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

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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 Cb(Rᵈ) for t>s, and describe the asymptotic behavior of G(t,s)f as t-s goes to infinity in terms of a family of measures muₛ, s in R, solution of the associated Fokker-Planck equation.