Let $B_t$ denote a standard Brownian motion, and $0 < l < u$. I am wondering if the law of $\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}}$ is atomless, that is, $\mathbb{P}\left(\sup_{l \leq t \leq u} \frac{|B_t|}{\sqrt{t}} = c \right) = 0$ for all constant $c$.
Any help will be greatly appreciated!
This seems to follow easily from the strong Markov property. Informally: if this probability were positive for some $c$, then we could stop the process when $t^{-1/2} |B_t| = c$ for the first time, and get a contradiction with the oscillatory character of $B_t$ for small times.
Write $$M = \sup_{l\le t\le u} t^{-1/2} |B_t|$$ and $$\tau = \inf \{t \ge l : t^{-1/2} |B_t| = c\}.$$ Observe that if $M = c$, then $\tau \le u$.
We have $$\mathbb{P}(M = c, \tau = u) \le \mathbb{P}(M = c, u^{-1/2} |B_u| = c) \le \mathbb{P}(u^{-1/2} |B_u| = c) = 0.$$ It is thus enough to show that $\mathbb{P}(M = c, \tau < u) = 0$. To this end, we show that for any $\varepsilon > 0$, $\mathbb{P}(M = c, \tau < u - \varepsilon) = 0$.
We have $$ \mathbb{P}(M = c, \tau < u - \varepsilon) \le \mathbb{P}(\sup_{\tau \le t\le \tau + \varepsilon} t^{-1/2} |B_t| = c, \tau < u) . $$ By the strong Markov property, the right-hand side is equal to $\mathbb{E}(\phi(\tau, X_\tau))$, where $$ \phi(t, x) = \mathbb{P}(\sup_{0 \le s\le \varepsilon} (t + s)^{-1/2} |x + B_s| = c) . $$ However, $\phi(t, x) = 0$ if $t^{-1/2} |x| = c$ (by the law of the iterated logarithm, for example), and therefore $\phi(\tau, X_\tau) = 0$ almost surely.
