On the structure of Gaussian random variables

We study when a given Gaussian random variable on a given probability space $(\Omega, {\cal{F}}, P)$ is equal almost surely to $\beta_{1}$ where $\beta$ is a Brownian motion defined on the same (or possibly extended) probability space. As a consequences of this result, we prove that the distribution of a random variable (satisfying in addition a certain property) in a finite sum of Wiener chaoses cannot be normal. This result also allows to understand better some characterization of the Gaussian variables obtained via Malliavin calculus.