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

Jonas Teuwen

doi:10.1142/S0219025716500302

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

Jonas Teuwen

Analysis

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Abstract

This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup e tL etL. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers L N LN of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for L L. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.

Original language	English
Article number	1650030
Pages (from-to)	1-13
Number of pages	13
Journal	Infinite Dimensional Analysis, Quantum Probability and Related Topics
Volume	19
Issue number	4
DOIs	https://doi.org/10.1142/S0219025716500302
Publication status	Published - 2016

Keywords

Ornstein–Uhlenbeck
Mehler kernel
Gaussian measure
Hermite polynomials
Caldéron reproducing formula

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10.1142/S0219025716500302

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abstract = "This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup e tL etL. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers L N LN of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for L L. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.",

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AB - This paper presents a closed-form expression for the integral kernels associated with the derivatives of the Ornstein–Uhlenbeck semigroup e tL etL. Our approach is to expand the Mehler kernel into Hermite polynomials and apply the powers L N LN of the Ornstein–Uhlenbeck operator to it, where we exploit the fact that the Hermite polynomials are eigenfunctions for L L. As an application we give an alternative proof of the kernel estimates by Ref. 10, making all relevant quantities explicit.

KW - Ornstein–Uhlenbeck

KW - Mehler kernel

KW - Gaussian measure

KW - Hermite polynomials

KW - Caldéron reproducing formula

U2 - 10.1142/S0219025716500302

DO - 10.1142/S0219025716500302

M3 - Article

VL - 19

SP - 1

EP - 13

JO - Infinite Dimensional Analysis, Quantum Probability and Related Topics

JF - Infinite Dimensional Analysis, Quantum Probability and Related Topics

SN - 0219-0257

IS - 4

M1 - 1650030

ER -

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

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Keywords

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