On the Sojourn Time of the Brownian Process in a Multidimensional Sphere

Abstract

We consider the Brownian motion process B^m(s) in the m-space and the distribution

F^m(t, x, a) = P{sup_{0 ≤ s ≤ t}|B^m(s) + ∞| < a}, where a > 0, x ∈ R^m, |x| < a.

There is a probability that a particle starting from the point x on the sphere S_r^m with the radius r = |x| < a will not be absorbed by the sphere S_a^m with a radius a before the epoch t.