Nonautonomous Kolmogorov parabolic equations with unbounded coefficients

We study a class of elliptic operators A with unbounded coefficients defined in I x R-d for some unbounded interval I subset of R. We prove that, for any s in I, the Cauchy problem u(s, .) = f in C-b(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.