For a transient Bessel process X let I(t) = in fs＞tX(s) and§(t) = inf{u≥ 2 t: X(u) = I(t)}. In this note we compute the joint distribution of I(t),§(t) and Xt.
Let X = {Xt,t >=0} be a d-dimensional (d>= 2) standard Brownian motion with drift c started at a fixed x, and BR = {x E Rd: |x| < R}, the ball centered at 0 with radius R. Consider the distributions of TR(t) and TR(∞), where TR(t) denotes the time spent by X in BR up to time t and TR(∞) the total time of X spent in BR. Explicit formulas for the Laplace transform of TR(∞) and the double Laplace transform of TR(t) are obtained.
