The supremum of Brownian local times on Holder curves

Abstract

For f : [maps the set] [0, 1] [into the set of real numbers] R, we consider L ([to the power of] f [subscript] t), the local time of spacetime Brownian motion on the curve f. Let S [subscript alpha] be the class of all functions whose Holder norm of order [alpha] is less than or equal to 1. We show that the supremum of L ([to the power of] f [subscript] 1) over f in S [subscript alpha] is finite if [alpha] > 1/2 and infinite if [alpha] < 1/2.