Université Paris 6Pierre et Marie Curie Université Paris 7Denis Diderot CNRS U.M.R. 7599 Probabilités et Modèles Aléatoires''

### Hardy's inequality in $L^{2}$ ([0,1]) and principal values of Brownian local times.

Auteur(s):

Code(s) de Classification MSC:

• 60F05 Central limit and other weak theorems
• 60F25 $L^p$-limit theorems
• 60G15 Gaussian processes
• 60G17 Sample path properties
• 60G48 Generalizations of martingales

Résumé: We present in a unified framework two examples of a random function $\phi \left( \omega ,s\right)$ on $\Re _{+}$ such that (a) the integral $% \int_{0}^{\infty }\phi \left( \omega ,s\right) g\left( s\right) ds$ is well defined and finite (at least, as a limit in probability) for every deterministic and square integrable function $g$, and (b) $\phi$ does not belong to $L^{2}\left( \Re ^{+},ds\right)$ with probability one. In particular, the second example is related to the existence of principal values of Brownian local times. Our key tools are Hardy's inequality, some semimartingale representation results for Brownian local times due to Ray, Knight and Jeulin, and the reformulation of certain theorems of Jeulin-Yor (1979) and Cherny (2001) in terms of bounded $L^{2}$ operators. We also establish, in the last paragraph, several weak convergence results.

Mots Clés: Brownian motion ; local times ; principal values ; Hardy inequality

Date: 2001-06-06

Prépublication numéro: PMA-679