Range of Brownian motion with drift

Let (B-delta(t))(t) >= 0 be a Brownian motion starting at 0 with drift delta > 0. Define by induction S-1 = -inf(t >=) (0) B-delta (t), rho(1) the last time such that B-delta(rho(1)) = -S-1, S-2 = sup(0) (<= t) (<=) rho(1) B-delta(t), rho(2) the last time such that B-delta(rho(2)) = S-2 and so on. Setting A(k) = S-k + Sk+1; k >= 1, we compute the law of (A(1),..., A(k)) and the distribution of (B-delta(t + rho(l)) - B-delta(rho(l)); 0 <= t <= (rho l - 1) (- rho l))(2) (<= l) (<= k) for any k >= 2, conditionally on (A(1),..., A(k)). We determine the law of the range R-delta(t) of (B-delta(s))(s) (>=) (0) at time t, and the first range time theta(delta)(a) (i.e. theta(delta)(a) = inf {t > 0; R-delta(t) > a}). We also investigate the asymptotic behaviour of theta(delta)(a) (resp. R-delta(t)) as a -> infinity (resp. t -> infinity).