Nonautonomous Kolmogorov parabolic equations with unbounded coefficients

We study a class of elliptic operators $A$ with unbounded coefficients defined in $I\times\mathbb{R}^d$ for some unbounded interval $I\subset\mathbb{R}$. We prove that, for any $s\in I$, the Cauchy problem $u(s,\cdot)=f\in C_b(\mathbb{R}^d)$ for the parabolic equation $D_tu=Au$ admits a unique bounded classical solution $u$. This allows to associate an evolution family $\{G(t,s)\}$ with $A$, in a natural way. We study the main properties of this evolution family and prove gradient estimates for the function $G(t,s)f$. Under suitable assumptions, we show that there exists an evolution system of measures for $\{G(t,s)\}$ and we study the first properties of the extension of $G(t,s)$ to the $L^p$-spaces with respect to such measures.