The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher

We consider the heat equation with a multiplicative Gaussian potential in dimensions $d\ge 3$. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic heat equation (SHE) with additive white noise, with an effective variance.