On a Brownian excursion law, I: convolution representations

TL;DR

This paper characterizes the law of Brownian motion with a minimal length excursion below a set level, providing explicit convolution-based representations and a layer construction approach.

Contribution

It introduces an explicit convolution representation for the law of Brownian excursions with minimal length, advancing understanding of their probabilistic structure.

Findings

Explicit convolution representations of the excursion law

Layer construction method for the process over time

Characterization of the process law in terms of sums of independent variables

Abstract

This paper studies Brownian motion subject to the occurrence of a minimal length excursion below a given excursion level. The law of this process is determined. The characterization is explicit and shows by a layer construction how the law is built up over time in terms of the laws of sums of a given set of independent random variables.