Let $(B_t)_{t\geq 0}$ the standard brownian motion (with $B_0=0$), $p$ be a real number greater than $1$ and $q$ its conjugate number. Prove that $X_p=\sup _{t\geq 0}(|B_t|-t^{p/2})$ is a.s. strictly positive and finite and has the same law as $\sup_{t\geq 0}\left(\frac{|B_t|}{1+t^{p/2}} \right)^q$.
By the law of the iterated logarithm we know that a.s. $|B_t|<t^{p/2}$ for sufficiently large $t$, so $X_p<+\infty $ a.s. It is also obvious that $X_p\geq 0$. But I have no idea in order to show that $X_p\sim \sup _{t\geq 0}\left(\frac{|B_t|}{1+t^{p/2}}\right)^q$.