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 ≥ 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.