Compactness and asymptotic behavior in nonautonomous linear parabolic equations with unbounded coefficients in ℝd

We consider a class of second-order linear nonautonomous parabolic equations in ℝd with time periodic unbounded coefficients. We give sufficient conditions for the evolution operator G(t, s) be compact in C b(ℝd) for t > s, and describe the asymptotic behavior of G(t, s)f as t – s → ∞ in terms of a family of measures μs, s ϵ ℝ, solution of the associated Fokker-Planck equation