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! 



Brownian motion in $n$ dimensions
The process $\|B(t)\|$ is called $n$-dimensional Bessel process (or Bessel process with parameter $\nu=\frac{n}{2}-1$). I think formula $\bf 4$.1.1.4 of Borodin-Salminen "Handbook of Brownian Motion -Facts and Formulae" is what you're looking for.


