On the Dirichlet and Neumann evolution operators in R^d_+

We prove some uniform and pointwise gradient estimates for the Dirichlet and the Neumann evolution operators $G_{\mathcal{D}}(t,s)$ and $G_{\mathcal{N}}(t,s)$ associated with a class of nonautonomous elliptic operators $\A(t)$ with unbounded coefficients defined in $I\times \Rd_+$ (where $I$ is a right-halfline or $I=\R$). We also prove the existence and the uniqueness of a tight evolution system of measures $\{\mu_t^{\mathcal{N}}\}_{t \in I}$ associated with $G_{\mathcal{N}}(t,s)$, which turns out to be sub-invariant for $G_{\mathcal{D}}(t,s)$, and we study the asymptotic behaviour of the evolution operators $G_{\mathcal{D}}(t,s)$ and $G_{\mathcal{N}}(t,s)$ in the $L^p$-spaces related to the system $\{\mu_t^{\mathcal{N}}\}_{t \in I}$.