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

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 $d\ge 3$, with an initial condition $u(0,x)=u_0(\epsilon x)$ and a suitably chosen $\lambda \in \mathbb{R}$. 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 $\bar{u}(t,x)$ 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 $w^{\epsilon }(t,x)=\bar{u}(t,x){\Psi }^{\epsilon }(t,x)+\epsilon u_1^{\epsilon }(t,x)$, where ${\Psi }^{\epsilon }(t,x)=\Psi (t/{\epsilon }^2,x/\epsilon )$, $\Psi$ is a solution of the SHE with constant initial conditions, and $u_1^{\epsilon }$ is an explicit corrector. We show that $\Psi (t,x)$ converges to a stationary process $\tilde{\Psi }(t,x)$ as $t\to \infty$, that $\mathbf{E}|u^{\epsilon }(t,x)-w^{\epsilon }(t,x){|}^2$ converges pointwise to $0$ 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, 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 $u_1^{\epsilon }(t,x)$ is constructed using a seemingly new approximation scheme on a mesoscopic time scale.

38 pages; to appear in Archive for Rational Mechanics and Analysis

References in corpus (7)

• An optimal error estimate in stochastic homogenization of discrete elliptic equations
• Quantitative stochastic homogenization and large-scale regularity
• An optimal variance estimate in stochastic homogenization of discrete elliptic equations
• Quantitative results on the corrector equation in stochastic homogenization
• The Edwards-Wilkinson limit of the random heat equation in dimensions three and higher
• On the moments of the (2+1)-dimensional directed polymer and stochastic heat equation in the critical window
• Higher-order pathwise theory of fluctuations in stochastic homogenization

Cited by in corpus (2)

• A forward-backward SDE from the 2D nonlinear stochastic heat equation
• A quenched local limit theorem for stochastic flows