Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 021, 35 pages      arXiv:2507.23737      https://doi.org/10.3842/SIGMA.2026.021
Contribution to the Special Issue on Asymptotics, Randomness and Noncommutativity

Renormalisation of Singular SPDEs with Correlated Coefficients

Nicolas Clozeau a and Harprit Singh b
a) Université de Toulon, Av. de l'Université, 83130 La Garde, France
b) SISSA, Via Bonomea 265, 34136 Trieste TS, Italy

Received September 01, 2025, in final form February 20, 2026; Published online March 07, 2026

Abstract
We show local well-posedness of the g-PAM and the $\phi^{K+1}_2$-equation for $K\geq 1$ on the two-dimensional torus when the coefficient field is random and correlated to the driving noise. In the setting considered here, even when the model in the sense of Hairer (2014) is stationary, naive use of renormalisation constants in general leads to variance blow-up. Instead, we prove convergence of renormalised models choosing random renormalisation functions analogous to the deterministic variable coefficient setting. The main technical contribution are stochastic estimates on the model in this correlated setting which are obtained by a combination of heat kernel asymptotics, Gaussian integration by parts formulae and Hairer-Quastel type bounds.

Key words: singular SPDEs; renormalisation in random environments; regularity structures.

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