THE SUPREMUM OF BROWNIAN LOCAL TIMES ON HÖLDER CURVES LE SUPREMUM DU TEMPS LOCAUX D’UN MOUVEMENT BROWNIEN SUR LES COURBES HOLDERIENNES Short title: Brownian local times
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چکیده
For f : [0, 1]→ R, we consider Lft , the local time of spacetime Brownian motion on the curve f . Let Sα be the class of all functions whose Hölder norm of order α is less than or equal to 1. We show that the supremum of Lf1 over f in Sα is finite is α > 12 and infinite if α < 1 2 . Abstrait: Soit Wt un mouvement brownien et soit L f t le temps local du processus (t,Wt) pour le courbe f : [0, 1] → R, c’est à dire, Lft = limε→0 1 2ε ∫ t 0 1]f(s)−ε,f(s)+ε[(Ws)ds. Soit Sα la classe de toutes fonctions telle que la norme holderienne du ordre α est moins de 1. Nous démontrons que supf∈Sα L f 1 <∞ p.s. si α > 12 et ce supremum est infini p.s. si α < 1 2 . AMS subject classification: 60J65, 60J55 Research partially supported by NSF grant DMS-9700721. 1
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