  The discretization of overdamped Langevin dynamics, through schemes such as
the Euler-Maruyama method, can be corrected by some acceptance/rejection rule,
based on a Metropolis-Hastings criterion for instance. In this case, the
invariant measure sampled by the Markov chain is exactly the Boltzmann-Gibbs
measure. However, rejections perturb the dynamical consistency of the resulting
numerical method with the reference dynamics. We present in this work some
modifications of the standard correction of discretizations of overdamped
Langevin dynamics on compact spaces by a Metropolis-Hastings procedure, which
allow us to either improve the strong order of the numerical method, or to
decrease the bias in the estimation of transport coefficients characterizing
the effective dynamical behavior of the dynamics. For the latter approach, we
rely on modified numerical schemes together with a Barker rule for the
acceptance/rejection criterion.
