Corrector theory for elliptic equations with long-range correlated random potential (Q2891100)

Let \(X\) be an open subset of \(\mathbb{R}^d\) and let \(P(x, D)\) be a pseudo-differential operator on \(\mathbb{R}^d\). In this paper the authors consider pseudo-differential equations with random potentials of the form NEWLINE\[NEWLINE P(x, D)u_{\varepsilon}(x, \omega) +\left(q_0(x)+q\left(\frac{x}{\varepsilon}, \omega\right)\right)u_{\varepsilon}(x, \omega)=f(x), \quad x\in X NEWLINE\]NEWLINE with zero Dirichlet boundary condition on \(\partial X\), where \(q_0\) is a smooth function bounded from below by a positive constant \(\gamma\), and \(q\) is defined by \(q(x)=\Phi(g(x))\) where \(\Phi\) is a real-valued function on \(\mathbb{R}\) satisfying \(|\Phi|\leq \gamma\) and \(\int^\infty_{-\infty}\Phi(s)e^{-s^2/2}ds=0\), \(g(x)\) is a centered Gaussian random field with unit variance and heavy-tailed correlation function of the form NEWLINE\[NEWLINE {\mathbb E}(g(y)g(y+x))\sim \kappa_g|x|^{-\alpha}, \quad |x|\to\infty NEWLINE\]NEWLINE for some positive constant \(\kappa_g\) and \(\alpha\in (0, d)\). Let \(u_0\) be the solution to the same equation above with \(q(\frac{x}{\varepsilon}, \omega)\) replaced by its zero average. Then \(u_{\varepsilon}-u_0\) is called the corrector, \({\mathbb E}(u_{\varepsilon})-u_0\) the deterministic corrector and \(u_{\varepsilon}-{\mathbb E}(u_{\varepsilon})\) the stochastic corrector. The authors first establish a result on the main order of the corrector in \(L^2(X)\), and then they characterize the deterministic and stochastic correctors. They also show that, with proper scaling, the mean-zero stochastic corrector converges to a Gaussian process in probability.