On the distribution of the Brownian motion process on its way to hitting zero

TL;DR

This paper investigates the distributional properties of a Brownian motion process conditioned on hitting zero, providing functional characterizations of related distributions.

Contribution

It introduces functional versions of recent distributional results for Brownian motion hitting zero, extending univariate to functional distributions.

Findings

Derived new functional distribution formulas for Brownian motion hitting zero.

Extended univariate distribution results to functional settings.

Provided insights into the behavior of Brownian paths approaching zero.

Abstract

We present functional versions of recent results on the univariate distributions of the process $V_{x,u} = x + W_{u\tau(x)},$ $0\le u\le 1$, where $W_\bullet$ is the standard Brownian motion process, $x>0$ and $\tau (x) =\inf\{t>0 : W_{t}=-x\}$.