Un processus ponctuel associ\'e aux maxima locaux du mouvement brownien

Abstract

Let $B = (B_t)_{t \in {\bf R}}$ be a symmetric Brownian motion, i.e. $(B_t)_{t \in {\bf R}_+}$ and $(B_{-t})_{t \in {\bf R}_+}$ are independent Brownian motions starting at $0$. Given $a \ge b>0$, we describe the law of the random set $${\cal M}_{a,b} = \{t \in {\bf R} : B_t = \max_{s \in [t-a,t+b]} B_s\},$$ and we describe the L\'evy measure of a subordinator whose closed range is the regenerative set $${\cal R}_a = \{t \in {\bf R}\_+ : B_t = \max_{s \in [(t-a)_+,t]} B_s\}.$$