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

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.