Abstract
It is known that if $B_t$ is a standard Wiener process then $\sup_t\lim \inf_{h\rightarrow 0+}(B_{t + h} - B_t)h^{-1/2} = 1 a.s.$ Here this is sharpened to $P(\exists t: \lim \inf_{h\rightarrow 0+}(B_{t+h} - B_t)h^{-1/2} = 1) = 1$, and $P(\exists t: B_{t + h} - B_t \geq h^{1/2}\forall h \in (0, \alpha)$ for some $\alpha > 0) = 0$. A number of other theorems of the same flavor are proved. Our results for the critical case for slow points are not as complete as the above.
Citation
Burgess Davis. Edwin Perkins. "Brownian Slow Points: The Critical Case." Ann. Probab. 13 (3) 779 - 803, August, 1985. https://doi.org/10.1214/aop/1176992908
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