# Dual representations of Laplace transforms of Brownian excursion and generalized meanders

## TL;DR

This paper establishes dual representations of Laplace transforms for Brownian excursions and generalized meanders, linking their distributions to an auxiliary Markov process with interchanged roles of arguments and times.

## Contribution

It introduces a novel duality framework expressing Laplace transforms of these stochastic processes via an auxiliary Markov process with different initial laws.

## Key findings

- Laplace transform of Brownian excursion distribution expressed via an auxiliary Markov process.
- Similar duality established for generalized meanders.
- Provides new insights into the structure of these stochastic processes.

## Abstract

The Laplace transform of the $d$-dimensional distribution of Brownian excursion is expressed as the Laplace transform of the $(d+1)$-dimensional distribution of an auxiliary Markov process, started from a $\sigma$-finite measure and with the roles of arguments and times interchanged. A similar identity holds for the Laplace transform of a generalized meander, which is expressed as the Laplace transform of the same auxiliary Markov process, with a different initial law.