Nonautonomous Kolmogorov parabolic equations with unbounded coefficients

We study a class of elliptic operators $A$ with unbounded coefficients defined in $I\times {\text{CR}}^d$ for some unbounded interval $I\subset \text{CR}$. We prove that, for any $s\in I$, the Cauchy problem $u(s,\cdot )=f\in C_b({\text{CR}}^d)$ for the parabolic equation $D_{t}u=Au$ 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.