The Random Heat Equation in Dimensions Three and Higher: The Homogenization Viewpoint.

We consider the stochastic heat equation ∂ s u = 1 2 Δ u + (β V (s , y) - λ) u , with a smooth space-time-stationary Gaussian random field V(s, y), in dimensions d ≥ 3 , with an initial condition u (0 , x) = u 0 (ε x) and a suitably chosen λ ∈ R . It is known that, for β small enough, the diffusively rescaled solution u ε (t , x) = u (ε - 2 t , ε - 1 x) converges weakly to a scalar multiple of the solution u ¯ (t , x) of the heat equation with an effective diffusivity a, and that fluctuations converge, also in a weak sense, to the solution of the Edwards-Wilkinson equation with an effective noise strength ν and the same effective diffusivity. In this paper, we derive a pointwise approximation w ε (t , x) = u ¯ (t , x) Ψ ε (t , x) + ε u 1 ε (t , x) , where Ψ ε (t , x) = Ψ (t / ε 2 , x / ε) , Ψ is a solution of the SHE with constant initial conditions, and u 1 ε is an explicit corrector. We show that Ψ (t , x) converges to a stationary process Ψ ~ (t , x) as t → ∞ , that E | u ε (t , x) - w ε (t , x) | 2 converges pointwise to 0 as ε → 0 , and that ε - d / 2 + 1 (u ε - w ε) converges weakly to 0 for fixed t. As a consequence, we derive new representations of the diffusivity a and effective noise strength ν . Our approach uses a Markov chain in the space of trajectories introduced in [17], as well as tools from homogenization theory. The corrector u 1 ε (t , x) is constructed using a seemingly new approximation scheme on a mesoscopic time scale. [ABSTRACT FROM AUTHOR]