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 $P^z(\tau_{U^+}>s\mid \tau_{R^+}>t)\leq P^z(\tau_{U}>s\mid \tau_{R}>t)$ for every $s,t>0$ and every $z\in U^+$.