We study frequent hypercyclicity in the context of strongly continuous semigroups of operators. More precisely, we give a criterion (sufficient condition) for a semigroup to be frequently hypercyclic, whose formulation depends on the Pettis integral. This criterion can be verified in certain cases in terms of the infinitesimal generator of semigroup. Applications are given for semigroups generated by Ornstein-Uhlenbeck operators, and especially for translation semigroups on weighted spaces of p-integrable functions, or continuous functions that, multiplied by the weight, vanish at infinity. MSC2010: 47A16, 47D06

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The chaotic and hypercyclic behavior of the C0-semigroups of operators generated by a perturbation of the Ornstein-Uhlenbeck operator with a multiple of the identity in L 2 (R N ) is investigated. Negative and positive results are presented, depending on the signs of the real parts of the eigenvalues of the matrix appearing in the drift of the operator.Mathematics Subject Classification (2010). Primary 47A16; Secondary 47D06, 47D07.

We study different qualitative properties of the semigroup generated by some degenerate differential elliptic operators on the standard simplex of R d . Some methods are new and are based on the representation formulas of the semigroup in terms of iterates of suitable positive operators. The main result is the ultracontractivity property which is obtained in the setting of weighted L p -spaces. We describe the asymptotic behavior of the semigroup and obtain the compactness property in the same setting and also in spaces of continuous functions.