Brownian Slow Points: The Critical Case

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.