  We consider non-reversible perturbations of reversible diffusions that do not
alter the invariant distribution and we ask whether there exists an optimal
perturbation such that the rate of convergence to equilibrium is maximized. We
solve this problem for the case of linear drift by proving the existence of
such optimal perturbations and by providing an easily implementable algorithm
for constructing them. We discuss in particular the role of the prefactor in
the exponential convergence estimate. Our rigorous results are illustrated by
numerical experiments.
