  We study asymptotic behavior in a class of non-autonomous second order
parabolic equations with time periodic unbounded coefficients in $\mathbb
R\times \mathbb R^d$. Our results generalize and improve asymptotic behavior
results for Markov semigroups having an invariant measure. We also study
spectral properties of the realization of the parabolic operator $u\mapsto
{\cal A}(t) u - u_t$ in suitable $L^p$ spaces.
