Local times in a Brownian excursion

TL;DR

This paper characterizes the distribution of local times at various levels during a Brownian excursion and applies these results to estimate certain integrals, advancing understanding of Brownian motion properties.

Contribution

It provides a detailed distributional analysis of local times in Brownian excursions and demonstrates their application in integral estimation.

Findings

Distribution of local times at each level is explicitly determined.

Results enable improved estimation of L^1 integrals on the real line.

Enhances understanding of Brownian excursion properties.

Abstract

Let $\{B(t), t \geq 0\}$ be a standard Brownian motion in $\mathbb{R}$. Let $T$ be the first return time to 0 after hitting 1, and $\{L(T,x), x \in \mathbb{R}\}$ be the local time process at time $T$ and level $x$. The distribution of $L(T,x)$ for each $x \in \mathbb{R}$ is determined. This is applied to the estimation of a $L^1$ integral on $\mathbb{R}$.