On the shape of the ground state eigenfunction for stable processes

Abstract

We prove that the ground state eigenfunction for symmetric stable processes of order $\alpha \in (0,2)$ killed upon leaving the interval $(-1,1)$ is concave on $(-1/2,1/2)$. We call this property "mid--concavity." A similar statement holds for rectangles in ${\mathbb{R}}^d$, $d>1$. These result follow from similar results for finite dimensional distributions of Brownian motion and subordination.