Asymptotic behavior in time periodic parabolic problems with unbounded coefficients

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.