Range of Brownian motion with drift

Let (Bδ(t))t≥0 be a Brownian motion starting at 0 with drift δ > 0. Define by induction S1 = − inf t≥0 Bδ(t), ρ1 the last time such that Bδ(ρ1) = −S1, S2 = sup 0≤t≤ρ1 Bδ(t), ρ2 the last time such that Bδ(ρ2) = S2 and so on. Setting Ak = Sk + Sk+1 ; k ≥ 1, we compute the law of (A1, · · · , Ak) and the distribution of ((Bδ(t + ρl) − Bδ(ρl) ; 0 ≤ t ≤ ρl−1 − ρl))2≤l≤k for any k ≥ 2, conditionally on (A1, · · · , Ak). We determine the law of the range Rδ(t) of (Bδ(s))s≥0 at time t, and the first range time θδ(a) (i.e. θδ(a) = inf{t > 0 ; Rδ(t) > a}). We also investigate the asymptotic behaviour of θδ(a) (resp. Rδ(t)) as a→∞ (resp. t→∞).