TITLE: Brownian motion in $n$ dimensions
QUESTION [5 upvotes]: Consider a particle starting at the origin in $\mathbb{R}^n$ and undergoing Brownian motion. Is there an expression known for the probability of the particle hitting the sphere $S^{n - 1}_r = \{x \in \mathbb{R}^n : \Vert x\Vert = r\}$ within time $t$ (obviously such an expression, if known, will be in terms of $r$ and $t$)? In other words, I am looking for a formula for $\mathbb{P}(\sup _{s \leq t}\Vert B(s)\Vert \geq r)$. Thanks!

REPLY [2 votes]: The distributions and properties  of the maximum of a Bessel processes and  Bessel bridges are discussed in details in
Jim Pitman: The law of of the maximum of a Bessel bridge