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

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

2005-02-02

arxiv.org/abs/math/0502057v2

Mathematics

Probability

12 pages

Scientific paper

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 $z\in U^+$.

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