We consider a semilinear parabolic partial differential equation in R+ × [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.