The supremum of Brownian local times on Holder curves

TL;DR

This paper investigates the maximum local time of space-time Brownian motion along functions with bounded H"older norm, establishing finiteness for exponents greater than 1/2 and divergence for smaller exponents.

Contribution

It characterizes the supremum of Brownian local times on H"older curves, revealing a phase transition at the critical exponent 1/2.

Findings

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

Supremum is infinite for H"older exponent < 1/2.

Identifies a critical H"older exponent at 1/2.

Abstract

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