Homogenization of Elliptic Operators with Randomly Heterogeneous Coefficients in Unbounded Domains

This paper investigates the stochastic homogenization of second-order linear elliptic operators with randomly heterogeneous coefficients defined over unbounded spatial domains. Unlike classical bounded domain settings where standard Dirichlet or Neumann boundary conditions constrain the global behavior, unbounded domains require specialized functional analytic frameworks to manage behavior at infinity and control large-scale spatial fluctuations. By employing weighted Sobolev spaces and regularized corrector fields on stationary and ergodic probability spaces, we establish rigorous qualitative H1-weak and L2-strong homogenization limits for the random elliptic system. Furthermore, under quantitative mixing assumptions on the coefficient field—specifically finite-range dependence and algebraic decay of correlations—we derive explicit convergence rates for the homogenized solutions in terms of the scale parameter epsilon. To complement the theoretical developments, we propose a computational truncation scheme utilizing artificial boundary conditions and Monte Carlo finite element discretizations. Numerical experiments in two and three dimensions validate the theoretical convergence rates, demonstrating the robustness of our stochastic corrector regularizations in unbounded geometries.