Let $B_t$, $t \in [0,1]$ be a standard Brownian motion. We call a time $t$ fast up if $$ \limsup_{h \searrow 0} \frac{B(t+h) - B(t)}{\sqrt{2 h \ln(1/h)}} =1. $$ (Note the absence of absolute value here.) Likewise call a time fast down if the $\liminf$ is $-1$.

Question: Are there times that are fast up but not fast down? (I think yes, but got stuck in the proof. Perhaps naively, I guess all fast times are either up or down.)

This is vaguely related to Can a Brownian motion be fast at its extrema? (which itself was motivated by Location of maximum of Brownian motion with rough drift. The relevance to the original question is that I suspect the set of fast down only times contains the argmax of the sum, with positive probability, so there is no density.)


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