  We consider a semilinear parabolic partial differential equation in
$\mathbf{R}_+\times [0,1]^d$, where $d=1, 2$ or $3$, with a highly oscillating
random potential and either homogeneous Dirichlet or Neumann boundary
condition. If the amplitude of the oscillations has the right size compared to
its typical spatiotemporal scale, then the solution of our equation converges
to the solution of a deterministic homogenised parabolic PDE, which is a form
of law of large numbers. Our main interest is in the associated central limit
theorem. Namely, we study the limit of a properly rescaled difference between
the initial random solution and its LLN limit. In dimension $d=1$, that
rescaled difference converges as one might expect to a centred
Ornstein-Uhlenbeck process. However, in dimension $d=2$, the limit is a
non-centred Gaussian process, while in dimension $d=3$, before taking the CLT
limit, we need to subtract at an intermediate scale the solution of a
deterministic parabolic PDE, subject (in the case of Neumann boundary
condition) to a non-homogeneous Neumann boundary condition. Our proofs make use
of the theory of regularity structures, in particular of the very recently
developed methodology allowing to treat parabolic PDEs with boundary conditions
within that theory.
