An elementary construction of the renormalised generalised parabolic Anderson model in dimension two and of the parabolic Anderson model in dimension three

The parabolic Anderson model (PAM), the stochastic heat equation $\partial_t u = \Delta u + u\,\xi$ driven by a spatial white noise $\xi$, is ill-posed in dimensions two and higher and requires renormalisation. Hairer and Labbé (2015) gave an elementary construction of the renormalised solution on the whole plane $\mathbf{R}^2$, exploiting the linearity of the equation through an exponential change of variables built from the Green function of the Laplacian, together with weighted Hölder spaces and a Picard fixed point, and avoiding the machinery of regularity structures and paracontrolled calculus, the two general theories of singular stochastic PDEs; the paper states that this construction applies neither to the generalised parabolic Anderson model $\partial_t u = \Delta u + g(u)\,\xi$ nor to the analogous problem in dimension three. The subsequent literature has since established existence in both target settings - the three-dimensional whole-space model by regularity structures in weighted Besov spaces (Besov spaces form a scale of function spaces between Hölder and Sobolev spaces) in 2018, and the generalised model globally on the two-dimensional torus (2024) and on the whole plane (2026) by paracontrolled calculus - but every known construction of either target uses one of the two elaborate theories, while the transform-based elementary programme has succeeded in adjacent targets: the dynamical $\Phi^4_3$ model (a nonlinear singular SPDE arising from three-dimensional $\Phi^4$ Euclidean field theory) on the torus, the elliptic Anderson operator (the random Hamiltonian $-\Delta+\xi$) in dimensions two and three, and several nonlinear Schrödinger equations with spatial white noise on full space. The surviving open core is precise and nonempty: an elementary, machinery-free construction of the renormalised generalised parabolic Anderson model in dimension two and of the three-dimensional parabolic Anderson model, in the spirit of the 2015 exponential transform.

Background

The parabolic Anderson model (PAM) is the stochastic heat equation with a random potential given by white noise: formally $\partial_t u = \Delta u + u\,\xi$ with initial condition $u_0$, where $\xi$ is a white noise on space, i.e. a centred generalised Gaussian random distribution whose covariance is the Dirac mass, here taken constant in time. White noise on $\mathbf{R}^d$ has Hölder regularity just below $-d/2$, and in dimension $d \ge 2$ the product $u\,\xi$ is classically undefined: the solution $u$ is expected to have Hölder regularity just below $2-d/2$, so the regularities of $u$ and $\xi$ sum to a negative number, outside the scope of classical product theories. The equation nevertheless becomes meaningful after renormalisation: one replaces $\xi$ by a mollification $\xi_\varepsilon$, subtracts a diverging constant $C_\varepsilon$ from the potential, solves the well-posed equation $\partial_t u_\varepsilon = \Delta u_\varepsilon + u_\varepsilon(\xi_\varepsilon - C_\varepsilon)$, and shows that $u_\varepsilon$ converges as $\varepsilon \to 0$ to a limit that does not depend on the mollifier. Two general theories provide such constructions on the two-dimensional torus: regularity structures (Hairer 2014), which replace Taylor expansions by abstract models of distributions together with a reconstruction theorem, and paracontrolled calculus (Gubinelli, Imkeller and Perkowski 2015), which decomposes the solution into a paraproduct of a rough noise-dependent field plus a smoother remainder. Both are elaborate machineries; on a torus in dimension three the linear PAM follows from the scaling-argument methods of Hairer and Pardoux (2015), again with machinery. Hairer and Labbé (2015) gave an elementary construction of the linear PAM on the whole plane $\mathbf{R}^2$ that uses neither. The trick exploits the linearity of the equation. Writing $G$ for the logarithmic Green function of the two-dimensional Laplacian, the field $Y = G \ast \xi$ (the formally time-stationary solution of the additive stochastic heat equation) has a gradient of Hölder regularity just below $0$, and the change of unknown $v = u\,e^{Y}$ turns the renormalised equation into a linear equation for $v$ whose only singular coefficient is the renormalised square $Z = |\nabla Y|^{2} - \mathbf{E}|\nabla Y|^{2}$, which converges in the second homogeneous Wiener chaos, the space of double stochastic integrals with respect to the noise (renormalised squares of Gaussian fields are also called Wick products). The equation for $v$ is then solved by a Picard fixed point using classical Schauder estimates in Hölder spaces of distributions weighted at infinity - polynomial weights for the noise and time-increasing exponential weights for the solution - which is what makes the unbounded domain tractable. The construction, however, is specific to the linear equation in dimension two, and the source states the limitation explicitly: as it stands it applies neither to the generalised parabolic Anderson model (gPAM) considered in the two torus constructions cited above - the equation $\partial_t u = \Delta u + g(u)\,\xi$ in which the white noise multiplies a smooth nonlinear function $g$ of the solution, subcritical (in the sense that the regularity gained from the heat operator outweighs what the nonlinearity costs, so that finitely many counterterms suffice) only in dimension at most two, whose renormalisation counterterm (the term subtracted from the equation, reducing in the linear case to the constant $C_\varepsilon$) involves both $g$ and its derivative - nor to the linear PAM in dimension three, where the noise has regularity just below $-3/2$ and the solution regularity just below $1/2$. Existence in both target settings is known today, but only through the elaborate theories. The question is whether the elementary, transform-based route can be extended to them.

Problem Statement

Determine whether the elementary construction of Hairer and Labbé (2015) can be extended beyond the linear two-dimensional setting to the two target settings named in the source, or establish an obstruction showing that this elementary, transform-based route cannot be extended. (a) The generalised parabolic Anderson model in dimension two: an elementary construction of the renormalised solution of $\partial_t u = \Delta u + g(u)\,\xi$ for nonlinearities $g\in C_b^2(\mathbf{R})$ on the whole plane $\mathbf{R}^2$, the source's own setting; this covers the two-dimensional torus, the setting of the existing paracontrolled and regularity-structures constructions. (b) The parabolic Anderson model in dimension three: an elementary construction of the renormalised solution of $\partial_t u = \Delta u + u\,\xi$ with spatial white noise in dimension three on the whole space $\mathbf{R}^3$, the source's own setting, where the weights at infinity must be handled; this covers the three-dimensional torus case, which was already accessible at source time through the scaling-argument methods of Hairer and Pardoux (2015). For each setting, an admissible construction must: (i) exhibit the mollified approximating equations and the explicit renormalisation constants, which must coincide with the counterterms already identified by the elaborate theories for that setting; (ii) prove convergence in probability of the renormalised random objects and of the mollified solutions to a unique, mollifier-independent limit, and identify that limit with the renormalised equation, equivalently with the solution constructed by the machinery; and (iii) use only elementary tools in the sense of the source - Gaussian and Wiener-chaos estimates, classical Schauder-type parabolic estimates in weighted Hölder or Besov spaces, and fixed-point arguments - without the apparatus of regularity structures (models, reconstruction) or paracontrolled distributions (paracontrolled ansätze). The excluded apparatus in (iii) is the framework described in the Background: an ansatz decomposing the solution itself into a paraproduct of a rough noise-dependent field plus a smoother remainder, or regularity-structures models together with the reconstruction theorem; the use of classical Bony-type paraproduct estimates as pointwise analytic tools inside a transform-based construction is not thereby excluded. The boundary of 'elementary' here is advisory rather than an exhaustive whitelist: the operative restriction is the exclusion of the two frameworks named above, so any technique that does not invoke them is admissible (the source's own toolkit being illustrative). A negative resolution must be concrete and checkable: for the setting it addresses it must exhibit a specific estimate, bound, or invariant whose failure blocks the 2015 transform-based route there - for instance a failure of a weighted Hölder or Besov estimate on the transformed unknown, or of the control of the renormalised product $g(u)\,\xi$ together with its counterterm - so a structural obstruction confined to that route resolves the setting; a bare restatement of the source's stated limitation does not suffice. An obstruction that blocks the route for both (a) and (b) resolves both settings; otherwise each setting requires its own. The whole-space cases subsume the corresponding compact cases. The two settings are the two named deficiencies of one and the same method, and the common objective is a machinery-free solution theory for the renormalised parabolic Anderson model paralleling the 2015 two-dimensional linear construction. The question is methodological - existence and uniqueness in both settings are already established by the elaborate theories.

References

1. Martin Hairer and Cyril Labbé (2015), 'A simple construction of the continuum parabolic Anderson model on $\mathbf{R}^2$', Electronic Communications in Probability 20, paper no. 43. DOI: 10.1214/ECP.v20-4038, https://doi.org/10.1214/ECP.v20-4038; arXiv:1501.00692, https://arxiv.org/abs/1501.00692
2. Massimiliano Gubinelli, Peter Imkeller and Nicolas Perkowski (2015), 'Paracontrolled distributions and singular PDEs', Forum of Mathematics Pi 3, e6. DOI: 10.1017/fmp.2015.2, https://doi.org/10.1017/fmp.2015.2; arXiv:1210.2684, https://arxiv.org/abs/1210.2684
3. Martin Hairer (2014), 'A theory of regularity structures', Inventiones Mathematicae 198, no. 2, 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
4. Martin Hairer and Étienne Pardoux (2015), 'A Wong-Zakai theorem for stochastic PDEs', Journal of the Mathematical Society of Japan 67, no. 4. DOI: 10.2969/jmsj/06741551, https://doi.org/10.2969/jmsj/06741551; arXiv:1409.3138, https://arxiv.org/abs/1409.3138
5. Martin Hairer and Cyril Labbé (2018), 'Multiplicative stochastic heat equations on the whole space', Journal of the European Mathematical Society 20, no. 4, 1005-1054. DOI: 10.4171/JEMS/781, https://doi.org/10.4171/JEMS/781; arXiv:1504.07162, https://arxiv.org/abs/1504.07162
6. Ismael Bailleul, Frédéric Bernicot and Dorothee Frey (2015), 'Space-time paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations', arXiv:1506.08773, https://arxiv.org/abs/1506.08773
7. Ismael Bailleul, Arnaud Debussche and Martina Hofmanová (2019), 'Quasilinear generalized parabolic Anderson model equation', Stochastics and Partial Differential Equations: Analysis and Computations 7, no. 1, 40-63. DOI: 10.1007/s40072-018-0121-1, https://doi.org/10.1007/s40072-018-0121-1; arXiv:1610.06726, https://arxiv.org/abs/1610.06726
8. Alexander Dunlap (2020), 'The continuum parabolic Anderson model with a half-Laplacian and periodic noise', Electronic Communications in Probability 25. DOI: 10.1214/20-ECP342, https://doi.org/10.1214/20-ECP342; arXiv:2002.07142, https://arxiv.org/abs/2002.07142
9. Yu Gu and Jingyu Huang (2018), 'Chaos expansion of 2D parabolic Anderson model', Electronic Communications in Probability 23. DOI: 10.1214/18-ECP129, https://doi.org/10.1214/18-ECP129; arXiv:1711.07578, https://arxiv.org/abs/1711.07578
10. Aukosh Jagannath and Nicolas Perkowski (2023), 'A simple construction of the dynamical $\Phi^4_3$ model', Transactions of the American Mathematical Society 376, no. 3, 1507-1522. DOI: 10.1090/tran/8724, https://doi.org/10.1090/tran/8724; arXiv:2108.13335, https://arxiv.org/abs/2108.13335
11. Antoine Mouzard and El Maati Ouhabaz (2025), 'A simple construction of the Anderson operator via its quadratic form in dimensions two and three', Comptes Rendus Mathématique 363, no. G2, 183-197. DOI: 10.5802/crmath.712, https://doi.org/10.5802/crmath.712; arXiv:2309.02821, https://arxiv.org/abs/2309.02821
12. Yueh-Sheng Hsu and Cyril Labbé (2026), 'Construction and spectrum of the Anderson Hamiltonian with white noise potential on $\mathbf{R}^2$ and $\mathbf{R}^3$', Probability and Mathematical Physics 7, no. 1, 1-35. DOI: 10.2140/pmp.2026.7.1, https://doi.org/10.2140/pmp.2026.7.1; arXiv:2401.17900, https://arxiv.org/abs/2401.17900
13. Quentin Chauleur and Antoine Mouzard (2023), 'The logarithmic Schrödinger equation with spatial white noise on the full space', arXiv:2308.15814, https://arxiv.org/abs/2308.15814
14. Antoine Mouzard and Immanuel Zachhuber (2026), 'Nonlinear Schrödinger equations with spatial white noise potential on full space for $d \le 3$', arXiv:2604.15226, https://arxiv.org/abs/2604.15226
15. Samaël Mackowiak (2025), 'Wellposedness of the cubic Gross-Pitaevskii equation with spatial white noise on $\mathbb{R}^2$', Nonlinearity 38, no. 3, 035026. DOI: 10.1088/1361-6544/adb237, https://doi.org/10.1088/1361-6544/adb237; arXiv:2311.10405, https://arxiv.org/abs/2311.10405
16. Arnaud Debussche, Ruoyuan Liu, Nikolay Tzvetkov and Nicola Visciglia (2024), 'Global well-posedness of the 2D nonlinear Schrödinger equation with multiplicative spatial white noise on the full space', Probability Theory and Related Fields 189, no. 3-4, 1161-1218. DOI: 10.1007/s00440-024-01288-y, https://doi.org/10.1007/s00440-024-01288-y; arXiv:2301.10825, https://arxiv.org/abs/2301.10825
17. Ajay Chandra, Guilherme de Lima Feltes and Hendrik Weber (2026), 'A priori bounds for the generalised parabolic Anderson model', Communications on Pure and Applied Mathematics 79, no. 5, 1315-1394. DOI: 10.1002/cpa.70025, https://doi.org/10.1002/cpa.70025
18. Hao Shen, Rongchan Zhu and Xiangchan Zhu (2026), 'Global well-posedness for 2D generalized Parabolic Anderson Model via paracontrolled calculus', Stochastics and Partial Differential Equations: Analysis and Computations 14, no. 1, 133-160. DOI: 10.1007/s40072-025-00358-z, https://doi.org/10.1007/s40072-025-00358-z; arXiv:2402.19137, https://arxiv.org/abs/2402.19137
19. Hao Shen, Rongchan Zhu and Xiangchan Zhu (2026), 'Global well-posedness for generalized parabolic Anderson model on the whole plane', arXiv:2606.26681, https://arxiv.org/abs/2606.26681
20. Salvador Esquivel and Felix Schwarzfischer (2026), 'A priori bounds for the generalized parabolic Anderson model in the full subcritical regime', arXiv:2608.22911, https://arxiv.org/abs/2608.22911
21. Cyril Labbé (2019), 'The continuous Anderson Hamiltonian in $d \le 3$', Journal of Functional Analysis 277, no. 9, 3187-3235. DOI: 10.1016/j.jfa.2019.05.027, https://doi.org/10.1016/j.jfa.2019.05.027
22. Massimiliano Gubinelli, Baris Evren Ugurcan and Immanuel Zachhuber (2020), 'Semilinear evolution equations for the Anderson Hamiltonian in two and three dimensions', Stochastics and Partial Differential Equations: Analysis and Computations 8, no. 1, 82-149. DOI: 10.1007/s40072-019-00143-9, https://doi.org/10.1007/s40072-019-00143-9; arXiv:1807.06825, https://arxiv.org/abs/1807.06825
23. Abel Klein and Lawrence J. Landau (1981), 'Construction of a unique self-adjoint generator for a symmetric local semigroup', Journal of Functional Analysis 44, no. 2, 121-137. DOI: 10.1016/0022-1236(81)90007-0, https://doi.org/10.1016/0022-1236(81)90007-0