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Publication 18-CNA-030

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

Alexander Dunlap
Department of Mathematics
Stanford University
Stanford, CA 94305, USA

Yu Gu
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, USA

Lenya Ryzhik
Department of Mathematics
Stanford University
Stanford, CA 94305

Ofer Zeitouni
Department of Mathematics
Weizmann Institute of Science
POB 26, Rehovot 76100, Israel

Abstract: We consider the stochastic heat equation $\partial_{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda)u$, driven by a smooth space-time stationary Gaussian random field $V(s,y)$, in dimensions $d\geq 3$, with an initial condition $u(0,x)=u_0(\epsilon x)$. It is known that the diffusively rescaled solution $u^{\epsilon}(t,x)=u(\epsilon^{-2}t,\epsilon^{-1}x)$ converges weakly to a scalar multiple of the solution $\bar{u}$ of a homogenized heat equation with an effective diffusivity $a$, and that fluctuations converge (again, in a weak sense) to the solution of the Edwards-Wilkinson equation with an effective noise strength $\nu$. In this paper, we derive a pointwise approximation $w^{\epsilon}(t,x)=\bar u(t,x)\Psi^{\epsilon}(t,x)+ \epsilon u_1^{\epsilon}(t,x)$, where $\Psi^{\epsilon}(t,x)=\Psi(t/\epsilon^2,x/\epsilon)$, $\Psi$ is a solution of the SHE with constant initial conditions, and $u_1$ is an explicit corrector. We show that $\Psi(t,x)$ converges to a stationary process $\tilde \Psi(t,x)$ as $t\to\infty$, that $w^{\epsilon}(t,x)$ converges pointwise (in $L^1$) to $u^{\epsilon}(t,x)$ as $\epsilon\to 0$, 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~$\nu$. Our approach uses a Markov chain in the space of trajectories introduced in [Gu-Ryzhik-Zeitouni 17], as well as tools from homogenization theory. The corrector $u_1^\epsilon(t,x)$ is constructed using a seemingly new approximation scheme on "long but not too long time intervals".

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