What is the probability density function or probability distribution of the time when 3-dimensional Brownian motion (no drift) starting from origin hits a sphere (ball) centered at the origin for the first time?
Brownian motion is also known as the Wiener process. For simplicity and I think without loss of generality, the 3-dimensional normal Wiener process can be parameterized such a way that the random variable follows in each dimension a zero-mean normal distribution of variance $t$ as function of time $t$. Likewise, the sphere radius can be set to 1.
The first hitting time can also be viewed as the exit or killing time in a killed Wiener process. I'm looking for an answer that leads to a way to generate pseudo random numbers from the first exit time distribution.