Brownian Motion Conditioned to Spend Limited Time Below a Barrier

We condition a Brownian motion with arbitrary starting point $y \in $\mathbb{R}$$onspendingatmost$1$timeunitbelow$0$ and provide an explicit description of the resulting process. In particular, we provide explicit formulas for the distributions of its last zero $g=g^y$ and of its occupation time $\mathrm{\Gamma }={\mathrm{\Gamma }}^y$ below $0$ as functions of $y$. This generalizes a result of Benjamini and Berestycki from 2011, which covers the special case $y=0$. Additionally, we study the behavior of the distributions of $g^y$ and ${\mathrm{\Gamma }}^y$, respectively, for $y\to \pm \mathrm{\infty }$.