# Levels at which every brownian excursion is exceptional

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## Abstract

Levels at which every Brownian excursion is exceptional SÉMINAIRE DE PROBABILITÉS (STRASBOURG) MARTIN T. BARLOW EDWIN A. PERKINS Levels at which every Brownian excursion is exceptional Séminaire de probabilités (Strasbourg), tome 18 (1984), p. 1-28. <http://www.numdam.org/item?id=SPS_1984__18__1_0> © Springer-Verlag, Berlin Heidelberg New York, 1984, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml), implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Levels at which every Brownian excursion is exceptional M.T. Barlow, E. Perkins Statistical Laboratory 16 Mill Lane Cambridge CB2 1SB England Dept. of Mathematics University of British Columbia Vancouver British Columbia V6T 1Y4 Canada 1. Introduction. Let Bt be a Brownian motion starting at 0 , defined on a complete filtered probability space (Q, F, F , P) which satisfies the usual conditions. For each x , let AX(B) be the set of starting times of excursions of B above the level x, , and let A(B) be the set of starting times of excursions of B above some level: that is 2 = AX(B) = {t : : Bt = x, Bs > x for t s t + e for some e > 0}, ’ A = A(B) = u AX(B) . . x Let t E , and f : : -~tR+ be a continuous strictly increasing function with f(0) - 0 . We will say that f is a lower function (respectively, upper function) for B at t if there is a 6(w) > 0 such that f(u) for all u e [0,6] (respectively, ~t~~ ~ for [O~D . ° If A c A , , and f is a lower (upper) function for B at t for all tEA, , we will say that f is a uniform lower (upper) function for B on

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