A Sharp Inequality for Conditional Distribution of the First Exit Time of Brownian Motion

Abstract

Let $U$ be a domain, convex in $x$ and symmetric about the y-axis, which is contained in a centered and oriented rectangle $R$. \linebreak If ${\tau }_A$ is the first exit time of Brownian motion from $A$ and $A^{+}=A\cap \{(x,y):x>0\}$, it is proved that $P^z({\tau }_{U^{+}}>s\mid {\tau }_{R^{+}}>t)\le P^z({\tau }_U>s\mid {\tau }_R>t)$ for every $s,t>0$ and every $z\in U^{+}$.

12 pages