On invariant measures associated to weakly coupled systems of Kolmogorov equations

In this paper, we deal with weakly coupled elliptic systems $\boldsymbol{\mathcal A}$ with unbounded coefficients. We prove the existence and characterize all the systems of invariant measures for the semigroup $({\bf T}(t))_{t\ge 0}$ associated to $\boldsymbol{\mathcal A}$ in $C_b(\mathbb R^d;\mathbb R^m)$. We also show some relevant properties of the extension of $({\bf T}(t))_{t\ge 0}$ to the $L^p$-spaces related to systems of invariant measures. Finally, we study the asymptotic behaviour of $({\bf T}(t))_{t\ge 0}$ as $t$ tends to $+\infty$.