Poisson kernels on the half-plane are bell-shaped

TL;DR

This paper proves that the Poisson kernel associated with a class of elliptic operators in the half-plane exhibits a bell-shaped profile, with derivatives changing sign in a structured manner, indicating unimodality and specific concavity properties.

Contribution

The paper establishes the bell-shaped nature of the Poisson kernel for a class of elliptic operators, revealing detailed derivative sign changes and concavity features.

Findings

Poisson kernel is bell-shaped with derivatives changing sign n times

Poisson kernel is unimodal with two inflection points

Kernel exhibits concave, convex, then concave behavior

Abstract

Consider a second-order elliptic operator $L$ in the half-plane $\mathbb R \times (0, \infty)$ with coefficients depending only on the second coordinate. The Poisson kernel for $L$ is used in the representation of positive $L$-harmonic functions, that is, solutions of $L u = 0$. In probabilistic terms, the Poisson kernel is the density function of the distribution of the diffusion in $\mathbb R \times (0, \infty)$ with generator $L$ at the hitting time of the boundary. We prove that the Poisson kernel for $L$ is bell-shaped: its $n$th derivative changes sign $n$ times. In particular, it is unimodal and it has two inflection points (it is concave, then convex, then concave again).