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

TL;DR

This paper characterizes the distribution of local maxima of a symmetric Brownian motion within specific intervals and describes the associated Lévy measure of a subordinator linked to these maxima, revealing new structural insights.

Contribution

It provides a detailed description of the law of local maxima sets and the Lévy measure of a related subordinator for symmetric Brownian motion, a novel analysis in stochastic process theory.

Findings

Law of the set of local maxima within specified intervals

Lévy measure of the subordinator associated with maxima

Structural properties of the regenerative set of maxima

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 $${\cal R}_a = \{t \in {\bf R}\_+ : B_t = \max_{s \in [(t-a)_+,t]} B_s\}.$$