Center for                           Nonlinear Analysis CNA Home People Seminars Publications Workshops and Conferences CNA Working Groups CNA Comments Form Summer Schools Summer Undergraduate Institute PIRE Cooperation Graduate Topics Courses SIAM Chapter Seminar Positions Contact Publication 18-CNA-030 The random heat equation in dimensions three and higher: the homogenization viewpoint Alexander DunlapDepartment of Mathematics Stanford University Stanford, CA 94305, USAajdunl2@stanford.edu Yu GuDepartment of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213, USAyug2@andrew.cmu.edu Lenya RyzhikDepartment of Mathematics Stanford University Stanford, CA 94305ryzhik@math.stanford.edu Ofer ZeitouniDepartment of Mathematics Weizmann Institute of Science POB 26, Rehovot 76100, Israelofer.zeitouni@weizmann.ac.ilAbstract: 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". Get the paper in its entirety as  18-CNA-030.pdf