density. In the linear setting, an operator having at least one orbit with that property is said to be *frequently hypercyclic*. This notion was introduced in [@BG3] and extensively studied since then; see the books [@BM] and [@GP] for more information). Hence, to find an ergodic measure with full support is an efficient way of showing that a given operator is hypercyclic, which comes as a measure-theoretic counterpart to the more traditional Baire category approach.

Throughout the paper, we shall restrict ourselves to the best understood infinite-dimensional measures, the so-called *Gaussian* measures. Moreover, the underlying topological vector space $X$ will always be a complex separable Fréchet space. (The reason for considering *complex* spaces only will become clear in the next few lines). In this setting, a Borel probability measure on $X$ is Gaussian if and only if it is the distribution of an almost surely convergent random series of the form $\xi=\sum_0^\infty g_n x_n$, where $(x_n)\subset X$ and $(g_n)$ is a sequence of independent, standard complex Gaussian variables. Given any property (P) relative to measure-preserving transformations, we shall say that an operator $T\in\mathfrak L(X)$ has property (P) *in the Gaussian sense* if there exists some Gaussian probability measure $\mu$ on 