Un processus ponctuel associé aux maxima locaux du mouvement brownien

Abstract

Let $B=(B_t{)}_{t\in \mathbf{R}}$ be a symmetric Brownian motion, i.e. $(B_t{)}_{t\in {\mathbf{R}}_{+}}$ and $(B_{-t}{)}_{t\in {\mathbf{R}}_{+}}$ are independent Brownian motions starting at $0$. Given $a\ge b>0$, we describe the law of the random set $${\mathcal{M}}_{a,b}=\{t\in \mathbf{R}:B_t=\underset{s\in [t-a,t+b]}{\max }{B}_s\},$$ and we describe the Lévy measure of a subordinator whose closed range is the regenerative set $${\mathcal{R}}_a=\{t\in \mathbf{R}\_+:B_t=\underset{s\in [(t-a{)}_{+},t]}{\max }{B}_s\}.$$

20 pages, 2 figures.