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

Abstract

We study the Hodge–Dirac operators $${\mathscr {D}}$$ associated with a class of non-symmetric Ornstein–Uhlenbeck operators $${\mathscr {L}}$$ in infinite dimensions. For $$p\in (1,\infty )$$ we prove that $$i{\mathscr {D}}$$ generates a $$C_0$$ -group in $$L^p$$ with respect to the invariant measure if and only if $$p=2$$ and $${\mathscr {L}}$$ 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 $${\mathscr {L}}$$ , both in the self-adjoint and the non-self-adjoint case.