Controle de la Norme $H^p$ D'Une Martingale par des Maximums de Temps Locaux

Abstract

Let $B$ be a brownian motion starting at 0. We denote by $L^\ast_t = \max_{x\in\mathbb{R}} L^\ast_t$ the maximum of local times at time $t$. The Barlow-Yor inequalities tell us that for every $p > 0$, there are constants $C_p > c_p > 0$ such that for every stopping time $\tau$, $c_p\mathbb{E}\lbrack\tau^{p/2}\rbrack \leq \mathbb{E}\lbrack L^{\ast p}_\tau\rbrack \leq C_p\mathbb{E}\lbrack\tau^{p/2}\rbrack.$ Given a fixed closed set $F \subset \mathbb{R}$, we give a condition on $F$ which is necessary and sufficient to derive similar inequalities with $\max_{x\in F}L^x_\tau$ instead of $L^\ast_\tau$ and we prove various related results.