  We consider slowly time-dependent singular stochastic partial differential
equations on the two-dimensional torus, driven by weak space-time white noise,
and renormalised in the Wick sense. Our main results are concentration results
on sample paths near stable equilibrium branches of the equation without noise,
measured in appropriate Besov and H\"older norms. We also discuss a case
involving a pitchfork bifurcation. These results extend to the two-dimensional
torus those obtained in [Berglund and Gentz, Proability Theory and Related
Fields, 2002] for finite-dimensional SDEs, and in [Berglund and Nader,
Stochastics and PDEs, 2022] for SPDEs on the one-dimensional torus.
