Consider a Brownian Motion starting at the center of a ball. I commonly read the following statement:
By rotational symmetry of Brownian Motion, it follows $P(B_{\tau})$ is a uniformly distributed probability measure on the ball.
(Here $P$ is the gaussian with mean at the origin of the ball and $B_{\tau}$ is Brownian motion that terminates on the surface of the ball at time equals $\tau$)
Intuitively, I think of a cube. The probability of hitting one face of the cube is 1/6, because for every path hitting one face, there will be 5 other reflected paths that hit the other faces. (That's where symmetry is doing the work). Now I somehow increase the number of faces to get a ball.
What's a formal argument for the highlighted statement though? I'm having some trouble with this.