601.05945. https://arxiv.org/abs/2601.05945 (preprint)