  In this article, basing upon 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.
