Mid-concavity of survival probability for isotropic Lévy processes

Abstract

Let $X$ be a symmetric, pure jump, unimodal Lévy process in $\mathbb{R} $ with an infinite Lévy measure. We prove that for any fixed $t > 0$ the survival probability $P^x(\tau _{(-a,a)} > t)$ is nondecreasing on $(-a,0]$, nonincreasing on $[0,a)$ and concave on $(-a/2,a/2)$, where $a > 0$ and $\tau _{(-a,a)}$ is the first exit time of the process $X$ from $(-a,a)$. We also show a similar statement for sets $(-a,a) \times F \subset \mathbb{R} ^d$.