# On the integral kernels of derivatives of the Ornstein–Uhlenbeck semigroup

Research paper by Jonas Teuwen

Indexed on: 15 Dec '16Published on: 23 Nov '16Published in: Infinite Dimensional Analysis, Quantum Probability and Related Topics

#### Abstract

Infinite Dimensional Analysis, Quantum Probability and Related Topics, Ahead of Print. This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup [math]. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers [math] of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for [math]. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.