Homogenization of the Lévy-type Operators

In \(L_2(\mathbb R^d)\), we consider a selfadjoint operator \({\mathbb A}_\varepsilon\), \(\varepsilon >0\), of the form

where \(0< \alpha < 2\). It is assumed that a function \(\mu(\mathbf{x},\mathbf{y})\) is bounded, positive definite, periodic in each variable, and is such that \(\mu(\mathbf{x},\mathbf{y})=\mu(\mathbf{y},\mathbf{x})\). A rigorous definition of the operator \({\mathbb A}_\varepsilon\) is given in terms of the corresponding quadratic form. It is proved that the resolvent \(({\mathbb A}_\varepsilon+I)^{-1}\) converges in the operator norm on \(L_2(\mathbb R^d)\) to the operator \(({\mathbb A}^0+I)^{-1}\) as \(\varepsilon\to 0\). Here, \({\mathbb A}^0\) is an effective operator of the same form with the constant coefficient \(\mu^0\) equal to the mean value of \(\mu(\mathbf{x},\mathbf{y})\). We obtain an error estimate of order \(O(\varepsilon^\alpha)\) for \(0< \alpha < 1\), \(O(\varepsilon (1+| \operatorname{ln} \varepsilon|)^2)\) for \( \alpha=1\), and \(O(\varepsilon^{2- \alpha})\) for \(1< \alpha < 2\). In the case where \(1< \alpha < 2\), the result is refined by taking the correctors into account.