On the Sojourn Time of the Brownian Process in a Multidimensional Sphere - Steišūnas, S. (Nonlinear Analysis: Modelling and Control, 2009)
We consider the Brownian motion process Bm(s) in the m-space and the distribution Fm(t, x, a) = Psup0 ≤ s ≤ t|Bm(s) + ∞| < a, where a > 0, x ∈ Rm, |x| < a. There is a probability that a particle starting from the point x on the sphere Srm with the radius r = |x| < a will not be absorbed by the sphere Sam with a radius a before the epoch t.