Periodic homogenization of a class of weakly coupled systems of linear PDEs

In this article, based on probabilistic methods, we discuss periodic homogenization of a class of weakly coupled systems of linear elliptic and parabolic partial differential equations. Under the assumption that the systems have rapidly periodically oscillating coefficients, we first prove that the appropriately centered and scaled continuous component of the associated regime-switching diffusion process converges weakly to a Brownian motion with covariance matrix given in terms of the coefficients of the systems. The homogenization results then follow by employing probabilistic representation of the solutions to the systems and the continuous mapping theorem. The presented results generalize the well-known results related to periodic homogenization of the classical elliptic boundary-value problem and the classical parabolic initial-value problem for a single equation.