  Uncertainty quantification plays an important role in applications that
involve simulating ensembles of trajectories of dynamical systems. Conrad et
al. (Stat. Comput., 2017) proposed randomisation of deterministic time
integration methods as a strategy for quantifying uncertainty due to time
discretisation. We consider this strategy for systems that are described by
deterministic, possibly non-autonomous operator differential equations defined
on a Banach space or a Gelfand triple. We prove pathwise and expected error
bounds on the random trajectories, given an assumption on the local truncation
error of the underlying deterministic time integration and an assumption that
the absolute moments of the random variables decay with the time step. Our
analysis shows that the error analysis for differential equations in
finite-dimensional Euclidean space carries over to infinite-dimensional
settings.
