Brownian Crossings via Regeneration Times

Abstract

Let {B t , t ≥ 0} be a standard one-dimensional Brownian motion. For each t < 0 let σ t be the last entrance time before t into the interval (a,b), d t the time of the first exit from (a,b) after t and Yt:=Bt−Bσt. In this paper we study i) the limit behaviour of the normalised occupation times of the process (Y t ), ii) the limiting joint distribution of (t − σ t , d t  − t) and (dt−t,Bdt−Bt), conditioned on the event {B t  ∈ (a,b)}, as t → ∞ and iii) derive renewal equations satisfied by the probabilities ϕ(t):=Pa{0<t−σt<u, 0<Bt−Bσt<y} and γ(t):=Pa{0<dt−t<u, 0<Bdt−Bt<y}.