Poisson kernels on the half‐plane are bell‐shaped

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