  We consider a Preconditioned Overdamped Langevin algorithm that does not
alter the invariant distribution (up to controllable discretisation errors) and
ask whether preconditioning improves the standard model in terms of reducing
the asymptotic variance and of accelerating convergence to equilibrium. We
present a detailed study of the dependence of the asymptotic variance on
preconditioning in some elementary toy models related to molecular simulation.
Our theoretical results are supported by numerical simulations.
