In this paper, we deal with weakly coupled elliptic systems with unbounded coefficients. We prove the existence and characterize all the systems of invariant measures for the associated semigroup associated C_b(R^d;R^m). We also show some relevant properties of the extension of the semigroup to the L^p-spaces related to systems of invariant measures. Finally, we study the asymptotic behaviour of the semigroup as t tends to infinity.
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