Finite speed of propagation and off-diagonal bounds for Ornstein-Uhlenbeck operators in infinite dimensions

TL;DR

This paper investigates the properties of Ornstein-Uhlenbeck operators in infinite dimensions, demonstrating conditions for the generation of $C_0$-groups, their finite speed of propagation, and off-diagonal estimates, with implications for understanding their spectral behavior.

Contribution

It establishes the precise conditions under which the associated Hodge-Dirac operators generate $C_0$-groups and provides explicit representations and propagation speed results in infinite-dimensional settings.

Findings

The $i\mathcal{D}$ operator generates a $C_0$-group in $L^p$ only when $p=2$ and $\mathcal{L}$ is self-adjoint.

An explicit $L^2$ representation of the $C_0$-group is provided.

Finite speed of propagation and off-diagonal estimates are proven for these operators.

Abstract

We study the Hodge-Dirac operators $\mathcal{D}$ associated with a class of non-symmetric Ornstein-Uhlenbeck operators $\mathcal{L}$ in infinite dimensions. For $p\in (1,\infty)$ we prove that $i\mathcal{D}$ generates a $C_0$-group in $L^p$ with respect to the invariant measure if and only if $p=2$ and $\mathcal{L}$ is self-adjoint. An explicit representation of this $C_0$-group in $L^2$ is given and we prove that it has finite speed of propagation. Furthermore we prove $L^2$ off-diagonal estimates for various operators associated with $\mathcal{L}$, both in the self-adjoint and the non-self-adjoint case.