The random heat equation in dimensions three and higher: the homogenization viewpoint

TL;DR

This paper analyzes the stochastic heat equation in dimensions three and higher, deriving pointwise approximations and new representations for effective diffusivity and noise strength using homogenization and Markov chain techniques.

Contribution

It introduces a novel pointwise approximation scheme for the solution, connecting homogenization theory with the Edwards-Wilkinson limit in higher dimensions.

Findings

Convergence of the approximation to the true solution as epsilon tends to zero.

New formulas for effective diffusivity and noise strength.

Weak convergence results for scaled differences.

Abstract

We consider the stochastic heat equation $\partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u$, with a smooth space-time stationary Gaussian random field $V(s,y)$, in dimensions $d\geq 3$, with an initial condition $u(0,x)=u_0(\varepsilon x)$ and a suitably chosen $\lambda\in{\mathbb R}$. It is known that, for $\beta$ small enough, the diffusively rescaled solution $u^{\varepsilon}(t,x)=u(\varepsilon^{-2}t,\varepsilon^{-1}x)$ converges weakly to a scalar multiple of the solution $\bar u(t,x)$ 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 $\nu$ and the same effective diffusivity. In this paper, we derive a pointwise approximation $w^\varepsilon(t,x)=\bar u(t,x)\Psi^\varepsilon(t,x)+\varepsilon u_1^\varepsilon(t,x)$, where $\Psi^\varepsilon(t,x)=\Psi(t/\varepsilon^2,x/\varepsilon)$, $\Psi$ is a solution of the SHE with constant initial conditions, and $u^\varepsilon_1$ is an explicit corrector. We show that $\Psi(t,x)$ converges to a stationary process $\tilde \Psi(t,x)$ as $t\to\infty$, that $\mathbf{E}|u^\varepsilon(t,x)-w^\varepsilon(t, x)|^2$ converges pointwise to $0$ as $\varepsilon\to 0$, and that $\varepsilon^{-d/2+1}(u^\varepsilon-w^\varepsilon)$ converges weakly to $0$ for fixed $t$. As a consequence, we derive new representations of the diffusivity $a$ and effective noise strength $\nu$. Our approach uses a Markov chain in the space of trajectories introduced in Gu, Ryzhik, and Zeitouni, "The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher," as well as tools from homogenization theory. The corrector $u_1^\varepsilon(t,x)$ is constructed using a seemingly new approximation scheme on a mesoscopic time scale.