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We extend to the vector-valued situation some earlier work of Ciesielski andRoynette 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. Itturns out that a Brownian motion, in this interpretation, is a Gaussian randomvariable with some pathological properties. We prove estimates for the firstmoment of the Besov norm of a Brownian motion. To obtain such results weestimate expressions of the form $\E \sup_{n\geq 1}\|\xi_n\|$, where the$\xi_n$ are independent centered Gaussian random variables with values in aBanach space. Using isoperimetric inequalities we obtain two-sided inequalitiesin terms of the first moments and the weak variances of $\xi_n$.

On Besov regularity of Brownian motions in infinite dimensions