The Random Heat Equation in Dimensions Three and Higher: The Homogenization Viewpoint

We consider the stochastic heat equation $$\partial _{s}u =\frac{1}{2}\Delta u +(\beta V(s,y)-\lambda )u$$ ? s = 1 2 ? + ( ? V , y ) - ? with a smooth space-time-stationary Gaussian random field V(s, y), in dimensions $$d\ge 3$$ d ? 3 an initial condition $$u(0,x)=u_0(\varepsilon x)$$ 0 x ? and suitably chosen $$\lambda \in {\mathbb {R}}$$ ? R . It is known that, for $$\beta $$ small enough, diffusively rescaled solution $$u^{\varepsilon }(t,x)=u(\varepsilon ^{-2}t,\varepsilon ^{-1}x)$$ t converges weakly to scalar multiple of $${\bar{u}}(t,x)$$ ¯ effective diffusivity a, that fluctuations converge, also weak sense, Edwards-Wilkinson noise strength $$\nu ? same diffusivity. In this paper, we derive pointwise approximation $$w^\varepsilon (t,x)={\bar{u}}(t,x)\Psi ^\varepsilon (t,x)+ \varepsilon u_1^\varepsilon (t,x)$$ w ? where $$\Psi (t,x)=\Psi (t/\varepsilon ^2,x/\varepsilon )$$ / SHE constant conditions, $$u^\varepsilon _1$$ explicit corrector. show stationary process $${\tilde{\Psi }}(t,x)$$ ~ as $$t\rightarrow \infty ? ? $${\mathbf {E}}|u^\varepsilon (t,x)-w^\varepsilon (t,x)|^2$$ E | $$\varepsilon \rightarrow 0$$ ^{-d/2+1}(u^\varepsilon -w^\varepsilon fixed t. As consequence, new representations Our approach uses Markov chain space trajectories introduced [17], well tools from homogenization theory. The corrector $$u_1^\varepsilon constructed using seemingly scheme on mesoscopic time scale.