The Hausdorff dimension of quasi-all Brownian paths (Q801406)

The Hausdorff dimension of range of w of d-dimensional standard Wiener space W (d\(\geq 2)\) is 2 with probability 1. A refinement of this property is given: the Hausdorff dimension of range of w is 2 except on a set of zero capacity, where the capacity is the one induced by a Dirichlet form related to the Ornstein-Uhlenbeck operator on W. The main point of proof is to estimate the Hausdorff dimension of range from below. This is done through showing that \(\int^{1}_{0}\int^{1}_{0}| w(t)- w(s)|^{-\alpha}dsdt\) \((\alpha <2)\) has finite Dirichlet norm and that this functional is finite except on a set of zero capacity.