UN PROCESSUS PONCTUEL ASSOCIÉ AUX MAXIMA LOCAUX DU MOUVEMENT BROWNIEN

Christophe Leuridan (1)

1 IF - Institut Fourier

Let $B = (B_t)_{t \in \rrf}$ be a symmetric brownian motion, so $(B_t)_{t \in \rrf_+}$ and $(B_{-t})_{t \in \rrf_+}$ are indépendent brownian motions. Given $a \ge b>0$, we give the law of the random set $${\cal M}_{a,b} = \{t \in {\bf R} : B_t = \max_{s \in [t-a,t+b]} B_s\}.$$ Using the relation connecting this set with the closed regenerative set $${\cal R}_a = \{t \in {\bf R}_+ : B_t = \max_{s \in [(t-a)_+,t]} B_s\},$$ we describe the Lévy measure of a subordinator whose closed range is ${\cal R}_a$.

Mots clés: subordinator

Mots clés: brownian motion; local maxima; point process; renewal process; closed régénérative set; subordinator.

Domaines: Probabilités [math.PR]

Christophe Leuridan. UN PROCESSUS PONCTUEL ASSOCIÉ AUX MAXIMA LOCAUX DU MOUVEMENT BROWNIEN. Probability Theory and Related Fields, 2010, 148 (3-4), pp.457-477. ⟨hal-00294870⟩