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 $Y_t := B_t - B_{\sigma_t}$. 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 $(d_t-t, B_{d_t}- B_t)$, conditioned on the event {B _t  ∈ (a,b)}, as t → ∞ and iii) derive renewal equations satisfied by the probabilities $\phi (t) := P_a \{ 0 < t-\sigma_t < u,~ 0< B_t -B_{\sigma_t} < y\}$ and $\gamma(t) := P_a\{0 < d_t - t < u, ~0 < B_{d_t} -B_{t}< y \}$.