Let $W_t$ be an one-dimensional standard Brownian motion, and $\theta_s$ is the shift such that $\theta_s( W_t)=W_{t+s}-W_s$, then are there any reference available regarding the distribution of the following random variable: $$\sup_{s} \theta_s(W_t)=\sup_{s} (W_{t+s}-W_s).$$
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$\begingroup$ Over what range does $s$ vary? $\endgroup$– Martin HairerCommented Nov 1, 2013 at 10:01
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$\begingroup$ We can consider s range over a finite domain first, say, $s\in [0,T-t]$? $\endgroup$– YueCommented Nov 1, 2013 at 10:19
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$\begingroup$ For $t$ fixed and large $T$, the supremum will then behave like $\sqrt{\log T}$. $\endgroup$– Martin HairerCommented Nov 1, 2013 at 13:53
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$\begingroup$ I wonder what is the distribution for $\lim_{s\to \infty}\theta_s W_t$. $\endgroup$– YueCommented Nov 1, 2013 at 14:35
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$\begingroup$ How about $\lim_{t\to 0}\sup_{s}\theta_s(W_t)$? $\endgroup$– YueCommented Nov 11, 2013 at 13:41
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