A probabilistic approach to the homogenization of divergence-form operators in periodic media (Q2774033)

For periodic uniformly elliptic matrix \({\mathbf a}=(a_{ij}(x))\) and bounded vectors \({\mathbf b} =(b_i(x))\) and \({\mathbf c}=(c_i(x))\), let \(L^\varepsilon\) be the operator defined by NEWLINE\[NEWLINEL^\varepsilon=(1/2)\sum_{i,j=1}^d (\partial/\partial x_i)\left(a_{ij} (\cdot/\varepsilon)\partial/\partial x_j\right) + \left((1/\varepsilon) b_i(\cdot/\varepsilon)+c_i(\cdot/\varepsilon)\right)\partial/\partial x_iNEWLINE\]NEWLINE and \(X^\varepsilon\) the associated diffusion process. In the first part of this article, the homogenization principle, that is, the convergence of \(X^\varepsilon-\beta t/\varepsilon\) to the diffusion process corresponding to the deterministic operator NEWLINE\[NEWLINE\overline{L}=(1/2)\overline{a}_{ij}(\partial^2/\partial x_i \partial x_j) + \overline{c}_i(\partial/\partial x_i)NEWLINE\]NEWLINE is explained. In the second part, the homogenization problem with lower order differential term is considered. In the proof, the probabilistic methods including Girsanov and Feynman-Kac transformations are used. Since the regularity of the coefficients is not assumed, the stochastic calculus related to Dirichlet forms is used. In the general theory of Dirichlet forms, one cannot specify properties for every fixed starting points. Using the analytic results, the author gives the result for any starting points.