Range of Brownian Motion with Drift

Let (B δ (t)) t ≥ 0 be a Brownian motion starting at 0 with drift δ > 0. Define by induction S 1=− inf t ≥ 0 B δ (t), ρ1 the last time such that B δ (ρ1)=−S 1, S 2=sup0≤ t ≤ρ 1 B δ (t), ρ2 the last time such that B δ (ρ2)=S 2 and so on. Setting A k =S k +S k+1; k ≥ 1, we compute the law of (A 1,...,A k ) and the distribution of (B δ (t+ρ l) − B δ (ρ l ); 0 ≤ t ≤ ρ l-1 − ρ l )2 ≤ l ≤ k for any k ≥ 2, conditionally on (A 1,...,A k ). We determine the law of the range R δ (t) of (B δ (s)) s≥ 0 at time t, and the first range time θδ (a) (i.e. θδ (a)=inf{t > 0; R δ (t) > a}). We also investigate the asymptotic behaviour of θ δ (a) (resp. R δ (t)) as a → ∞ (resp. t → ∞).