Strong mixing Gaussian measures for chaotic semigroups

In this paper we will be concerned with the problem of the existence of an invariant mixing measure considering its connection with the chaotic behavior of linear semigroups on separable Banach spaces. We first prove an identity characterizing invariant Gaussian measure involving its covariance operator and the infinitesimal generator of the semigroup. This gives an answer to a question raised by Rudnicki in his inspiring review paper [32]. Under suitable conditions, we use the proved identity to give an invariant mixing Gaussian measure as distribution of a Wiener integral.