Suppose that (B1(t), B2(t)) is a Brownian motion in R2, and define τa = inf{t : B1(t) = a}. Then X(a) = B2(τ a ) is the value of B2 when B1 hits a. The process {X(a) : a ≥ 0} is, of course, a Brownian...

Suppose that (B1(t), B2(t)) is a Brownian motion in R2, and define τa = inf{t : B1(t) = a}. Then X(a) = B2(τ_a) is the value of B2 when B1 hits a. The process {X(a) : a ≥ 0} is, of course, a Brownian motion directed by {τ_a
: a ≥ 0}. Show that X has stationary independent increments and that X(a) has a Cauchy distribution with density