Given a path $x:[0,T]\to\mathbb{R}$, and $p\ge 1$, we define the $p$-variation of $x$ by $$\|x\|_p := \sup_{\pi}\left(\sum_{[s,t]\in\pi} |x_t - x_s|^p\right)^{1/p},$$ where the supremum is taken over all (finite) partitions of $[0,T]$. (Note that the mesh size of the partition is not assumed to go to $0$!)
If $B:[0,T]\to\mathbb{R}$ is a Brownian motion, it is well-known that the $p$-variation of $B$ is almost surely infinite for $p\le 2$, but almost surely finite for $p>2$. Is there anything known about the distribution of $\|B\|_p$, like the existence of moments/exponential moments?
