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
ajdunl2@stanford.edu
Yu Gu
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213, USA
yug2@andrew.cmu.edu
Lenya Ryzhik
Department of Mathematics
Stanford University
Stanford, CA 94305
ryzhik@math.stanford.edu
Ofer Zeitouni
Department of Mathematics
Weizmann Institute of Science
POB 26, Rehovot 76100, Israel
ofer.zeitouni@weizmann.ac.il
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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