On the Hausdorff Dimension of the Set Generated by Exceptional Oscillations of a Two-Parameter Wiener Process

S. Orey and S. J. Taylor (1974, Proc. London Math. Soc.28, 174–192) proved that for 0⩽λ⩽1 the set E(λ)={t∈[0, 1] : lim suph↓0(2h log(1/h))−1/2 (W′(t+h)−W′(t))⩾λ} has Hausdorff dimension dim E(λ)=1−λ2 a.s. where W′(t) is a standard Wiener process. A corresponding result is obtained when W′ is replaced by a two-parameter Wiener process.