The supremum of Brownian local times on H\"older curves, II

TL;DR

This paper investigates the maximum local time of space-time Brownian motion along H"older continuous curves, establishing finiteness conditions based on the H"older exponent.

Contribution

It proves that the supremum of local times over H"older curves with exponent greater than 1/2 is finite, extending understanding of Brownian local times on irregular paths.

Findings

Supremum of local times is finite for H"older exponent > 1/2.

The result characterizes the regularity needed for bounded local times.

Provides a quantitative threshold for local time behavior on H"older curves.

Abstract

For $f: [0,1]\to \mathbb R$, we consider $L^f_t$, the local time of space-time Brownian motion on the curve $f$. Let ${\cal S}_\alpha$ be the class of all functions whose H\"older norm of order $\alpha$ is less than or equal to 1. We show that the supremum of $L^f_1$ over $f$ in ${\cal S}_\alpha$ is finite if $\alpha>\frac12$.