On the integral kernels of derivatives of the Ornstein-Uhlenbeck semigroup

TL;DR

This paper derives explicit formulas for the integral kernels of derivatives of the Ornstein-Uhlenbeck semigroup using Hermite polynomial expansions, providing an alternative proof of existing kernel estimates.

Contribution

It introduces a closed-form expression for these kernels by expanding the Mehler kernel into Hermite polynomials and applying powers of the Ornstein-Uhlenbeck operator.

Findings

Explicit kernel formulas for derivatives of the Ornstein-Uhlenbeck semigroup

Alternative proof of kernel estimates by Portal (2014)

All relevant quantities made explicit

Abstract

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein-Uhlenbeck semigroup $e^{tL}$. Our approach is to expand the Mehler kernel into Hermite polynomials and applying the powers $L^N$ of the Ornstein-Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for $L$. As an application we give an alternative proof of the kernel estimates by Portal [2014], making all relevant quantities explicit.