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

Abstract

We study the Hodge–Dirac operators D associated with a class of non-symmetric
Ornstein–Uhlenbeck operators L in infinite dimensions. For p ∈ (1,∞) we prove that iD
generates a C0-group in L p with respect to the invariant measure if and only if p = 2 and
L is self-adjoint. An explicit representation of this C0-group in L2 is given, and we prove
that it has finite speed of propagation. Furthermore, we prove L2 off-diagonal estimates for
various operators associated with L, both in the self-adjoint and the non-self-adjoint case.