Renormalisation of Singular SPDEs with Correlated Coefficients

TL;DR

This paper establishes local well-posedness for certain singular stochastic PDEs with correlated random coefficients on a two-dimensional torus, introducing novel renormalisation techniques to handle variance blow-up.

Contribution

It develops a new renormalisation approach for singular SPDEs with correlated coefficients, ensuring convergence of models despite variance issues.

Findings

Proves local well-posedness for g-PAM and φ^{K+1}_2 equations with correlated randomness.

Introduces stochastic estimates combining heat kernel asymptotics and Gaussian integration by parts.

Demonstrates convergence of renormalised models with random renormalisation functions.

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.