What is the measure of the amount of time during which a Brownian motion $B_t$ is increasing?
I know that the expectation of this amount of time is $\frac{t}{2}$, and I was thinking to write this set as a limit, $\lim_{dt \rightarrow 0} \{ t: dB_{t+dt} > dB_t\}$.
I know Green measure is used for finding such measures, but this would be the Green measure of a highly disconnected set, so I don't know how to proceed.