  We consider a two-dimensional stochastic heat equation with Gaussian noise of
correlation length $\rho^{1/2} \ll 1$ and strength
$\sigma(v)/\sqrt{\log\rho^{-1}}$ depending nonlinearly on the solution $v$. The
first author and Gu showed that, if $\operatorname{Lip}(\sigma)$ lies below a
critical value, the one-point statistics of $v$ converge in distribution as
$\rho\to 0$ to the law of a random variable expressed in terms of a
forward-backward stochastic differential equation (FBSDE). In this paper, we
show (for a slightly different smoothing mechanism) that this result can hold
even when $\operatorname{Lip}(\sigma)$ exceeds the critical value. More
precisely, the FBSDE characterizes the limiting one-point statistics of the
SPDE so long as the square root of the decoupling function of the FBSDE remains
Lipschitz. In addition, we extend the results to coupled systems of SPDEs and
provide a quantitative rate of convergence.
