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WROCŁAW UNIVERSITY
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Contents of PMS, Vol. 28, Fasc. 1,
pages 143 - 162
 

ON BESOV REGULARITY OF BROWNIAN MOTIONS IN INFINITE DIMENSIONS

Tuomas P. Hytönen
Mark C. Veraar

Abstract: 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 $\xi _n$ 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 $\xi _n$.

2000 AMS Mathematics Subject Classification: Primary: 60J65; Secondary: 28C20, 46E40, 60G17.

Key words and phrases: Gaussian random variable, maximal estimates, Besov-Orlicz norm, non-separable Banach space, sample path.

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