  We consider a class of singular perturbations to the stochastic heat equation
or semilinear variations thereof. The interesting feature of these
perturbations is that, as the small parameter epsilon tends to zero, their
solutions converge to the 'wrong' limit, i.e. they do not converge to the
solution obtained by simply setting epsilon = 0. A similar effect is also
observed for some (formally) small stochastic perturbations of a deterministic
semilinear parabolic PDE.
  Our proofs are based on a detailed analysis of the spatially rough component
of the equations, combined with a judicious use of Gaussian concentration
inequalities.
