  Open quantum systems weakly coupled to the environment are modeled by
completely positive, trace preserving semigroups of linear maps. The generators
of such evolutions are called Lindbladians. In the setting of quantum many-body
systems on a lattice it is natural to consider Lindbladians that decompose into
a sum of local interactions with decreasing strength with respect to the size
of their support. For both practical and theoretical reasons, it is crucial to
estimate the impact that perturbations in the generating Lindbladian, arising
as noise or errors, can have on the evolution. These local perturbations are
potentially unbounded, but constrained to respect the underlying lattice
structure. We show that even for polynomially decaying errors in the
Lindbladian, local observables and correlation functions are stable if the
unperturbed Lindbladian has a unique fixed point and a mixing time which scales
logarithmically with the system size. The proof relies on Lieb-Robinson bounds,
which describe a finite group velocity for propagation of information in local
systems. As a main example, we prove that classical Glauber dynamics is stable
under local perturbations, including perturbations in the transition rates
which may not preserve detailed balance.
