三维及更高维的随机heat equation：均匀化观点
The random heat equation in dimensions three and higher: the homogenization viewpoint
摘要：
考虑随机热方程$\partial_{s}u =\frac{1}{2}Δu +(βV(s,y)-λ)u$，具有光滑的时空平稳高斯随机场$V(s,y)$，维数为$d\geq 3$，初始条件为$u(0,x)=u_0(\varepsilon x)$，$λ\in{\mathbb R}$是适当选择的.众所周知，对于足够小的$β$，扩散重新标度的解$u^{\varepsilon}(t,x)=u(\varepsilon^{-2}t,\varepsilon^{-1}x)$弱收敛到具有有效扩散率$a$的热方程的解$\bar u(t,x)$的标量倍数，并且波动也在弱意义上收敛到具有有效噪声强度$ν$和相同有效扩散率的爱德华-威尔金森方程的解。本文导出了一个逐点逼近$w^\varepsilon(t,x)=\bar u(t,x)Ψ^\varepsilon(t,x)+\varepsilon u_1^\varepsilon(t,x)$，其中$Ψ^\varepsilon(t,x)=Ψ(t/\varepsilon^2,x/\varepsilon)$，$Ψ$是具有常初始条件的SHE的解，$u^\varepsilon_1$是一个显式校正子.我们发现，$Ψ(t,x)$收敛到一个平稳过程$\tilde Ψ(t,x)$ $t\to\infty$，即$\mathbf{E}|u^\varepsilon(t,x)-w^\varepsilon(t, x)|^2$逐点收敛到$0$ $\varepsilon\to 0$，和$\varepsilon^{-d/2+1}(u^\varepsilon-w^\varepsilon)$弱收敛到$0$固定$t$。因此，我们得到新的表示的扩散率$a$和有效的噪声强度$ν$。我们的方法在Gu，Ryzhik和Zeitouni的“三维及更高维度随机热方程的爱德华-威尔金森极限”中引入的轨迹空间中使用马尔可夫链，以及来自均匀化理论的工具。校正器$u_1^\varepsilon(t,x)$是使用一个看似新的近似方案构造的介观时间尺度。
英文摘要：
We consider the stochastic heat equation $\partial_{s}u =\frac{1}{2}Δu +(βV(s,y)-λ)u$, 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 $λ\in{\mathbb R}$. It is known that, for $β$ 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 $ν$ and the same effective diffusivity. In this paper, we derive a pointwise approximation $w^\varepsilon(t,x)=\bar u(t,x)Ψ^\varepsilon(t,x)+\varepsilon u_1^\varepsilon(t,x)$, where $Ψ^\varepsilon(t,x)=Ψ(t/\varepsilon^2,x/\varepsilon)$, $Ψ$ is a solution of the SHE with constant initial conditions, and $u^\varepsilon_1$ is an explicit corrector. We show that $Ψ(t,x)$ converges to a stationary process $\tilde Ψ(t,x)$ 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 $ν$. 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.
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