  We extend to the vector-valued situation some earlier work of Ciesielski and
Roynette on the Besov regularity of the paths of the classical Brownian motion.
We also consider a Brownian motion as a Besov space valued random variable. It
turns out that a Brownian motion, in this interpretation, is a Gaussian random
variable with some pathological properties. We prove estimates for the first
moment of the Besov norm of a Brownian motion. To obtain such results we
estimate expressions of the form $\E \sup_{n\geq 1}\|\xi_n\|$, where the
$\xi_n$ are independent centered Gaussian random variables with values in a
Banach space. Using isoperimetric inequalities we obtain two-sided inequalities
in terms of the first moments and the weak variances of $\xi_n$.
