On the shape of the ground state eigenfunction for stable processes

摘要

We prove that the ground-state eigenfunction for symmetric stable processes of order alpha is an element of (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.