On Besov Regularity of Brownian Motions in Infinite Dimensions

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 supn≥1 ‖ξn‖, where the ξn 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 ξn.