Hausdorff measure of the level sets of multi-parameter wiener processes in one dimension

LetZ _N (t) be anN-parameter Wiener process in one dimension, and $$E(x,T) = \left\{ {t = (t_1 , \cdot \cdot \cdot ,t_N ):Z_N (t) = x,0 \leqslant t_1 , \cdot \cdot \cdot ,t_N \leqslant T} \right\}$$ . Then we obtain that with probability one, the Hausdorff measure function ofE(x,T) is $$\psi _N (r) = r^{N - \tfrac{1}{2}} (\log \log r^{ - 1} )^{\tfrac{1}{2}} ,\forall r \in (0,\frac{1}{4})$$ for anyx∈R ¹ andT>0.