The supremum of Brownian local times on Holder curves

Abstract

For $f:[0,1]\to \mathbb{R}$, we consider $L_t^f$, the local time of space-time Brownian motion on the curve $f$. Let $\sS_\al$ be the class of all functions whose Hölder norm of order $\al$ is less than or equal to 1. We show that the supremum of $L_1^f$ over $f$ in $\sS_\al$ is finite is $\al>\frac12$ and infinite if $\al<\frac12$.

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