On the shape of the ground state eigenfunction for stable processes
  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 $\R^d$, $d>1$. These result follow from similar results for finite dimensional distributions of Brownian motion and subordination. 
Rodrigo Banuelos, Tadeusz Kulczycki and Pedro J. Mendez-Hernandez