On invariant measures associated to weakly coupled systems of Kolmogorov equations

Abstract

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