On a nonsymmetric Ornstein-Uhlenbeck semigroup and its generator

TL;DR

This paper introduces a nonsymmetric Ornstein-Uhlenbeck operator on complex Wiener space by adding a rotation term, explores its eigenfunctions, and demonstrates hypercontractivity properties similar to classical results.

Contribution

It extends the classical Ornstein-Uhlenbeck framework by incorporating a rotation term, resulting in a nonsymmetric operator with explicit eigenfunctions and hypercontractivity.

Findings

Eigenfunctions of the nonsymmetric operator are explicitly characterized.

Hypercontractivity for the nonsymmetric Ornstein-Uhlenbeck semigroup is established.

The operator remains normal despite nonsymmetry.

Abstract

If we add a simple rotation term to both the Ornstein-Uhlenbeck semigroup and the definition of the H-derivative, then analogue to the classical Malliavin calculus on the real Wiener space [I. Shigekawa, Stochastic analysis, 2004], we get a normal but nonsymmetric Ornstein-Uhlenbeck operator $L$ on the complex Wiener space. The eigenfunctions of the operator $L$ are given. In addition, the hypercontractivity for the nonsymmetric Ornstein-Uhlenbeck semigroup is shown.