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

We consider the stochastic heat equation partial derivative(s)u = 1/2 Delta u + (beta V(s, y) - lambda)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)(epsilon x) and a suitably chosen lambda is an element of R. It is known that, for beta small enough, the diffusively rescaled solution u(epsilon)(t, x) = u(epsilon(-2)t, epsilon(-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. and the same effective diffusivity. In this paper, we derive a pointwise approximation w(epsilon)(t, x) = (u) over bar (t, x)Psi(epsilon)(t, x)+epsilon u(1)(epsilon)(t, x), where Psi(epsilon)(t, x) = Psi (t/epsilon(2), x/epsilon), Psi is a solution of the SHE with constant initial conditions, and u(1)(epsilon) is an explicit corrector. We show that Psi (t, x) converges to a stationary process (Psi) over tilde (t, x) as t -> infinity, that E|u(epsilon)(t, x)- w(epsilon)(t, x)|(2) converges pointwise to 0 as epsilon -> 0, and that epsilon(-d/2+1)(u(epsilon) - w(epsilon)) converges weakly to 0 for fixed t. As a consequence, we derive new representations of the diffusivity a and effective noise strength.. 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)(epsilon)(t, x) is constructed using a seemingly new approximation scheme on a mesoscopic time scale.